$\displaystyle=d\omega_{G}(\partial_{\dot{\lambda}}\xi,0)+d(\
\partial_{\dot{\lambda}}\xi)(\dot{p})=$
$\displaystyle=d(\partial_{\dot{\lambda}}\xi)(\dot{p}).$

From the last equation and $(\ref{eq 5.3})$, the thesis follows. ∎

####  7.3 Proof of Theorem 7.1.2

We are now able to discuss the holomorphic dependence on the immersion data
for immersions into $\mathbb{X}_{n+1}$.

###### Proof of Theorem 7.1.2.

We prove that for all $(\lambda_{0},p_{0})\in\Lambda\times M$ there exists an
open neighbourhood $U\times V$ and a $\sigma\colon U\times
V\to\mathbb{X}_{n+1}$ as in the statement of Theorem 7.1.2 and that, for any
other $\sigma^{\prime}$ satisfying the thesis like $\sigma$,
$\sigma^{\prime}(\lambda,p)=\phi(\lambda)\circ\sigma(\lambda,p)$ with
$\phi\colon U\mapsto Isom(\mathbb{X}_{n})$ holomorphic. Then, since
$M\times\Lambda$ is simply connected, the proof of the Theorem follows by the
standard analytic continuation method as in "Step 2" of the proof of Theorem
6.4.2.

Choose the neighbourhood $U\times V$ so that there exists on $V$ a
$g_{\lambda_{0}}$-orthonormal frame $(X_{1},\dots X_{n})$, then the Gram-
Schmidt algorithm applies on $U\times V$ to this frame in order to have for
all $\lambda\in U$ a $g_{\lambda}$-orthonormal frame
$(e_{1;\lambda},\dots,e_{n;\lambda})$ on $V$ in the form

$e_{i;\lambda}(p)=\sum_{j}T_{i}^{j}(\lambda,p)X_{j}(p)$

where

$T=(T^{j}_{i})_{i,j}\colon U\times V\to Mat(n,{\mathbb{C}})$

is such that $T(\cdot,p)$ is holomorphic for all $p\in V$. Define
$\\{\theta_{\lambda}^{i}\\}_{i}$ as the dual coframe. Both the
$e_{i;\lambda}$’s and the $\theta^{i}_{\lambda}$’s are (pointwise) holomorphic
in $\lambda$.

We recall the steps in the construction of the immersions from the immersion
data, as we showed in Section 6.4, and show the holomorphic dependence is
preserved at each step.

  * •

From $(g_{\lambda},\Psi_{\lambda})$, locally construct the form
$\omega_{\lambda}\in\Omega^{1}(V,\mathfrak{o}(n+2,{\mathbb{C}}))$

$\omega_{\lambda}=\begin{pmatrix}{\scalebox{2.0}{$\Theta_{\lambda}$}}&\begin{matrix}-\Psi_{\lambda}^{1}&-i\theta_{\lambda}^{1}\\\
\dots&\dots\\\ -\Psi_{\lambda}^{n}&-i\theta_{\lambda}^{n}\end{matrix}\\\
\begin{matrix}\Psi_{\lambda}^{1}&\dots&\Psi_{\lambda}^{n}\\\
i\theta_{\lambda}^{1}&\dots&i\theta_{\lambda}^{n}\end{matrix}&\begin{matrix}0\text{
}&0\text{ }\\\ 0\text{ }&0\text{ }\end{matrix}\end{pmatrix}$

where $(\theta^{i}_{\lambda})_{i=1}^{n}$ is the dual of
$(e_{i;\lambda})_{i=1}^{n}$,
$\Theta_{\lambda}=(\theta^{i}_{j;\lambda})_{i,j=1}^{n}$ is the matrix of the
Levi-Civita connection forms w.r.t. the $\theta_{\lambda}^{i}$’s and
$\Psi_{\lambda}=\Psi_{\lambda}^{i}\otimes e_{i;\lambda}$.

Let us show that the map $\lambda\mapsto\omega_{\lambda}$ is holomorphic in
$\lambda$.

One can easily check in coordinate charts that the $d\theta^{i}_{\lambda}$’s
are holomorphic in $\lambda$. Defining

$d\theta^{i}=:\alpha^{i}_{j,k;\lambda}\theta^{j}\wedge\theta^{k},$

the coefficients $\alpha^{i}_{j,k;\lambda}$’s are holomorphic in $\lambda$,
hence, since one can straightforwardly show that

$\theta^{i}_{k;\lambda}:=-\alpha^{i}_{j,k;\lambda}\theta^{j},$

we conclude that $\Theta_{\lambda}$ is holomorphic in $\lambda$; finally, the
$\Psi_{\lambda}^{i}$’s are holomorphic in $\lambda$ since
$\Psi_{\lambda}=\Psi_{\lambda}^{i}\otimes e_{i;\lambda}$.

  * •

Up to shrinking $U$ and $V$, assume they are simply connected. By Proposition
7.2.1, there exists a smooth map $\Phi\colon U\times V\to
SO(n+2,{\mathbb{C}})$ such that $\Phi(\cdot,p_{0})\equiv e$ and
$\Phi(\lambda,\cdot)^{*}\omega_{G}=\omega_{\lambda}$ where $\omega_{G}$ is the
Maurer-Cartan form of $G$. We prove that $\Phi$ is holomorphic w.r.t.
$\lambda$.

Define $\xi$ as in Equation $(\ref{def Sigma})$. For all $\dot{\lambda}\in
T_{\lambda_{0}}\Lambda$, using Lemma 7.2.2 and the fact that the
$\omega_{\lambda}$’s vary in a holomorphic way in $\lambda$, one has

$d(\partial_{i\dot{\lambda}}\xi)=Ad(\Phi_{\lambda_{0}})\circ\partial_{i\dot{\lambda}}{\omega}=Ad(\Phi_{\lambda_{0}})\circ\big{(}i\partial_{\dot{\lambda}}{\omega}\big{)}=i\
d(\partial_{\dot{\lambda}}\xi).$

As a result, both $\partial_{i\dot{\lambda}}\xi$ and
$i\partial_{\dot{\lambda}}\xi$ solve the Cauchy problem (7.7) for
$i\dot{\lambda}$, hence they coincide.

By $(\ref{def Sigma})$, one has that for all $p\in V$ and $\lambda\in U$

$\displaystyle(\partial_{i\dot{\lambda}}\Phi)(\lambda_{0},p)$
$\displaystyle=d(L_{\Phi(\lambda_{0},p)})((\partial_{i\dot{\lambda}}\xi)(\lambda_{0},p))=d(L_{\Phi(\lambda_{0},p)})(i(\partial_{\dot{\lambda}}\xi)(\lambda_{0},p))=$
$\displaystyle=i(\partial_{\dot{\lambda}}\Phi)(\lambda_{0},p).$

  * •

By the proof of Theorem 6.4.2, defining the map

$\sigma\colon U\times V\to\mathbb{X}_{n+1}$

by

$\sigma(\lambda,p)=\Phi(\lambda,p)\cdot e,$

one has that $\sigma(\lambda,\cdot):V\to\mathbb{X}_{n+1}$ is an immersion with
immersion data $(g_{\lambda},\Psi_{\lambda})$. Finally, $\sigma$ is
holomorphic with respect to $\lambda$ since $\Phi$ is.

  * •

By Theorem 6.4.2, if $\sigma^{\prime}$ satisfies the statement of Theorem
7.1.2 like $\sigma$, then
$\sigma^{\prime}(\lambda,p)=\phi(\lambda)\cdot\sigma(\lambda,p)$ for a unique
function $\phi\colon U\to Isom_{0}(\mathbb{X}_{n})\cong SO(n+2,{\mathbb{C}})$.

We prove that $\phi$ is holomorphic.

Fix $p\in V$ and a basis $X_{1},\dots,X_{n}\in T_{p}V$. Then, the functions
$A,A^{\prime}\colon U\to GL(n+2,{\mathbb{C}})$ defined by

$\begin{split}A(\lambda)&=\begin{pmatrix}d_{p}(\sigma(\lambda,\cdot))(X_{1})\
\ |&\dots\ \ |&d_{p}(\sigma(\lambda,\cdot))(X_{n})\ \
|&d_{p}(\sigma(\lambda,\cdot))(\nu_{\sigma(\lambda,\cdot)})\ \
|&\sigma(p)\end{pmatrix}\\\ &\text{and}\\\
A^{\prime}(\lambda)&=\begin{pmatrix}d_{p}(\sigma^{\prime}(\lambda,\cdot))(X_{1})\
\ |&\dots\ \ |&d_{p}(\sigma^{\prime}(\lambda,\cdot))(X_{n})\ \
|&d_{p}(\sigma^{\prime}(\lambda,\cdot))(\nu_{\sigma^{\prime}(\lambda,\cdot)})\
\ |&\sigma^{\prime}(p)\end{pmatrix}\end{split}$

are holomorphic with respect to $\lambda$: this is an elementary consequence
of the assumptions on $\sigma$ and $\sigma^{\prime}$ and of the fact that
vector fields on $U$ and on $V$ commute with each other when seen as vector
fields on $U\times V$. By definition of $\phi$,
$\phi(\lambda)=A^{\prime}(\lambda)(A(\lambda))^{-1}$, hence, by the chain
rule, $\phi$ is holomorphic.

By the analytic continuation argument mentioned above, one gets that for all
$\lambda$ there exists a neighbourhood $U\subset\Lambda$ such that there
exists $\sigma\colon U\times\widetilde{M}\to\mathbb{X}_{n+1}$ as in the
statement.

Moreover, by fixing a point $p_{0}\in\widetilde{M}$ and setting
$\Phi(\lambda,p_{0})\equiv e_{G}$ for all $\lambda$, one gets that the 1-jet
of the immersion $\sigma(\lambda,\cdot)$ at $p_{0}$ is the same for all
$\lambda$. As a result, different $\sigma$’s defined on domains of the form
$U\times\widetilde{M}$ with the same 1-jet on $p_{0}$ glue together, proving
that $\sigma$ extends globally on $\Lambda\times\widetilde{M}$.

It is now simple to check that the monodromy of $\sigma(\lambda,\cdot)$ is
holomorphic with respect to $\lambda$. Indeed, for all $\alpha\in\pi_{1}(M)$,
the map

$mon_{\alpha}\colon\Lambda\to SO(n+2,{\mathbb{C}})$

is defined by

$\sigma(\lambda,\alpha(p))=mon_{\alpha}(\lambda)\circ\sigma(\lambda,p):$

since both $\sigma$ and $\sigma\circ\alpha$ satisfy the conditions in the
statement of the Theorem for the data $(g,\Psi)$, by the previous step
$mon_{\alpha}$ is holomorphic in $\lambda$.

∎

###### Proof of Corollary 7.1.3.

Since $g_{\lambda}$ is holomorphic in $\lambda$, the set
$\\{(g_{\lambda},0)\\}_{\lambda\in\Lambda}$ is a holomorphic family of
immersion data for $M$. Then, Theorem 7.1.2 ensures that there exists
$\overline{\sigma}\colon\Lambda\times\widetilde{M}\to\mathbb{X}_{n+1}$ with
$\overline{\sigma}(\cdot,p)$ holomorphic and with
$\overline{\sigma}(\lambda,\cdot)$ an immersion with immersion data
$(g_{\lambda},0)$.

Denoting $\nu_{\lambda}$ as the normal vector field of $\sigma_{\lambda}$, one
has that $\nu_{\lambda}$ is a constant independent from $p\in\widetilde{M}$,
as we showed in the proof of Theorem 6.5.2.

Now, one can construct a map
$\phi\colon\Lambda\to\mathrm{SO}(n+2,{\mathbb{C}})$ such that
$\phi(\lambda)(\nu_{\lambda})\equiv\begin{pmatrix}0&\dots&0&1\end{pmatrix}$:
indeed, clearly one can construct $\phi$ locally on $\Lambda$ by extending
$\nu_{\lambda}$ to a basis of ${\mathbb{C}}^{n+2}$, then applying Gram-Schmidt
algorith to get a orthonormal basis, and then choosing $\phi(\lambda)$ as the
unique isometry of $({\mathbb{C}}^{n+2},\langle\cdot,\cdot\rangle_{0})$ that
sends this basis to the canonical basis (one has
$\phi(\lambda)\in\mathrm{SO}(n+2)$ up to rearranging the basis); to see that
$\phi$ can be chosen globally, apply the standard analytic continuation
methods we already used in the proofs of Theorem 7.1.2 and of Theorem 6.4.2.

Then, under the canonical inclusion $\mathbb{X}_{n}\to\mathbb{X}_{n+1}$ given
by $p\mapsto(p,0)$, the map
$\sigma\colon\Lambda\times\widetilde{M}\to\mathbb{X}_{n}$ defined by
$\sigma(\lambda,p)=\psi(\lambda)\circ\sigma(\lambda,p)$ is the researched
function. The monodromy of $\sigma(\lambda,\cdot)$ is holomorphic in $\lambda$
since $SO(n+1,{\mathbb{C}})$ is a complex submanifold of
$SO(n+2,{\mathbb{C}})$. ∎

####  7.4 An application: the complex landslide is holomorphic

In this section, we would like to provide a direct application of Theorem
7.1.2: we show an alternative proof for the fact that the holonomy of the
complex landslide, defined in [BMS13], is holomorphic.

We briefly recall some basic notions on projective structures, the notions of
landslide, smooth grafting and complex landslide. One can use as main
references [Dum09] and [BMS13].

#####  7.4.1 Projective structures and holonomy

Let us quickly recall the definition of
${\mathrm{PSL}(2,\mathds{C})}$-character variety, projective structures and of
their holonomy. For a more complete survey on these topics, see [Dum09].

Let $S$ be an oriented closed surface of genus $\mathrm{g}\geq 2$.

Let us define the character variety of ${\mathrm{PSL}(2,\mathds{C})}$. The
general definition of character variety is more complex but for immersions of
$\pi_{1}(S)$ into ${\mathrm{PSL}(2,\mathds{C})}$ it has a simple description.
The reader might refer to [CS83, Gun67, Gol88] for more detailed treatments.

Fixing a set of generators $\\{\alpha_{1},\dots,\alpha_{2\mathrm{g}}\\}$, the
set

$\mathcal{R}(\pi_{1}(S),{\mathrm{PSL}(2,\mathds{C})}):=\mathrm{Hom}(\pi_{1}(S),{\mathrm{PSL}(2,\mathds{C})})$

can be seen naturally as a subset of
${\mathrm{PSL}(2,\mathds{C})}^{\times(2\mathrm{g})}$, and inherits an affine
complex structure.

Define on $\mathcal{R}$ the equivalence relation

$\rho_{1}\sim\rho_{2}\qquad\text{if and only if}\qquad
tr^{2}(\rho_{1}(\alpha))=tr^{2}(\rho_{2}(\alpha))\quad\text{for all
$\alpha\in\pi_{1}(S)$}$

with $tr^{2}$ denoting the square of the trace of a matrix. This equivalence
relation induces a quotient

$\mathcal{X}(S):={\mathcal{R}(\pi_{1}(S),{\mathrm{PSL}(2,\mathds{C})})}\mathbin{/\mkern-6.0mu/}{{\mathrm{PSL}(2,\mathds{C})}}$
(7.12)

that we denote as _the character variety of_ ${\mathrm{PSL}(2,\mathds{C})}$.
For a suitable choice of a set of generators of $\pi_{1}(S)$, the evaluation
of $tr^{2}$ provides an injective map into ${\mathbb{C}}^{N}$: the image is an
affine variety, and this allows us to say that $\mathcal{X}(S)$ is an
algebraic variety.

Let us now focus on complex projective structures. In terms of
$(G,X)$-structures, a _(complex) projective structure_ on $S$ is a
$({\mathrm{PSL}(2,\mathds{C})},\mathbb{C}\mathbb{P}^{1})$-structure on $S$. A
projective structure induces a complex structure and an orientation, we will
stick to projective structures compatible with the orientation on $S$. As a
$({\mathrm{PSL}(2,\mathds{C})},\mathbb{C}\mathbb{P}^{1})$-structure, a
projective structure is determined by a
$(\pi_{1}(S),{\mathrm{PSL}(2,\mathds{C})})$-equivariant developing map
$\widetilde{S}\to\mathbb{C}\mathbb{P}^{1}$, which is unique up to post-
composition with elements in ${\mathrm{PSL}(2,\mathds{C})}$. We therefore
define

$\begin{split}\widetilde{\mathcal{P}}(S)&:=\\{\text{projective structures on
$S$}\\}=\\\ &=\faktor{\left\\{(f,\rho)\ \bigg{|}\
\begin{split}&\rho\colon\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})},\\\
&f\colon\widetilde{S}\to\mathbb{C}\mathbb{P}^{1}\text{ $\rho$-equiv.
orientation-preserving local
diffeo}\end{split}\right\\}}{{\mathrm{PSL}(2,\mathds{C})}}\\\
\mathcal{P}(S)&:=\faktor{\widetilde{\mathcal{P}}(S)}{\text{Diff}_{0}(S)}\end{split}$

Equip the set of pairs $\\{(f,\rho)\\}$ with the compact-open topology, and
$\widetilde{\mathcal{P}}(S)$ and $\mathcal{P}(S)$ with the quotient topology.
The space $\mathcal{P}(S)$ is a topological manifold, over which one can
define a smooth and complex structure - compatible with the topology - via the
Schwarzian parametrization (e.g. see [Dum09]).

Now, one can define the holonomy map

$\displaystyle\widetilde{Hol}\colon\widetilde{\mathcal{P}}(S)$
$\displaystyle\to\mathcal{X}(S)$ $\displaystyle[(f,\rho)]$
$\displaystyle\mapsto[\rho]\ ,$

which passes to the quotient to a map

$Hol\colon{\mathcal{P}}(S)\to\mathcal{X}(S).$

In fact, one can prove that the image of $Hol$ lies into a Zariski-open subset
of $\mathcal{X}(S)$ corresponding to representations
$\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})}$ whose action on $\mathbb{H}^{3}$
and $\partial\mathbb{H}^{3}$ does not fix neither an internal point, nor a
point in the boundary, nor an unoriented geodesic ([Gun67]). As a result, the
image of $Hol$ is contained in a smooth complex region of $\mathcal{X}(S)$. In
this context, the following Theorem holds.

######  Theorem 7.4.1.

(Hejhal [Hej75], Hubbard [Hub81], Earle [Ear81]) The holonomy map $Hol$ is a
local biholomorphism.

#####  7.4.2 The complex landslide

Let $\text{\Capricorn}(S)$ be the Teichmüller space of $S$.

In [BMS13], the authors define two smooth functions satisfying several
interesting properties: the _Landslide map_

$L\colon\text{\Capricorn}(S)\times\text{\Capricorn}(S)\times{\mathbb{R}}\to\text{\Capricorn}(S)\times\text{\Capricorn}(S)$

and the _Smooth Grafting map_

$SGr\colon\text{\Capricorn}(S)\times\text{\Capricorn}(S)\times{\mathbb{R}}^{+}\to\mathcal{P}(S)$

which turn out to have several interesting geometric properties and are
related respectively to the earthquake map and to the grafting map. (For
brevity we are not going to discuss earthquakes and grafting in this thesis.)

There are several equivalent definitions of $L$ and $SGr$, we mention the ones
most fitted to our pourposes.

Let $h$ be a hyperbolic metric. As a consequence of some works by Schoen
[Sch93] and Labourie [Lab92], it has been proved that the map

$\begin{split}\left\\{\begin{array}[]{l}b\colon TS\to TS\\\ \text{bundle
isomorphism}\\\ \text{such that:}\end{array}\Bigg{|}\begin{array}[]{l}\text{
$h$-self adjoint,}\\\ d^{\nabla_{h}}b=0,\\\
det(b)=1\end{array}\right\\}&\xrightarrow{\sim}\text{\Capricorn}(S)\\\
b&\mapsto[h(b\cdot,b\cdot)]\end{split}$ (7.13)

is well-posed and bijective. We will denote the tensors in the domain as _$h$
-regular_.

  * •

Given
$([h],[h(b\cdot,b\cdot)])\in\text{\Capricorn}(S)\times\text{\Capricorn}(S)$
with $b$ $h$-regular, define for all $t\in{\mathbb{R}}$ the operator

$\beta_{t}:=\cos(\frac{t}{2})id+\sin(\frac{t}{2})Jb$

where $J$ is the complex structure induced by $h$. The _landslide map_ is
defined as

$L([h],[h(b\cdot,b\cdot)],t)=\bigg{(}\Big{[}h\Big{(}\beta_{t}\
\cdot,\beta_{t}\cdot\Big{)}\Big{]},\Big{[}h\Big{(}\beta_{t+\pi}\
\cdot,\beta_{t+\pi}\ \cdot\Big{)}\Big{]}\bigg{)}$

  * •

Given
$([h],[h(b\cdot,b\cdot)])\in\text{\Capricorn}(S)\times\text{\Capricorn}(S)$,
observe that for all $s\in{\mathbb{R}}$ the pair

$(\mathrm{I}_{s},B_{s}):=\bigg{(}\cosh^{2}\Big{(}\frac{s}{2}\Big{)}h,-\tanh\Big{(}\frac{s}{2}\Big{)}b\bigg{)}$
(7.14)

satisfies Gauss-Codazzi equations for $\mathbb{H}^{3}$. As a result, for all
$s$ there exists a unique $\pi_{1}(S)$-equivariant isometric immersion

$\sigma_{s}\colon(\widetilde{S},\widetilde{g}_{s})\to\mathbb{H}^{3}$

with shape operator $-\tanh\big{(}\frac{s}{2}\big{)}\widetilde{B_{s}}$.

We will discuss convex hypersurfaces in $\mathbb{H}^{n+1}$ later on in
Definition 9.3.8. For now, let us just mention that $\sigma_{s}$ is convex,
i.e. the tensor $b_{s}$ is negative definite, if and only if $s>0$, and, as a
result of convexity (see Lemma 11.3.1), for $s>0$ the map $\sigma_{s}$ induces
a map

$G^{+}_{s}\colon\widetilde{S}\to\partial\mathbb{H}^{3}\cong\mathbb{C}\mathbb{P}^{1}$

which is a diffeomorphism onto its image and which is defined by
$G^{+}_{s}(p)$ being the endpoint of the geodesic ray starting at $\sigma(p)$
with tangent direction $\nu(p)$. The map $G^{+}_{s}$ is equivariant inducing
the same monodromy into ${\mathrm{PSL}(2,\mathds{C})}$ as $\sigma_{s}$, thus
it defines a projective structure on $S$.

In conclusion, the _smooth grafting_ is defined as the map

$Sgr([h],[h(b\cdot,b\cdot)],s)=[G_{s}^{+}].$

Finally, in the same article, the authors define a _complex landslide_ as the
composition of landslides and smooth grafting: more precisely, for every
hyperbolic metric $h$ and $h$-regular tensor $b$, they define a map

$\displaystyle P_{h,b}\colon H$ $\displaystyle\to\mathcal{P}(S)$ (7.15)
$\displaystyle z=t+is$ $\displaystyle\mapsto
Sgr\bigg{(}L\Big{(}[h],[h(b\cdot,b\cdot)],t\Big{)},s\bigg{)}$ (7.16)

where $H\subset{\mathbb{C}}$ is the upper half-plane.

######  Theorem 7.4.2 (Bonsante - Mondello - Schlenker).

$P_{h,b}$ is holomorphic.

The argument for the proof in [BMS13] uses analytic methods. Here we provide
an alternative proof using Theorem 7.1.2 and Proposition 6.7.1.

######  Theorem 7.4.3.

The holonomy of the projective structure $P_{h,b}(z)$ is equal to the
monodromy of the immersion $\widetilde{S}\to{\mathbb{G}}$ with induced complex
metric

$g_{z}=h\bigg{(}(\cos(z)id-\sin(z)Jb)\
\cdot,(\cos(z)id-\sin(z)Jb)\cdot\bigg{)}$

with constant curvature $-1$.

As a consequence of Theorem 7.1.2 and Theorem 7.4.1, $P_{h,b}$ is holomorphic.

###### Proof.

In order to make the notation more simple, observe that there exists a natural
right action of the group $\Gamma(GL(TS))$, with the operation of composition
pointwise, over the space $\Gamma(Sym^{2}(T^{*}S))$ given by

$\begin{split}\Gamma(Sym^{2}(T^{*}S))\times\Gamma(GL(TS))&\to\Gamma(Sym^{2}(T^{*}S))\\\
\tau\ast\alpha&=\tau(\alpha\cdot,\alpha\cdot)\ ,\end{split}$ (7.17)

and the same holds replacing $TS$ with ${\mathbb{C}}TS$ and $T^{*}S$ with
${\mathbb{C}}T^{*}S$, and we will adopt the same notation.

Let $z=t+is$ and let us try to compute the holonomy of
$P_{h,b}(z)=Sgr\bigg{(}L\Big{(}[h],[h(b\cdot,b\cdot)],t\Big{)},s\bigg{)}$. In
order to do it, we need to compute some elements.

  * •

For all $x\in{\mathbb{R}}$, define $\bar{h}_{x}:=h\ast\beta_{x}$. Then

$L\Big{(}[h],[h(b\cdot,b\cdot)],t\Big{)}=\bigg{(}[\bar{h}_{t}],[\bar{h}_{t+\pi}]\bigg{)}.$

  * •

By a straightforward computation on a $h$-orthonormal frame diagonalizing $b$,
one gets that $JbJb=-id$, and as a consequence that

$\displaystyle\beta_{t}\circ\beta_{s}$
$\displaystyle=(\cos\Big{(}\frac{t}{2}\Big{)}id+sin\Big{(}\frac{t}{2}\Big{)}Jb)\circ(\cos\Big{(}\frac{s}{2}\Big{)}id+sin\Big{(}\frac{s}{2}\Big{)}Jb)=$
$\displaystyle=\cos\Big{(}\frac{t+s}{2}\Big{)}id+\sin\Big{(}\frac{t+s}{2}\Big{)}Jb=$
$\displaystyle=\beta_{t+s}$

for all $t,s\in{\mathbb{R}}$. Hence, one has that

$\displaystyle\bar{h}_{t+\pi}$
$\displaystyle=h\ast\beta_{t+\pi}=(h\ast\beta_{\pi})\ast\beta_{t}=$
$\displaystyle=(h\ast
b)\ast\beta_{t}=(h\ast\beta_{t})\ast(\beta_{-t}b\beta_{t})=$
$\displaystyle=\bar{h}_{t}\ast(\beta_{-t}b\beta_{t}).$

Let us define $\bar{b}_{t}:=(\beta_{-t}b\beta_{t})$. One can directly prove
that $\bar{b}_{t}$ is self-adjoint and Codazzi for $\bar{h}_{t}$ and has
determinant $1$.

  * •

By a direct computation, one gets that the complex structure of $\bar{h}_{t}$
is

$\bar{J}_{t}=\beta_{-t}J\beta_{t}.$

In order to compute the monodromy of
$Sgr([\bar{h}_{-t}],[\bar{h}_{-t+\pi}],s)$, consider, as in Equation (7.14),
the immersion into $\mathbb{H}^{3}$ with immersion data
$(\mathrm{I}_{s},B_{s}):=(cosh^{2}(s)\bar{h}_{t},\tanh(s)\bar{b}_{t})$ and
apply Proposition 6.7.1 to conclude that it has the same monodromy as the
induced immersion into ${\mathbb{G}}$ with pull-back metric

$\displaystyle h_{z}:=\mathrm{I}_{s}\ast(id-iJ_{\mathrm{I}_{s}}\psi_{s})$
$\displaystyle=\cosh^{2}\Big{(}\frac{s}{2}\Big{)}\
\overline{h}_{t}\ast\Big{(}id+i\tanh\Big{(}\frac{s}{2}\Big{)}\overline{J}_{t}\overline{b}_{t}\Big{)}=$
$\displaystyle=\bar{h}_{t}\ast\Big{(}\cosh\Big{(}\frac{s}{2}\Big{)}id+i\sinh\Big{(}\frac{s}{2}\Big{)}\overline{J}_{t}\overline{b}_{t}\Big{)}=$
$\displaystyle=\bar{h}_{t}\ast\Big{(}\cosh\Big{(}\frac{s}{2}\Big{)}id+i\sinh\Big{(}\frac{s}{2}\Big{)}\beta_{-t}Jb\beta_{t}\Big{)}=$
$\displaystyle=\bar{h}_{t}\ast\bigg{(}\beta_{-t}\circ\Big{(}\cosh\Big{(}\frac{s}{2}\Big{)}id+i\sinh\Big{(}\frac{s}{2}\Big{)}Jb\Big{)}\circ\beta_{t}\bigg{)}=$
$\displaystyle=h\ast\bigg{(}(\cosh\Big{(}\frac{s}{2}\Big{)}id+i\sinh\Big{(}\frac{s}{2}\Big{)}Jb)\circ\beta_{t}\bigg{)}=$
$\displaystyle=h\ast\bigg{(}\Big{(}\cos\Big{(}i\frac{s}{2}\Big{)}id+\sin\Big{(}i\frac{s}{2}\Big{)}Jb\Big{)}\circ\Big{(}\cos\Big{(}\frac{t}{2}\Big{)}id+\sin\Big{(}\frac{t}{2}\Big{)}Jb\Big{)}\bigg{)}=$
$\displaystyle=h\ast\Big{(}\cos\Big{(}\frac{z}{2}\Big{)}id+\sin\Big{(}\frac{z}{2}\Big{)}Jb\Big{)}.$

By construction, each $g_{z}$ is a complex metric of curvature $-1$,
corresponding to an equivariant immersion of $\widetilde{S}$ into
$\mathbb{{\mathbb{G}}}$ for all $z$.

Clearly, for any $X,Y\in\Gamma(TS)$,

$g_{z}(X,Y)=\cos^{2}(z)h(X,X)+\sin^{2}(z)h(bX,bY)+\sin z\cos
z(h(X,JbY)+h(JbX,Y))$

is holomorphic in $z$, hence the family of complex metrics
$\\{g_{z}\\}_{z\in{\mathbb{C}}}$ is holomorphic. By Corollary 7.1.3, there
exists a map $\sigma\colon{\mathbb{C}}\times\widetilde{S}\to{\mathbb{G}}$ with
$\sigma=\sigma(z,p)$ holomorphic in $z$ and with $\sigma_{z}$ providing pull-
back metric $g_{z}$. The monodromy of $g_{z}$, hence the holonomy of
$P_{h,b}$, is holomorphic in $z$; by Theorem 7.4.1, the projective structure
$P_{h,b}(z)$ depends on $z$ holomorphicly. ∎

###  Chapter 8 A Gauss-Bonnet Theorem and a Uniformization Theorem for complex
metrics

####  8.1 Positive complex metrics

In this section we suggest an intrinsic study of complex metrics on surfaces.
To this aim, we will introduce a natural generalization of the concept of
complex structure for surfaces and we will present a uniformization theorem in
this setting.

In the following, $S$ denotes an oriented surface.

The natural inclusion $TS\hookrightarrow{\mathbb{C}}TS$ factors to a bundle
inclusion

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Fiberwise, for every $p\in S$, $\mathbb{P}_{\mathbb{R}}(T_{p}S)$ is mapped
homeomorphicly into a circle in
$\mathbb{P}_{\mathbb{C}}({\mathbb{C}}T_{p}S)\cong S^{2}$ whose complementary
is the disjoint union of two open discs. The conjugation map on
${\mathbb{C}}TS$ descends to a bundle isomorphism on
$\mathbb{P}_{\mathbb{C}}({\mathbb{C}}TS)$ that fixes
$\mathbb{P}_{\mathbb{R}}(TS)$ and that swaps the two discs fiberwise.

######  Definition 8.1.1.

A _bicomplex structure_ on a surface $S$ is a tensor J
$\in\Gamma({\mathbb{C}}T^{*}S\otimes{\mathbb{C}}TS)$ such that

  * •

$\textbf{J}^{2}=\textbf{J}\circ\textbf{J}=-id_{{\mathbb{C}}TS}$; as a result,
J is diagonalizable with eigenvalues $\pm i$ and eigenspaces
$V_{i}(\textbf{J}),V_{-i}(\textbf{J})\subset{\mathbb{C}}TS$ with complex
dimension $1$.

  * •

for all $p\in S$, the eigenspaces $V_{i}(\textbf{J}_{p})$ and
$V_{-i}(\textbf{J}_{p})$ of $\textbf{J}_{p}$ have trivial intersection with
$T_{p}S$ and are such that the points
$\mathbb{P}_{\mathbb{C}}(V_{i}(\textbf{J}_{p}))$ and
$\mathbb{P}_{\mathbb{C}}(V_{-i}(\textbf{J}_{p}))$ lie in different connected
components of
$\mathbb{P}_{\mathbb{C}}({\mathbb{C}}T_{p}S)\setminus\mathbb{P}_{\mathbb{R}}(T_{p}S)$.

######  Remark 8.1.2.

Clearly complex structures extend to bicomplex structures. Observe that a
bicomplex structure J is not a complex structure for $S$ in general, since it
may not restrict to a section of $TS\otimes T^{*}S$. In other words,
$\textbf{J}(\overline{X})\neq\overline{\textbf{J}(X)}$ in general.

Nevertheless, J induces two complex structures $J_{1},J_{2}$ defined by the
conditions

$V_{i}(\textbf{J})=V_{i}(J_{2})=\overline{V_{-i}(J_{2})}\qquad\text{and}\qquad
V_{-i}(\textbf{J})=V_{-i}(J_{1})=\overline{V_{i}(J_{1})}$

which totally characterize $J_{1}$ and $J_{2}$.

Also observe that $J_{1}$ and $J_{2}$ induce the same orientation. Indeed,
representing the set of complex structures on $S$ as $C(S)\sqcup
C(\overline{S})$, the map

$\displaystyle C(S)\sqcup C(\overline{S})$
$\displaystyle\to\mathbb{P}_{\mathbb{C}}({\mathbb{C}}T_{p}S)\setminus\mathbb{P}_{\mathbb{R}}(T_{p}S)$
$\displaystyle J_{0}$
$\displaystyle\mapsto\mathbb{P}_{\mathbb{C}}(V_{i}(J_{0})_{p})$

is continuous, hence it sends $C(S)$ and $C(\overline{S})$ onto the two
distinct connected component of the image.

As a result, every bicomplex structure induces an orientation. An orientation
of $S$ fixed, denote with $BC(S)$ the set of orientation-consistent bicomplex
strucures. With the notations above, we therefore have a bijection

$\begin{split}BC(S)&\xrightarrow{\sim}C(S)\times C(S)\\\
\mathbf{J}&\mapsto(J_{1},J_{2}).\end{split}$ (8.1)

Endow $BC(S)$ with the pull-back topology for which, in particular, it is
connected.

######  Definition 8.1.3.

Let $g$ be a complex metric on $S$.

  * •

Denote the set of isotropic vectors of $g$ as

$ll(g):=\\{v\in{\mathbb{C}}TS\ |\ g(v,v)=0\\}\setminus\\{0_{S}\\}$

and define $ll(g_{p}):=ll(g)\cap{\mathbb{C}}T_{p}S$. Notice that $ll(g_{p})$
is the union of two complex lines with trivial intersection.

  * •

A _positive complex metric_ on a surface $S$ is a complex metric $g$ such that
the two points $\mathbb{P}_{\mathbb{C}}(ll(g_{p}))$ lie in different connected
components of
$\mathbb{P}_{\mathbb{C}}({\mathbb{C}}T_{p}S)\setminus\mathbb{P}_{\mathbb{C}}(T_{p}S)$.

  * •

Denote with $CM^{+}(S)$ the space of positive complex metrics on $S$ with the
usual $C^{\infty}$-topology for spaces of sections.

For the rest of the chapter, we will mostly focus on positive complex metrics.
The main interest lies in the fact that positive complex metrics locally
determine a bicomplex structure.

######  Proposition 8.1.4.

Let $g$ be a positive complex metric on a surface $S$.

  * 1)

For all $p\in S$ there exist exactly two opposite endomorphisms
$\textbf{J}_{p},-\textbf{J}_{p}\in End({\mathbb{C}}T_{p}S)$ such that

$\begin{cases}g_{p}(\textbf{J}_{p}\cdot,\textbf{J}_{p}\cdot)=g_{p}(\cdot,\cdot),\\\
\textbf{J}_{p}^{2}=-id_{p}\end{cases}.$ (8.2)

The two solutions $\textbf{J}_{p},-\textbf{J}_{p}$ depend smoothly on $p$ and
on $g_{p}$.

  * 2)

For all $p_{0}\in S$, there exists a local neighbourhood $U=U(p_{0})$ over
which $\textbf{J}=\\{\textbf{J}_{p}\\}_{p\in U}$ and
$-\textbf{J}=\\{-\textbf{J}_{p}\\}_{p\in U}$ define two bicomplex structures
on $U$ and

$ll(g_{p})=V_{i}(\textbf{J}_{p})\cup V_{-i}(\textbf{J}_{p}).$

We will say that J and $-\textbf{J}$ are (locally) _compatible_ with $g$.

  * 3)

If J is compatible with $g$ and $\textbf{J}\equiv(J_{1},J_{2})$ in
correspondence $(\ref{BC(S) to C(S)xC(S)})$, then locally

$g=f\ dw_{1}\cdot d\overline{w}_{2}$

where $f$ is a ${\mathbb{C}}$-valued smooth function and $w_{1}$, $w_{2}$ are
local holomorphic coordinates for the complex structures $J_{1}$ and $J_{2}$
respectively.

  * 4)

If J is compatible with $g$ on $U$, for any norm-1 local vector field $X$ on
$U$, a local $g$-orthonormal frame on $U$ is given by $\\{X,\textbf{J}X\\}$.

###### Proof.

  * 1)

With respect to two local $g$-orthonormal vector fields $X_{1}$ and $X_{2}$,
$\textbf{J}_{p}$ is represented by a matrix $A_{p}\in Mat(2,{\mathbb{C}})$
satisfying:

$\begin{cases}A_{p}^{2}=-I_{2}\\\ {}^{t}A_{p}A_{p}=I_{2}\end{cases},$

which admits exactly two opposite constant solutions for $A_{p}$, namely
$A_{p}=\pm\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$, so the two solutions
$\textbf{J}_{p}$ and $-\textbf{J}_{p}$ are smooth.

  * 2)

The equation follows directly from $(\ref{compatibility})$. Since $g$ is
positive, $\mathbb{P}_{\mathbb{C}}(V_{i}(\textbf{J}_{p}))$ and
$\mathbb{P}_{\mathbb{C}}(V_{-i}(\textbf{J}_{p}))$ lie in different connected
components of
$\mathbb{P}_{\mathbb{C}}({\mathbb{C}}T_{p}S)\setminus\mathbb{P}_{\mathbb{R}}(T_{p}S)$,
so J and $-\textbf{J}$ are bicomplex structures.

  * 3)

A ${\mathbb{C}}$-bilinear form on a 2-dimensional ${\mathbb{C}}$-vector space
is determined up to a constant by its isotropic directions. For local
holomorphic coordinates $w_{1},w_{2}$ for $J_{1}$ and $J_{2}$, we have

$V_{i}(\textbf{J})=V_{i}(J_{2})={\mathbb{C}}\frac{\partial}{\partial
w_{2}}\qquad
V_{-i}(\textbf{J})=V_{-i}(J_{1})={\mathbb{C}}\frac{\partial}{\partial\overline{w}_{1}},$

hence the locally defined complex metrics $g$ and $dw_{1}\cdot
d\overline{w}_{2}=\frac{1}{2}(dw_{1}\otimes
d\overline{w}_{2}+d\overline{w}_{2}\otimes dw_{1})$ are pointwise equal up to
a constant since they have the same isotropic directions. Explicitly, we have

$g=g\bigg{(}\frac{\partial}{\partial
w_{1}},\frac{\partial}{\partial\overline{w}_{2}}\bigg{)}dw_{1}d\overline{w}_{2}$

  * 4)

The thesis follows directly by the compatibility of J with $g$.

∎

We just showed that positive complex metrics locally define a bicomplex
structure. In fact, the obstruction to the existence of a global compatible
bicomplex structure is only a matter of orientability.

######  Theorem 8.1.5.

Let $g$ be a positive complex metric on a surface $S$.

The following are equivalent:

  * •

$S$ is orientable;

  * •

there exists a global positive bicomplex structure J on $S$ compatible with
$g$;

  * •

there exists a complex $1$-form
$dA\in\Omega^{2}({\mathbb{C}}TS,{\mathbb{C}})$, unique up to a sign, such that
$\|dA\|^{2}_{g}=1$, i.e. for all local $g-$orthonormal frame $(X_{1},X_{2})$
one has $\nobreak{dA(X_{1},X_{2})\in\\{-1,1\\}}$.

We will call $dA$ the _area form_ of $g$.

###### Proof.

As we showed, a bicomplex structure induces two complex structures which
induce an orientation on $S$. Conversely, given an orientation on $S$, locally
there exists a unique bicomplex structure compatible with $g$ and inducing the
same orientation, hence there exists a unique global one.

If J is a bicomplex structure compatible with $g$, then
$dA=g(\textbf{J}\cdot,\cdot)$ is a 2-form of norm 1. Conversely, if J is a
local bicomplex structure compatible with $g$, the local $2$-forms
$g(\textbf{J}\cdot,\cdot)$ and $g(-\textbf{J}\cdot,\cdot)$ are opposite, have
module 1 and locally they are the only $2$-forms of $g$-module 1, since
$dim_{\mathbb{C}}\bigwedge^{2}{\mathbb{C}}T_{p}S=1$: as a result, for a given
$dA$, one has $dA=g(\textbf{J}\cdot,\cdot)$ for a unique global J.

∎

We will say that $\\{X,Y\\}$ is a _positive orthonormal frame_ for
${\mathbb{C}}TS$ with respect to the orientation induced by J if
$\textbf{J}(X)=Y$.

With the language of fiber bundles, a positive complex metric on a surface $S$
provides a reduction of the frame bundle of ${\mathbb{C}}TS\to S$ to the
structure group $O(2,{\mathbb{C}})$, while the pair of a positive complex
metric and an orientation on $S$ provides a reduction to $SO(2,{\mathbb{C}})$.

######  Corollary 8.1.6.

Given an oriented surface $S$, the map that sends a positive complex metric
$g$ to the orientation-consistent bicomplex structure J compatible with $g$
induces a homeomorphism

$\begin{split}\faktor{CM^{+}(S)}{C^{\infty}(S,{\mathbb{C}}^{*})}&\xrightarrow{\sim}BC(S)\cong
C(S)\times C(S)\\\ C^{\infty}(S,{\mathbb{C}}^{*})\cdot
g&\mapsto\textbf{J}_{g}\end{split}$ (8.3)

Conformal structures of Riemannian metrics correspond to elements of the
diagonal of $C(S)\times C(S)$.

###### Proof.

We showed that the map is continuous. Let us construct an inverse. Fix a real
volume form $\alpha\in\Omega^{2}(TS)$, which extends by
${\mathbb{C}}$-bilinearity to a form in
$\Omega^{2}({\mathbb{C}}TS,{\mathbb{C}})$. For any bicomplex structure J,
define the symmetric tensor

$g:=\alpha(\textbf{J}\cdot,\cdot)+\alpha(\cdot,\textbf{J}\cdot):$

$g$ is a complex metric as one can check on any basis $Y_{i},Y_{-i}$ of
eigenvectors for J with different wigenvalues; moreover $g$ induces J in turn,
hence it is a positive complex metric.

By choosing $\alpha$ as a real volume form for $S$, one gets an explicit
correspondence between the diagonal of $C(S)\times C(S)$ and the set of
conformal structures of Riemannian metrics. ∎

Finally, we remark that Proposition 6.5.4 can be restated in an interesting
way for positive complex metrics in terms of developing maps for projective
structures.

######  Theorem 8.1.7.

Let $S$ be an oriented surface and $g$ a positive complex metric on $S$.
Denote with $(\widetilde{S},\widetilde{g})$ the universal cover.

$(S,g)$ has constant curvature $-1$ if and only if there exists a smooth map

$(f_{1},f_{2})\colon\widetilde{S}\to{\mathbb{G}}=\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\setminus\Delta$

such that:

  * •

$\widetilde{g}=(f_{1},f_{2})^{*}\langle\cdot,\cdot\rangle=-\frac{4}{(f_{1}-f_{2})^{2}}df_{1}\cdot
df_{2}$

  * •

$f_{1}$ and $f_{2}$ are local diffeomorphisms, respectively preserving and
reversing the orientation.

  * •

$f_{1}$ and $f_{2}$ are
$(\pi_{1}(S),{\mathrm{PSL}(2,\mathds{C})})$-equivariant local diffeomorphisms
inducing two projective structures with the same representation

$hol_{f_{1}}=hol_{f_{2}}=mon_{(f_{1},f_{2})}\colon\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})}$

Moreover, if J is the bicomplex structure on $S$ induced by $g$, then, in
identification (8.1),

$\textbf{J}\equiv(f_{1}^{*}(\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}),f_{2}^{*}(-\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}))$

where $\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}$ is the standard complex
structure on $\mathbb{C}\mathbb{P}^{1}$.

###### Proof.

As we observed in Proposition 6.5.4, the fact that the complex metric $g$ is
positive implies that $f_{1}$ and $f_{2}$ are local diffeomorphisms.

Let $w_{k}$ be a local holomorphic chart on $p\in S$ for the pull-back complex
structure $f_{k}^{*}\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}$. Then, by Cauchy-
Riemann equations,

$Ker(df_{k}\colon{\mathbb{C}}T_{p}S\to{\mathbb{C}})=Span_{\mathbb{C}}\Big{(}\frac{\partial}{\partial\overline{w_{k}}}_{|p}\Big{)}$

so by the explicit description of the metric one has

$ll(g_{p})=\bigg{(}Span_{\mathbb{C}}\Big{(}\frac{\partial}{\partial\overline{w_{1}}}_{|p}\Big{)}\oplus
Span_{\mathbb{C}}\Big{(}\frac{\partial}{\partial\overline{w_{2}}}_{|p}\Big{)}\bigg{)}\setminus\\{0_{p}\\}$

and the two points of $\mathbb{P}_{\mathbb{C}}(ll(g_{p}))$ lie in two
different components of
$\mathbb{P}_{\mathbb{C}}({\mathbb{C}}T_{p}S)\setminus\mathbb{P}_{\mathbb{R}}(T_{p}S)$
if and only if the two pull-back complex structures via $f_{1}$ and $f_{2}$
induce opposite orientations.

The choice of $(f_{1},f_{2})$ inducing the same $\widetilde{g}$ is unique up
to post-composition with elements of ${\mathrm{PSL}(2,\mathds{C})}$, since the
composition with the diagonal swap $s:(x,y)\mapsto(y,x)$ on ${\mathbb{G}}$
would give an immersion with orientation-reversing first component.

All the elements of $\pi_{1}(S)$ are orientation-preserving, hence $\sigma$ is
$(\pi_{1}(S),{\mathrm{PSL}(2,\mathds{C})})$-equivariant and, since
${\mathrm{PSL}(2,\mathds{C})}$ acts diagonally on ${\mathbb{G}}$, both $f_{1}$
and $f_{2}$ are equivariant with the same holonomy.

From the description of $ll(g)$ above, one can conclude the last part of the
theorem. ∎

####  8.2 A Gauss-Bonnet Theorem for positive complex metrics

In this section, our aim is to prove a generalization of Gauss-Bonnet Theorem
in the setting of positive complex metrics.

Let $g$ be a positive complex metric on an oriented surface $S$. We defined
the area form $dA_{g}$ of $g$ as the unique 2-form with constant $g$-norm $1$
and compatible with the orientation on $S$, in the sense explained in Theorem
8.1.5. Locally,

$dA_{g}=\theta^{1}\wedge\theta^{2}$

for a local positive $g$-coframe $\\{\theta^{1},\theta^{2}\\}$.

######  Theorem 8.2.1.

Let $S$ be a closed oriented surface and $g$ a complex metric of constant
curvature $K_{g}$. Then

$\int_{S}K_{g}dA_{g}=2\pi\chi(S)$

where $\chi(S)$ is the Euler-Poincaré characteristic of $S$.

###### Proof.

Consider a real vector field $\mathtt{X}\in\Gamma(TS)$ with finite set of
zeros $\Lambda=\\{x_{1},\dots,x_{n}\\}$. Since the complex metric is positive,
$\|\mathtt{X}\|^{2}\neq 0$ over $S\setminus\Lambda$. Let the index $k$ vary in
$\\{1,\dots,n\\}$.

Consider for all $k$ two open disks $B_{k}$ and $\widetilde{B}_{k}$ in $S$,
such that

$x_{k}\in B_{k}\subset\overline{B}_{k}\subset\widetilde{B}_{k}$

and such that $B_{k}\cap B_{j}\neq\emptyset$ iff $k=j$.

Let J be the bicomplex structure induced by $g$ and by the orientation on $S$.
The $g$-orthonormal frame
$\bigg{(}\frac{\mathtt{X}}{\|\mathtt{X}\|},\textbf{J}\big{(}\frac{\mathtt{X}}{\|\mathtt{X}\|}\big{)}\bigg{)}$
can be defined locally on $S\setminus\Lambda$ but is defined globally on it
only up to a sign, and the same holds for the induced dual coframe
$\pm(\theta^{1},\theta^{2})$. Nevertheless, the area form
$dA_{g}=\theta^{1}\wedge\theta^{2}$ is globally well-defined on
$S\setminus\Lambda$ and so is the corresponding Levi-Civita connection form
$\omega$ defined by

$\displaystyle d\theta^{1}=\omega\wedge\theta^{2}$ $\displaystyle
d\theta^{2}=-\omega\wedge\theta^{1}.$

On the whole $S\setminus\Lambda$ we have $d\omega=-KdA$ as in standard
Riemannian Geometry (see [KN96a]).

On each $\widetilde{B}_{i}$ we fix a norm-$1$ complex vector field $e_{i}$ and
consider the coframe $(\theta^{1}_{i},\theta^{2}_{i})$ defined as the dual of
$(e_{i},\textbf{J}(e_{i}))$; let $\omega_{i}$ be the Levi-Civita connection
form with respect to $(\theta^{1}_{i},\theta^{2}_{i})$. Orient each
$\partial{\widetilde{B}_{i}}$ with the orientation induced by
$\widetilde{B}_{i}$. By Stokes formula,

$\displaystyle\int_{S}KdA$
$\displaystyle=\int_{S\setminus\bigcup_{i}B_{i}}KdA+\sum_{i=1}^{n}\int_{\overline{B}_{i}}KdA=$
$\displaystyle=$
$\displaystyle-\int_{S\setminus\bigcup_{i}\widetilde{B}_{i}}d\omega-\sum_{i=1}^{n}\int_{\overline{B}_{i}}d\omega_{i}=$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\int_{\partial
B_{i}}\omega-\omega_{i}.$

We prove that $\int_{S}KdA$ is a conformal invariant.

For all smooth $f\colon S\to{\mathbb{C}}$, $\overline{g}=fg$ is a positive
complex metric conformal to $g$. For some local choice of a square root of $f$
and of $g(\mathtt{X},\mathtt{X})$, one can define a local
$\overline{g}$-orthonormal frame
$\Big{(}\frac{\mathtt{X}}{\sqrt{f\|\mathtt{X},\mathtt{X}\|_{g}^{2}}},\frac{\textbf{J}\mathtt{X}}{\sqrt{f\|\mathtt{X},\mathtt{X}\|_{g}^{2}}}\Big{)}$
with $\overline{g}$-dual given by the coframe
$(\overline{\theta}^{1},\overline{\theta}^{2})$, which in turn defines a Levi-
Civita connection form $\overline{\omega}$ for $\overline{g}$ independent of
the choice of the root. On each $\widetilde{B}_{i}$, fix a square root
$\sqrt{f}$ of $f$, so a $\overline{g}$-orthonormal frame is given by
$(\frac{e_{1}}{\sqrt{f}},\frac{\textbf{J}(e_{1})}{\sqrt{f}})$ with associated
$\overline{g}$-coframe $(\overline{\theta}_{i}^{1},\overline{\theta}_{i}^{2})$
and Levi-Civita connection form $\overline{\omega}_{i}$. By an explicit
computation, one can check that on each
$\widetilde{B}_{i}\setminus\\{x_{i}\\}$

$\overline{\omega}-\omega=\overline{\omega}_{i}-\omega_{i}=-\frac{1}{2f}df\circ\textbf{J}.$

As a result,

$\int_{S}K_{\overline{g}}dA_{\overline{g}}=\sum_{i=1}^{n}\int_{\partial
B_{i}}\overline{\omega}-\overline{\omega}_{i}=\sum_{i=1}^{n}\int_{\partial
B_{i}}\omega-\omega_{i}=\int_{S}K_{g}dA_{g}.$

Now, on each $\widetilde{B}_{k}\setminus\\{x_{k}\\}$, the coframes
$(\theta^{1}_{i},\theta^{2}_{i})$ and $(\theta^{1},\theta^{2})$ (the last one
is well-defined only locally) are both $g$-orthonormal and J-oriented, so we
locally have

$\begin{pmatrix}\theta^{1}\\\
\theta^{2}\end{pmatrix}=\begin{pmatrix}\cos(\alpha_{k})&\sin(\alpha_{k})\\\
-\sin(\alpha_{k})&\cos(\alpha_{k})\end{pmatrix}\begin{pmatrix}\theta_{k}^{1}\\\
\theta_{k}^{2}\end{pmatrix}$

where $e^{i\alpha_{k}}\in{\mathbb{C}}^{*}$ is a locally defined smooth
function, defined only up to a sign on
$\widetilde{B}_{k}\setminus\\{x_{k}\\}$. Nevertheless,
$e^{2i\alpha_{k}}\colon\partial B_{k}\to S^{1}$ is a well defined smooth
function and $d\alpha_{k}$ is a well-defined one form on $\partial B_{k}$.

A standard calculation shows that $\omega-\omega_{k}=d\alpha_{k}$, hence

$\displaystyle\int_{S}KdA$ $\displaystyle=\sum_{k=1}^{n}\int_{\partial
B_{k}}\omega-\omega_{k}=\sum_{k=1}^{n}\int_{\partial B_{k}}d\alpha_{k}=$
$\displaystyle=\sum_{k=1}^{n}\int_{\partial
B_{k}}\frac{1}{2i}\frac{d(e^{2i\alpha_{k}})}{e^{2i\alpha_{k}}}=\sum_{k=1}^{n}\pi
deg(e^{2i\alpha_{k}})\in\pi{\mathds{Z}}.$

Recalling that $BC(S)$ is connected and that the quantity
$\int_{S}K_{g}dA_{g}$ is a conformal invariant, the map

$\int_{S}K\colon BC(S)\to\pi{\mathds{Z}}$

is well-posed and continuous, hence it is constant. Since
$\int_{S}KdA=2\pi\chi(S)$ for Riemannian metrics, the thesis follows. ∎

####  8.3 A Uniformization Theorem through Bers Theorem

The aim of this section is to prove a generalization of the classic Riemann’s
Uniformization Theorem for bicomplex structures and positive complex metrics.

Before stating the main theorem, we recall some notions about quasi-Fuchsian
representations.

A representation $\rho\colon\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})}$ is
_quasi-Fuchsian_ if its limit set
$\Lambda_{\rho}\subset\mathbb{C}\mathbb{P}^{1}$, i.e. the set of accumulation
points of any $\rho(\pi_{1}(S))$-orbit in $\mathbb{C}\mathbb{P}^{1}$, is a
Jordan curve in $\mathbb{C}\mathbb{P}^{1}$. The set of quasi-Fuchsian
representations defines $QF(S)$ an open subset of the character variety
$\mathcal{X}(S)$, defined in Equation (7.12).

A major result in the setting of quasi-Fuchsian representations is the Bers
Simultaneous Uniformization Theorem.

######  Theorem 8.3.1 (Bers Simultaneous Uniformization Theorem, [Ber60]).

Let $S$ be an oriented surface with $\chi(S)<0$. For all $J_{1},J_{2}$ complex
structures over $S$, there exists a unique quasi-Fuchsian representation
$\rho=\rho(J_{1},J_{2})\colon\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})}$ such
that, defined $\Lambda_{\rho}\subset\mathbb{C}\mathbb{P}^{1}$ as the limit set
of $\rho$ and $\Omega_{+},\Omega_{-}$ as the connected components of
$\mathbb{C}\mathbb{P}^{1}\setminus\Lambda_{\rho}$:

  * •

there exists a unique diffeomorphism $f_{1}\colon\widetilde{S}\to\Omega_{+}$,
which is $J_{1}$-holomorphic and $\rho$-equivariant;

  * •

there exists a unique diffeomorphism $f_{2}\colon\widetilde{S}\to\Omega_{-}$,
which is $J_{2}$-antiholomorphic and $\rho$-equivariant.

Moreover, $(\rho,f_{1},f_{2})$ are continuous functions of $(J_{1},J_{2})\in
BC(S)$.

This correspondence determines a homeomorphism in the quotient

$\mathfrak{B}\colon\text{\Capricorn}(S)\times\text{\Capricorn}(S)\xrightarrow{\sim}QF(S)$
(8.4)

where $\text{\Capricorn}(S)$ is the Teichmüller space of $S$.

Recall that in Subsection 7.4.1 we considered the holonomy maps
$\widetilde{Hol}$ and $Hol$ from the spaces $\widetilde{P}(S)$ and $P(S)$
respectively. Define the loci

$\displaystyle\widetilde{\mathcal{P}}_{QF}(S):=$
$\displaystyle\widetilde{Hol}^{-1}(QF(S))\subset\widetilde{\mathcal{P}}(S),$
$\displaystyle\mathcal{P}_{QF}(S):=$ $\displaystyle
Hol^{-1}(QF(S))\subset\mathcal{P}(S)$

which, by continuity of the holonomy maps, are open subsets of
$\widetilde{\mathcal{P}}(S)$ and of ${\mathcal{P}}(S)$ respectively.

######  Theorem 8.3.2 (W. Goldman, [Gol87]).

Let $S$ be a compact oriented surface of genus $g(S)\geq 2$.

An open connected component of the space $\widetilde{\mathcal{P}}_{QF}(S)$ is

$\displaystyle\widetilde{\mathcal{P}}_{0}(S):=\\{$ $\displaystyle
f\colon\widetilde{S}\to\mathbb{C}\mathbb{P}^{1},\text{$\rho$-equivariant for
some quasi-Fuchsian $\rho\colon\pi_{1}(S)\to\mathbb{C}\mathbb{P}^{1}$,}$
$\displaystyle\text{ with }f(\widetilde{S})\cap\Lambda_{\rho}=\emptyset\\}.$

Moreover, the restriction

$Hol\colon\faktor{\widetilde{\mathcal{P}}_{0}(S)}{\text{Diff}_{0}(S)}=:{\mathcal{P}}_{0}(S)\to
QF(S)$

is a homeomorphism.

With the notations of Theorem 8.3.1 and equation (8.4), the inverse is given
by

$[\rho]\mapsto[f_{1}(\rho)]=\Big{[}f_{1}\Big{(}\mathfrak{B}^{-1}([\rho])\Big{)}\Big{]}.$

We are now able to state a generalization of the Uniformization Theorem.

For an oriented surface $S$, define

$CM^{+}_{-1}(S):=\\{\text{Positive complex metrics with constant curvature
$-1$}\\}$

endowed with the subspace topology from $CM^{+}(S)$.

######  Theorem 8.3.3.

Let $S$ be an oriented surface with $\chi(S)<0$.

  * 1)

For all $g\in CM^{+}(S)$, there exists a smooth $f\colon S\to{\mathbb{C}}^{*}$
such that $f\cdot g$ has constant curvature $-1$ and quasi-Fuchsian monodromy.

More precisely, the map

$\displaystyle\mathfrak{{J}}\colon CM^{+}(S)$ $\displaystyle\to BC(S)$
$\displaystyle g$ $\displaystyle\mapsto[g]=\textbf{J}_{g}$

admits a continuous right inverse

$\mathfrak{U}\colon C(S)\times C(S)\cong BC(S)\to CM^{+}_{-1}(S)$

such that the diagram

${C(S)\times C(S)\cong
BC(S)}$${CM_{-1}^{+}(S)}$${\text{\Capricorn}(S)\times\text{\Capricorn}(S)}$${\mathcal{X}(S)}$$\scriptstyle{\mathfrak{U}}$$\scriptstyle{mon}$$\scriptstyle{\mathfrak{B}}$

commutes, where $\mathfrak{B}$ is defined as in (8.4).

  * 2)

If $S$ is closed, the image of $\mathfrak{U}$ is the connected component of
$CM^{+}_{-1}(S)$ containing Riemannian metrics.

###### Proof of Theorem 8.3.3.

  * 1)

Consider $\textbf{J}\in BC(S)$, which corresponds to two complex structures
$(J_{1},J_{2})\in C(S)\times C(S)$.

Applying Theorem 8.3.1 to the couple $(J_{1},J_{2})$, there exist:

    * –

$\rho=\rho(J_{1},J_{2})\colon\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})}$ quasi-
Fuchsian;

    * –

$f_{1}\colon\widetilde{S}\to\mathbb{C}\mathbb{P}^{1}$ $J_{1}$-holomorphic and
$\rho$-equivariant embedding;

    * –

$f_{2}\colon\widetilde{S}\to\mathbb{C}\mathbb{P}^{1}$ $J_{2}$-antiholomorphic
and $\rho$-equivariant embedding

with $Im(f_{1})\cap Im(f_{2})=\emptyset$. Hence, we have an admissible,
$\rho$-equivariant embedding

$\sigma=(f_{1},f_{2})\colon\widetilde{S}\to{{\mathbb{G}}}=\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\setminus\Delta.$

By Theorem 8.1.7, the pull-back complex metric on $\widetilde{S}$ is positive
and projects to a positive complex metric $g$ on $S$.

After we show that the constructed $g$ is compatible with J, the thesis
follows by defining $\mathfrak{U}(\textbf{J}):=g$. Let $w_{1}$ and $w_{2}$ be
the complex coordinates on $\widetilde{S}$ induced by $J_{1}$ and $J_{2}$
respectively. Then,

$\displaystyle g$
$\displaystyle=\frac{4}{(f_{1}-f_{2})^{2}}f_{1}^{*}dz_{1}\cdot
f_{2}^{*}dz_{2}=\frac{4}{(f_{1}-f_{2})^{2}}df_{1}\cdot df_{2}=$
$\displaystyle=\frac{4}{(f_{1}-f_{2})^{2}}\frac{\partial
f_{1}}{\partial{w_{1}}}\frac{\partial
f_{2}}{\partial\overline{w_{2}}}d{w_{1}}\cdot d\overline{w_{2}};$

hence $V_{i}(J_{2})=V_{i}(\textbf{J})$ and $V_{-i}(J_{1})=V_{-i}(\textbf{J})$
are the isotropic directions of $g$.

  * 2)

By regarding

$Im(\mathfrak{U})=\\{g\ |\ (\mathfrak{U}\circ\mathfrak{J})(g)=g\\}\subset
CM_{-1}^{+}(S),$

we have that $Im(\mathfrak{U})$ is a closed connected subset of
${\mathbb{C}}M_{-1}^{+}(S)$. We need to prove that it is also open.

As we stated in Theorem 8.1.7, for any $g\in CM_{-1}^{+}$ there exist two
developing maps for projective structures $f_{1}$ and $f_{2}$ on $S$ and
$\overline{S}$ respectively, uniquely defined up to post-composition with
elements in ${\mathrm{PSL}(2,\mathds{C})}$, such that
$\sigma=(f_{1},f_{2})\colon(\widetilde{S},\widetilde{g})\to{\mathbb{G}}$ is an
isometric immersion. We therefore have a map

$\displaystyle proj\colon CM_{-1}^{+}(S)$
$\displaystyle\to\mathcal{\widetilde{P}}(S)\times\mathcal{\widetilde{P}}(\overline{S})$
$\displaystyle g=(f_{1},f_{2})^{*}\langle\cdot,\cdot\rangle_{\mathbb{G}}$
$\displaystyle\mapsto([f_{1}],[f_{2}]).$

The thesis follows after we show that

$Im(\mathfrak{U})=proj^{-1}(\mathcal{\widetilde{P}}_{0}(S)\times\mathcal{\widetilde{P}}_{0}(\overline{S}))$

since $\mathcal{\widetilde{P}}_{0}(S)$ is an open subset of
$\mathcal{\widetilde{P}}_{QF}(S)$, hence of $\mathcal{\widetilde{P}}(S)$.

By construction of $\mathfrak{U}$, clearly

$Im(\mathfrak{U})\subseteq
proj^{-1}(\mathcal{\widetilde{P}}_{0}(S)\times\mathcal{\widetilde{P}}_{0}(\overline{S})).$

Conversely, assume
$g=(f_{1},f_{2})^{*}\langle\cdot,\cdot\rangle_{\mathbb{G}}\in
proj^{-1}(\mathcal{\widetilde{P}}_{0}(S)\times\mathcal{\widetilde{P}}_{0}(\overline{S}))$.
Since
$hol_{f_{1}}=hol_{f_{2}}\colon\pi_{1}(S)\to{\mathrm{PSL}(2,\mathds{C})}$,
$f_{1}$ and $f_{2}$, by Theorem 8.3.2, the maps $f_{1}$ and $f_{2}$ correspond
exactly to the maps one gets through Theorem 8.3.1 from the pair of complex
structures
$(f_{1}^{*}(\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}),f_{2}^{*}(-\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}))$,
hence

$(f_{1},f_{2})^{*}\langle\cdot,\cdot\rangle_{\mathbb{G}}=\mathfrak{U}(f_{1}^{*}(\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}}),f_{2}^{*}(-\mathbb{J}^{\mathbb{C}\mathbb{P}^{1}})).$

∎

##  Part III Integrable immersions into $\mathbb{G}_{n+1}$

###  Chapter 9 The Gauss map of hypersurfaces in $\mathbb{H}^{n+1}$

In this Chapter we will focus on the construction of the Gauss map of an
immersed hypersurface, its relation with the normal evolution and the geodesic
flow action on the unit tangent bundle, and provide several examples of great
importance for the rest of Part III.

After a discussion in the general setting, we will then focus on the case of
immersions with small principal curvatures.

####  9.1 Lift to the unit tangent bundle

We defined the Gauss map associated to an immersion into $\mathbb{H}^{3}$ in
Section 6.7. Let us formally recall it now for hypersurfaces in general
dimension, together with the notions of lift to the unit tangent bundle.

Recall that in Chapter 5 we defined a para-Sasaki metric
${\overline{Sas}_{n+1}}$ on $T^{1}\mathbb{H}^{n+1}$ and a para-Kähler
structure $({\mathbbm{g}},\mathbbm{J},\Omega)$ on the space
$\mathbb{G}_{n+1}$.

######  Definition 9.1.1.

Let $M$ be an oriented $n$-dimensional manifold, let
$\sigma:M\to\mathbb{H}^{n+1}$ be an immersion, and let $\nu$ be the unit
normal vector field of $\sigma$ compatible with the orientations of $M$ and
$\mathbb{H}^{n+1}$. We define the _lift_ of $\sigma$ as

$\zeta_{\sigma}:M\to
T^{1}\mathbb{H}^{n+1}\qquad\zeta_{\sigma}(p)=(\sigma(p),\nu(p))~{}.$

The _Gauss map_ of $\sigma$ is then the map

$G_{\sigma}:M\to\mathbb{G}_{n+1}\qquad
G_{\sigma}=\mathrm{p}\circ\zeta_{\sigma}~{}.$

In other words, the Gauss map of $\sigma$ is the map which associates to $p\in
M$ the geodesic $\ell$ of $\mathbb{H}^{n+1}$ orthogonal to the image of
$d_{p}\sigma$ at $\sigma(p)$, oriented compatibly with the orientations of $M$
and $\mathbb{H}^{n+1}$.

Recall that the _first fundamental form_ $\mathrm{I}$ of $\sigma$ is defined
as the pull-back Riemannian metric
$\mathrm{I}=\sigma^{*}{\langle\cdot,\cdot\rangle_{\mathbb{H}^{n+1}}}$ on $M$.

Also recall that the _shape operator_ $B$ of $\sigma$ can be defined as the
$(1,1)$-tensor on $M$ defined by

$d\sigma\circ B(W)=-D_{W}\nu~{},$ (9.1)

for $D$ the Levi-Civita connection of $\mathbb{H}^{n+1}$.

######  Proposition 9.1.2.

Given an oriented manifold $M^{n}$ and an immersion
$\sigma:M\to\mathbb{H}^{n+1}$, the lift of $\sigma$ is an immersion orthogonal
to the fibers of $\mathrm{p}:T^{1}\mathbb{H}^{n+1}\to\mathbb{G}_{n+1}$. As a
consequence $G_{\sigma}$ is an immersion.

###### Proof.

By a direct computation in the hyperboloid model, the differential of
$\zeta_{\sigma}$ has the expression

$d_{p}\zeta_{\sigma}(W)=(d_{p}\sigma(W),d_{p}\nu(W))=(d_{p}\sigma(W),-d_{p}\sigma(B(W)))~{},$
(9.2)

indeed, the ambient derivative in ${\mathbb{R}}^{n+1,1}$ of the vector field
$\nu$ equals the covariant derivative with respect to $D$, since $d_{p}\nu(W)$
is orthogonal to $\sigma(p)$ as a consequence of the condition
$\langle\sigma(p),\nu(p)\rangle=0$.

As both $d_{p}\sigma(W)$ and $d_{p}\sigma(B(W))$ are tangential to the image
of $\sigma$ at $p$, hence orthogonal to $\nu(p)$, $d_{p}\zeta_{\sigma}(W)$ can
be written as:

$d_{p}\zeta_{\sigma}(W)=d_{p}\sigma(W)^{\mathcal{H}}-d_{p}\sigma(B(W))^{\mathcal{V}}~{}.$
(9.3)

Therefore, for every $W\neq 0$, $d_{p}\zeta_{\sigma}(W)$ is a non-zero vector
orthogonal to $\chi_{\zeta(p)}$ by Definition 5.1.2. Since the differential of
$\mathrm{p}$ is a vector space isomorphism between $\chi_{\zeta(p)}^{\perp}$
and $T_{G_{\sigma}(p)}\mathbb{G}_{n+1}$, the Gauss map $G_{\sigma}$ is also an
immersion. ∎

As a consequence of Proposition 9.1.2, we can compute the first fundamental
form of the Gauss map $G_{\sigma}$, that is, the pull-back metric
$G_{\sigma}^{*}{\mathbbm{g}}$, which we denote by $\overline{\mathrm{I}}$.
Since the lift $\zeta_{\sigma}$ is orthogonal to $\chi$, it suffices to
compute the pull-back metric of ${\overline{Sas}_{n+1}}$ by $\zeta_{\sigma}$.
By Equation (9.3), we obtain:

$\overline{\mathrm{I}}=\mathrm{I}-\mathrm{I}\mathrm{I}\mathrm{I}~{},$ (9.4)

where $\mathrm{I}\mathrm{I}\mathrm{I}=\mathrm{I}(B\cdot,B\cdot)$ is the third
fundamental form of $\sigma$.

Let us now see that the orthogonality to the generator of the geodesic flow
essentially characterizes the lifts of immersed hypersurfaces in
$\mathbb{H}^{n+1}$, in the following sense.

######  Proposition 9.1.3.

Let $M^{n}$ be an orientable manifold and $\zeta:M\to T^{1}\mathbb{H}^{n+1}$
be an immersion orthogonal to the fibers of
$\mathrm{p}:T^{1}\mathbb{H}^{n+1}\to\mathbb{G}_{n+1}$. If
$\sigma:=\pi\circ\zeta$ is an immersion, then $\zeta$ is the lift of $\sigma$
with respect to an orientation of $M$.

###### Proof.

Let us write $\zeta=(\sigma,\nu)$. Choosing the orientation of $M$ suitably,
we only need to show that the unit vector field $\nu$ is normal to the
immersion $\sigma$. By differentiating, $d\zeta=(d\sigma,\mathrm{d}\nu)$ and
by (5.3) we obtain

$\langle\nu(p),d\sigma(W)\rangle+\langle\sigma(p),\mathrm{d}\nu(W)\rangle=0$

for all $W\in T_{p}M$. By the orthogonality hypothesis and the expression
$\chi_{\zeta(p)}=(\nu(p),\sigma(p))$ (Lemma 5.1.1) we obtain

$\langle\nu(p),d\sigma(W)\rangle-\langle\sigma(p),\mathrm{d}\nu(W)\rangle=0~{},$

hence $\langle\nu(p),\mathrm{d}\sigma(W)\rangle=0$ for all $W$. Since by
hypothesis the differential of $\sigma$ is injective, $\nu(p)$ is uniquely
determined up to the sign, and is a unit normal vector to the immersion
$\sigma$. ∎

In relation with Proposition 9.1.3, it is important to remark that there are
(plenty of) immersions in $T^{1}\mathbb{H}^{n+1}$ which are orthogonal to
$\chi$ but are _not_ the lifts of immersions in $\mathbb{H}^{n+1}$, meaning
that they become singular when post-composed with the projection
$\pi:T^{1}\mathbb{H}^{n+1}\to\mathbb{H}^{n+1}$. Some examples of this
phenomenon (Example 9.1.9), and more in general of the Gauss map construction,
are presented in Subsection 9.1.2 below.

#####  9.1.1 Geodesic flow and normal evolution

We develop here the construction of the normal evolution of hypersurfaces into
$\mathbb{H}^{n+1}$.

######  Definition 9.1.4.

Given an oriented manifold $M^{n}$ and an immersion
$\sigma:M\to\mathbb{H}^{n+1}$, the _normal evolution_ of $\sigma$ is the map

$\sigma_{t}:M\to\mathbb{H}^{n+1}\qquad\sigma_{t}(p)=\exp_{\sigma(p)}(t\nu(p))~{},$

where $\nu$ is the unit normal vector field of $\sigma$ compatible with the
orientations of $M$ and $\mathbb{H}^{n+1}$.

The relation between the normal evolution and the geodesic flow is encoded in
the following proposition.

######  Proposition 9.1.5.

Let $M^{n}$ be an orientable manifold and $\sigma:M\to\mathbb{H}^{n+1}$ be an
immersion. Suppose $\sigma_{t}$ is an immersion for some $t\in{\mathbb{R}}$.
Then $\zeta_{\sigma_{t}}=\varphi_{t}\circ\zeta_{\sigma}$.

###### Proof.

The claim is equivalent to showing that, if the differential of $\sigma_{t}$
is injective at $p$, then
$(\sigma_{t}(p),\nu_{t}(p))=\varphi_{t}(\sigma(p),\nu(p))$, where $\nu_{t}$ is
the normal vector of $\sigma_{t}$. The equality on the first coordinate holds
by definition of the geodesic flow, since $t\mapsto\gamma(t)=\sigma_{t}(p)$ is
precisely the parameterized geodesic such that $\gamma(0)=\sigma(p)$ with
speed $\gamma^{\prime}(0)=\nu(p)$. It thus remains to check that the normal
vector in $\sigma_{t}(p)$ equals $\gamma^{\prime}(t)$.

By the usual expression of the exponential map in the hyperboloid model of
$\mathbb{H}^{n+1}$, the normal evolution is:

$\sigma_{t}(p)=\cosh(t)\sigma(p)+\sinh(t)\nu(p)~{},$ (9.5)

and therefore

$d\sigma_{t}(V)=d\sigma\circ(\cosh(t)\mathrm{id}-\sinh(t)B)(V)$ (9.6)

for $V\in T_{p}M$. It is then immediate to check that

$\dot{\gamma}(t)=\sinh(t)\sigma(p)+\cosh(t)\nu(p)$

is orthogonal to $d\sigma_{t}(V)$ for all $V$. If $d\sigma_{t}$ is injective,
this implies that $\gamma^{\prime}(t)$ is the unique unit vector normal to the
image of $\sigma$ and compatible with the orientations, hence it equals
$\nu_{t}(p)$. This concludes the proof. ∎

A straightforward consequence, recalling that the Gauss map is defined as
$G_{\sigma}=\mathrm{p}\circ\zeta_{\sigma}$ and that
$\mathrm{p}\circ\varphi_{t}=\mathrm{p}$, is the following.

######  Corollary 9.1.6.

The Gauss map of an immersion $\sigma:M^{n}\to\mathbb{H}^{n+1}$ is invariant
under the normal evolution, namely $G_{\sigma_{t}}=G_{\sigma}$, as long as
$\sigma_{t}$ is an immersion.

######  Remark 9.1.7.

It follows from Equation (9.6) that, for any immersion
$\sigma:M^{n}\to\mathbb{H}^{n+1}$, the differential of $d\sigma_{t}$ at a
point $p$ is injective for small $t$. However, in general $\sigma_{t}$ might
fail to be a global immersion for all $t\neq 0$. In Section 9.3 we will
discuss the condition of small principal curvatures for $\sigma$, which is a
sufficient condition to ensure that $\sigma_{t}$ remains an immersion for all
$t$.

As a related phenomenon, it is possible to construct examples of immersions
$\zeta:M^{n}\to T^{1}\mathbb{H}^{n+1}$ which are orthogonal to the fibers of
$\mathrm{p}$ but such that $\pi\circ\varphi_{t}\circ\zeta$ fails to be an
immersion for all $t\in{\mathbb{R}}$. We will discuss this problem later on,
see Example 10.4.1.

#####  9.1.2 Fundamental examples

It is now useful to describe several explicit examples. All of them will
actually play a role in some of the proofs in the next sections.

######  Example 9.1.8 (Totally geodesic hyperplanes).

Let us consider a totally geodesic hyperplane $\mathcal{P}$ in
$\mathbb{H}^{n+1}$, and let $\sigma:\mathcal{P}\to\mathbb{H}^{n+1}$ be the
inclusion map. Since in this case the shape operator vanishes everywhere, from
Equation (9.4) the Gauss map is an isometric immersion (actually, an
embedding) into $\mathbb{G}_{n+1}$ with respect to the first fundamental form
of $\sigma$. Totally geodesic immersions are in fact the only cases for which
this occurs.

A remark that will be important in the following is that the lift
$\zeta_{\sigma}$ is horizontal: that is, by Equation (9.3),
$d\zeta_{\sigma}(w)$ equals the horizontal lift of $w$ for every vector $w$
tangent to $\mathcal{P}$ at $x$. Therefore for every $x\in\mathcal{P}$, the
image of $d\zeta_{\sigma}$ at $x$ is exactly the horizontal subspace
$\mathcal{H}^{0}_{(x,\nu(x))}$, for $\nu(x)$ the unit normal vector of
$\mathcal{P}$.

######  Example 9.1.9 (Spheres in tangent space).

A qualitatively opposite example is the following. Given a point
$x\in\mathbb{H}^{n+1}$, let us choose an isometric identification of
$T_{x}\mathbb{H}^{n+1}$ with the $(n+1)$-dimensional Euclidean space, and
consider the $n$-sphere $\mathbb{S}^{n}$ as a hypersurface in
$T_{x}\mathbb{H}^{n+1}$ by means of this identification. Then we can define
the map

$\zeta:\mathbb{S}^{n}\to T^{1}\mathbb{H}^{n+1}\qquad\zeta(v)=(x,v)~{}.$

The differential of $\zeta$ reads $d\zeta_{v}(w)=(0,w)=w^{\mathcal{V}}$ for
every $w\in T_{v}\mathbb{S}^{n}\cong v^{\perp}$, hence $\zeta$ is an
immersion, which is orthogonal to the fibers of $\mathrm{p}$. Actually,
$\zeta$ is vertical: this means that $d\zeta_{v}(w)$ is the vertical lift of
$w$ for every $w\in v^{\perp}$, and therefore
$d_{v}\zeta(T_{v}\mathbb{S}^{n})$ is exactly the vertical subspace
$\mathcal{V}^{0}_{(x,v)}$.

Clearly we are not in the situation of Propositions 9.1.2 and 9.1.3, as
$\pi\circ\zeta$ is a constant map. On the other hand, $\mathrm{p}\circ\zeta$
has image in $\mathbb{G}_{n+1}$ consisting of all the (oriented) geodesics
$\ell$ going through $x$. However, when post-composing $\zeta$ with the
geodesic flow, $\varphi_{t}\circ\zeta$ projects to an immersion in
$\mathbb{H}^{n+1}$ for all $t\neq 0$ and is in fact an embedding with image a
geodesic sphere of $\mathbb{H}^{n+1}$ of radius $|t|$ centered at $x$. As a
final remark, the first fundamental form of $\mathrm{p}\circ\zeta$, is _minus_
the spherical metric of $\mathbb{S}^{n}$, since by Equation (5.14)
${\overline{Sas}_{n+1}}(w^{\mathcal{V}},w^{\mathcal{V}})=-\langle w,w\rangle$.

The previous two examples can actually be seen as special cases of a more
general construction, which will be very useful in the next sections.

######  Example 9.1.10 (A mixed hypersurface in the unit tangent bundle).

Let us consider a totally geodesic $k$-dimensional submanifold $Q$ of
$\mathbb{H}^{n+1}$, for $0\leq k\leq n$. Consider the unit normal bundle

$\mathrm{N}^{1}Q=\\{(x,v)\in T^{1}\mathbb{H}^{n+1}\,|\,x\in Q,v\text{
orthogonal to }Q\\}~{},$

which is an $n$-dimensional submanifold of $T^{1}\mathbb{H}^{n+1}$, and let
$\iota$ be the obvious inclusion map of $\mathrm{N}^{1}Q$ in
$T^{1}\mathbb{H}^{n+1}$. Observe that $\pi\circ\iota$ is nothing but the
bundle map $\mathrm{N}^{1}Q\to Q$, hence not an immersion unless $k=n$ which
is the case we discussed in Example 9.1.8. The map $\mathrm{p}\circ\iota$ has
instead image in $\mathbb{G}_{n+1}$ which consists of all the oriented
geodesics $\ell$ orthogonal to $Q$. See Figure 9.1. Let us focus on its
geometry in $\mathbb{G}_{n+1}$.

Given $(x,v)\in N^{1}Q$, take an orthonormal basis $\\{w_{1},\ldots,w_{k}\\}$
of $T_{x}Q$. Clearly the $w_{i}$’s are orthogonal to $v$, and let us complete
them to an orthonormal basis $\\{w_{1},\ldots,w_{n}\\}$ of $v^{\perp}\subset
T_{x}\mathbb{H}^{n+1}$. Then $\\{w_{1},\ldots,w_{n}\\}$ identifies to a basis
of $T_{(x,v)}\mathrm{N}^{1}Q$. By repeating the arguments of the previous two
examples, $d\iota_{(x,v)}(w_{i})$ is the _horizontal_ lift of $w_{i}$ at
$(x,v)$ if $1\leq i\leq k$, and is the _vertical_ lift if $i>k$. In particular
they are all orthogonal to $\chi_{(x,v)}$, and therefore the induced metric on
$\mathrm{N}^{1}Q$ by the metric ${\overline{Sas}_{n+1}}$ coincides with the
first fundamental form of $\mathrm{p}\circ\iota$. This metric has signature
$(k,n-k)$, and $\\{w_{1},\ldots,w_{n}\\}$ is an orthonormal basis, for which
$w_{1},\ldots,w_{k}$ are positive directions and $w_{k+1},\ldots,w_{n}$
negative directions.

Similarly to the previous example, for all $t\neq 0$ the map
$\varphi_{t}\circ\iota$ has the property that, when post-composed with the
projection $\pi$, it gives an embedding with image the equidistant “cylinder”
around $Q$.

Figure 9.1: The normal bundle $N^{1}Q$ of a $k$-dimensional totally geodesic
submanifold $Q$ in $\mathbb{H}^{n+1}$ (here $k=1$ and $n=2$). On the right:
after composing with the geodesic flow $\varphi_{t}$ for $t\neq 0$, one
obtains an equidistant cylinder.

Let us now consider a final example, which allows also to prove the
integrability of the almost para-complex structure $\mathbbm{J}$ of
$\mathbb{G}_{n+1}$ we introduced in Lemma 5.2.1.

######  Example 9.1.11 (Horospheres).

Let us consider a horosphere $H$ in $\mathbb{H}^{n+1}$, and apply the Gauss
map construction of Definition 9.1.1 to the inclusion
$\sigma:H\to\mathbb{H}^{n+1}$.

It is known that the shape operator of $H$ is $\pm\mathrm{id}$ (the sign
varies according to the choice of the normal vector field, or equivalently by
the choice of orientation on $H$), a result we will also deduce later on from
our arguments in Remark 9.3.10. Define $\zeta_{\pm}$ as the lift of $\sigma$
induced by the choice of the normal vector field for which the shape operator
is $\pm\mathrm{id}$.

Now, by Proposition 9.1.2, the lift $\zeta_{\pm}$ is orthogonal to the fibers
of $\mathrm{p}$, and moreover, by Equation (9.3),
$d\zeta_{\pm}(w)=w^{\mathcal{H}}\pm w^{\mathcal{V}}$. As a result, by Equation
(5.18), one has in fact that the image of $d_{x}\zeta_{\pm}$ is the whole
$(\pm 1)$-eigenspace of $\mathrm{J}$ in
$T_{\zeta_{\pm}(x)}T^{1}\mathbb{H}^{n}$.

A direct application of Example 9.1.11 shows that the almost para-complex
structure $\mathbbm{J}$ is integrable:

######  Corollary 9.1.12.

The $(1,1)$-tensor $\mathbbm{J}$ is a para-complex structure on
$\mathbb{G}_{n+1}$.

###### Proof.

Given $(x,v)\in T^{1}\mathbb{H}^{n+1}$, consider the two horospheres $H^{\pm}$
containing $x$ with normal vector $v$ at $x$. The vector $v$ points to the
convex side of one of them, and to the concave side of the other. Let us
orient them in such a way that $v$ is compatible with the ambient orientation.
Then Example 9.1.11 shows that the Gauss maps of the horospheres $H^{\pm}$
have image integral submanifolds for the distributions
$T^{\pm}\mathbb{G}_{n+1}$ associated to the almost para-complex structure
$\mathbbm{J}$, which is therefore integrable. ∎

####  9.2 A consequence: $(\mathbb{G}_{3},\mathbbm{g})\cong
Re\big{(}({\mathbb{G}},\langle\cdot,\cdot\rangle_{\mathbb{G}})\big{)}$

We defined two different metrics on the space of geodesics of
$\mathbb{H}^{3}$, the pseudo-Riemannian metric ${\mathbbm{g}}$ and the
holomorphic Riemannian metric
$\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{\mathbb{G}}$.

By Proposition 6.7.1 and Equation 9.4, we can immediately observe that if
$\sigma\colon S\to\mathbb{H}^{3}$ is an immersion, then

$G_{\sigma}^{*}({\mathbbm{g}})=Re\Big{(}G_{\sigma}^{*}(\langle\cdot,\cdot\rangle_{\mathbb{G}})\Big{)}.$
(9.7)

We show that Equation 9.7, together with the examples considered in Subsection
9.1.2, allows to prove the following

######  Proposition 9.2.1.

On $\mathbb{G}_{3}$,

${\mathbbm{g}}=Re\Big{(}\langle\cdot,\cdot\rangle_{\mathbb{G}}\Big{)}$

###### Proof.

Let $\ell=\mathrm{p}(x,v)\in\mathbb{G}_{3}$, then in the hyperboloid model
$\chi_{(x,v)}=(v,0)$, and

$\chi^{\bot}_{(x,v)}=Span_{\mathbb{R}}((w_{1},0),(w_{2},0),(0,w_{1}),(0,w_{2}))\subset
T_{(x,v)}T^{1}\mathbb{H}^{3}$

with $(w_{1},w_{2},v)$ being a orthonormal basis for $T_{x}M$. Then,

$d\mathrm{p}(w_{1},0),d\mathrm{p}(w_{2},0),d\mathrm{p}(0,w_{1}),d\mathrm{p}(0,w_{2})\in
T_{\ell}\mathbb{G}_{3}$

generate the tangent space and are mutually ${\mathbbm{g}}$-orthonormal, with
the former two ones having ${\mathbbm{g}}$-norm $1$ and the latter two ones
having ${\mathbbm{g}}$-norm $-1$. We prove that $Re\langle\cdot,\cdot\rangle$
coincides with ${\mathbbm{g}}$ on this basis.

  * •

As in Example 9.1.8, let
$\sigma_{1}\colon\mathbb{H}^{2}\hookrightarrow\mathbb{H}^{3}$ be the totally
geodesic inclusion passing by $x$ and with normal vector $v$. Then
$\zeta_{\sigma_{1}}(x)=\ell$ and

$d\zeta_{\sigma_{1}}(T_{x}\mathbb{H}^{2})=Span_{{\mathbb{R}}}((w_{1},0),(w_{2},0)).$

As a result, by Proposition 9.2.1, the metrics ${\mathbbm{g}}$ and
$Re\langle\cdot,\cdot\rangle$ coincide on
$Span_{{\mathbb{R}}}(d\mathrm{p}(w_{1},0),d\mathrm{p}(w_{2},0))$, hence
$(w_{1},0)$ and $(w_{2},0)$ have norm $1$ and are orthogonal for both metrics.

  * •

As in Example 9.1.9, there exists an embedded sphere $\sigma_{2}\colon
S^{2}\hookrightarrow\mathbb{H}^{3}$ passing by $x$ and with $\ell$ as normal
geodesic at $x$ such that

$d\zeta_{\sigma_{2}}(T_{x}S^{2})=Span_{{\mathbb{R}}}((0,w_{1}),(0,w_{2})),$

hence, similarly as above, $d\mathrm{p}(0,w_{1})$ and $d\mathrm{p}(0,w_{2})$
have norm $-1$ and are orthogonal for both metrics.

  * •

Using Example 9.1.10, one can take an embedded cylinder $\sigma_{3}\colon
S^{1}\times{\mathbb{R}}\hookrightarrow\mathbb{H}^{3}$ passing by $x$ with
normal geodesic $\ell$ such that

$d\zeta_{\sigma_{3}}(T_{x}(S^{1}\times{\mathbb{R}}))=Span_{{\mathbb{R}}}((w_{1},0),(0,w_{2})).$

Hence $d\mathrm{p}(w_{1},0)$ and $d\mathrm{p}(0,w_{2})$ are orthogonal for
$Re\langle\cdot,\cdot\rangle$ too, and similarly for $(w_{2},0)$ and
$(0,w_{1})$.

  * •

We are left to prove that $Re\langle
d\mathrm{p}(w_{k},0),d\mathrm{p}(0,w_{k})\rangle_{\mathbb{G}}=0$, with
$k=1,2$. By Example 9.1.11, the horosphere $\sigma_{4}\colon
H\hookrightarrow\mathbb{H}^{3}$ passing by $x$ and orthogonal to $v$ can be
oriented so that

$d\zeta_{\sigma_{4}}(T_{x}H)=Span_{\mathbb{R}}((w_{1},w_{1}),(w_{2},w_{2})).$

Hence, clearly $d\zeta_{\sigma_{4}}(T_{x}H)$ is isotropic for
${\overline{Sas}_{3}}$, so $dG_{\sigma_{4}}(T_{x}H)\subset
T_{\ell}\mathbb{G}_{3}$ is isotropic for both ${\mathbbm{g}}$ and
$Re{\langle\cdot,\cdot\rangle}$. As a result, for $k=1,2$,

$\displaystyle 0$ $\displaystyle=Re\langle
d\mathrm{p}(w_{k},w_{k}),d\mathrm{p}(w_{k},w_{k})\rangle=$
$\displaystyle=Re\langle
d\mathrm{p}(w_{k},0),d\mathrm{p}(w_{k},0)\rangle+Re\langle
d\mathrm{p}(0,w_{k}),d\mathrm{p}(0,w_{k})\rangle+2Re\langle
d\mathrm{p}(w_{k},0),d\mathrm{p}(0,w_{k})\rangle=$ $\displaystyle=2Re\langle
d\mathrm{p}(w_{k},0),d\mathrm{p}(0,w_{k})\rangle,$

and the proof follows.

∎

####  9.3 Immersions with small principal curvatures

In this section we define and study the properties of immersed hypersurfaces
in $\mathbb{H}^{n+1}$ with small principal curvatures and their Gauss map.

#####  9.3.1 Extrinsic geometry of hypersurfaces

Let us start by defining our condition of small principal curvatures. Recall
that the principal curvatures of an immersion of a hypersurface in a
Riemannian manifold (in our case the ambient manifold is $\mathbb{H}^{n+1}$)
are the eigenvalues of the shape operator, which was defined in (9.1).

######  Definition 9.3.1.

An immersion $\sigma\colon M^{n}\to\mathbb{H}^{n+1}$ has _small principal
curvatures_ if its principal curvatures at every point lie in
$(-1,1)\subset{\mathbb{R}}$.

As a consequence of Equation (9.4), we have a direct characterization of
immersions with small principal curvatures in terms of their Gauss map:

######  Proposition 9.3.2.

Given an oriented manifold $M^{n}$ and an immersion
$\sigma:M\to\mathbb{H}^{n+1}$, $\sigma$ has small principal curvatures if and
only if its Gauss map $G_{\sigma}$ is a Riemannian immersion.

We recall that an immersion into a pseudo-Riemannian manifold is _Riemannian_
if the pull-back of the ambient pseudo-Riemannian metric, which in our case is
the first fundamental form
$\overline{\mathrm{I}}=G_{\sigma}^{*}{\mathbbm{g}}$, is a Riemannian metric.

###### Proof of Proposition 9.3.2.

The condition that the Gauss map is a Riemannian immersion is equivalent to
${\overline{\mathrm{I}}}(W,W)>0$ for every $W\neq 0$. By Equation (9.4), this
is equivalent to $\|B(W)\|^{2}<\|W\|^{2}$ for the norm on $M$ induced by
$\mathrm{I}$, and this is equivalent to the eigenvalues of $B$ (that is, the
principal curvatures) being strictly smaller than 1 in absolute value. ∎

######  Remark 9.3.3.

Let us observe that a consequence of the hypothesis of small principal
curvatures is that the first fundamental form of $\sigma$ has negative
sectional curvature. Indeed, if $V,W$ is a pair of orthonormal vectors on
$T_{p}M$, then by the Gauss’ equation the sectional curvature of the plane
spanned by $V$ and $W$ is:

$K_{\mathrm{Span}(V,W)}=-1+\mathrm{I}\mathrm{I}(V,V)\mathrm{I}\mathrm{I}(W,W)-\mathrm{I}\mathrm{I}(V,W)^{2}~{}.$

Since the principal curvatures of $\sigma$ are less than one in absolute
value, we have $\|B(V)\|<\|V\|$ and the same for $W$. Moreover $V$ and $W$ are
unit vectors, hence both $|\mathrm{I}\mathrm{I}(V,V)|$ and
$|\mathrm{I}\mathrm{I}(W,W)|$ are less than one and the sectional curvature is
negative.

Recall that we introduced in Definition 9.1.4 the normal evolution
$\sigma_{t}$ of an immersion $\sigma:M\to\mathbb{H}^{n+1}$, for $M$ an
oriented $n$-manifold. An immediate consequence of Proposition 9.3.2 is the
following:

######  Corollary 9.3.4.

Given an oriented manifold $M^{n}$ and an immersion
$\sigma:M\to\mathbb{H}^{n+1}$ with small principal curvatures, for all
$t\in{\mathbb{R}}$ the normal evolution $\sigma_{t}$ is an immersion with
small principal curvatures.

###### Proof.

It follows from Equation (9.6) that $\sigma_{t}$ is an immersion if the shape
operator $B$ of $\sigma$ satisfies $\|B(W)\|^{2}<\|W\|^{2}$ for every $W\neq
0$, that is, if $\sigma$ has small principal curvatures. Since the Gauss map
is invariant under the normal evolution by Corollary 9.1.6, $\sigma_{t}$ has
small principal curvatures for all $t$ as a consequence of Proposition 9.3.2.
∎

It will be useful to describe more precisely, under the hypothesis of small
principal curvatures, the behaviour of the principal curvatures under the
normal evolution.

######  Lemma 9.3.5.

Let $M^{n}$ be an oriented manifold and let $\sigma:M\to\mathbb{H}^{n+1}$ be
an immersion. If $\sigma_{t}$ is an immersion, the principal directions (i.e.
the eigenvectors of the shape operator) of $\sigma_{t}$ are the same as those
of $\sigma$ with principal curvatures

$\lambda_{i;t}=\frac{\lambda_{i}-\tanh(t)}{1-\tanh(t)\lambda_{i}}~{}.$ (9.8)

where $\lambda_{1}(p),\ldots,\lambda_{n}(p)$ denote the principal curvatures
of $\sigma$ at $p$.

As a consequence, if $\sigma$ has small principal curvatures, the function

$\kappa_{\sigma}(p):=\frac{1}{n}\sum_{i=1}^{n}\mathrm{arctanh}(\lambda_{i}(p))~{},$
(9.9)

satisfies

$\kappa_{\sigma_{t}}=\kappa_{\sigma}-t$

for every $t\in{\mathbb{R}}$.

###### Proof.

We showed in the proof of Proposition 9.1.5 that in the hyperboloid model of
$\mathbb{H}^{n+1}$ the normal vector of $\sigma_{t}$, compatible with the
orientations of $M$ and $\mathbb{H}^{n+1}$, has the expression

$\nu_{t}(p)=\sinh(t)\sigma(p)+\cosh(t)\nu(p)~{},$

where $\nu=\nu_{0}$ is the normal vector for $\sigma=\sigma_{0}$. Using also
Equation (9.6), the shape operator $B_{t}$ of $\sigma_{t}$, whose defining
condition is $d\sigma_{t}\circ B_{t}(W)=-D_{W}\nu_{t}$ as in Equation (9.1),
is:

$B_{t}=(\mathrm{id}-\tanh(t)B)^{-1}\circ(B-\tanh(t)\mathrm{id})~{}.$ (9.10)

First, Equation (9.10) shows that if $V$ is a principal direction (i.e. an
eigenvalue of $B$) for $\sigma$, then it is also for $\sigma_{t}$. Second, if
$\lambda_{i}$ is a principal curvature of $\sigma$, then the corresponding
principal curvature for $\sigma_{t}$ is

$\lambda_{i;t}=\frac{\lambda_{i}-\tanh(t)}{1-\tanh(t)\lambda_{i}}=\tanh(\mu_{i}-t)~{},$
(9.11)

where $\mu_{i}=\mathrm{arctanh}(\lambda_{i})$.

Observe that if the $\lambda_{i}^{\prime}s$ are lower than 1 in absolute
value, then $\mathrm{arctanh}{\lambda_{i}}$ is well-posed and one has

$\lambda_{i;t}=\tanh(\mathrm{arctanh}(\lambda_{i})-t).$

The formula $\kappa_{\sigma_{t}}=\kappa_{\sigma}-t$ follows. ∎

######  Remark 9.3.6.

Assume $\sigma$ has small principal curvatures. Although the principal
curvatures of $\sigma$ are not smooth functions, the function
$\kappa_{\sigma}$ defined in (9.9) is smooth as long as $\sigma$ has small
principal curvatures. Indeed, using the expression of the inverse hyperbolic
tangent in terms of the elementary functions, we may express:

$\displaystyle\kappa_{\sigma}(p)$
$\displaystyle=\frac{1}{2n}\left(\sum_{i=1}^{n}\log\left(\frac{1+\lambda_{i}(p)}{1-\lambda_{i}(p)}\right)\right)=\frac{1}{2n}\log\left(\frac{\prod_{i=1}^{n}(1+\lambda_{i}(p))}{\prod_{i=1}^{n}(1-\lambda_{i}(p))}\right)=$
$\displaystyle=\frac{1}{2n}\log\left(\frac{\det(\mathrm{id}+B)}{\det(\mathrm{id}-B)}\right)~{},$

where $B$ is the shape operator of $\sigma$ as usual. This proves the
smoothness of $\kappa_{\sigma}$, which will be implicitly used in Proposition
10.5.7.

#####  9.3.2 Comparison horospheres

Our next goal is to discuss global injectivity of immersions with small
principal curvatures (Proposition 9.3.15) and of their Gauss maps (Proposition
9.3.16), under the following completeness assumption.

######  Definition 9.3.7.

An immersion $\sigma\colon M^{n}\to\mathbb{H}^{n+1}$ is _complete_ if the
first fundamental form $\mathrm{I}$ is a complete Riemannian metric.

Here we provide some preliminary steps.

######  Definition 9.3.8.

Given an oriented manifold $M^{n}$ and an immersion
$\sigma:M\to\mathbb{H}^{n+1}$, let $B=-D\nu$ be its shape operator with
respect to the unit normal vector field $\nu$ compatible with the orientations
of $M$ and $\mathbb{H}^{n+1}$. We say that $\sigma$ is (_strictly_) _convex_
if $B$ is negative semi-definite (resp. definite), and, conversely, that it is
(_strictly_) _concave_ if $B$ is positive semi-definite (resp. definite).

When $\sigma$ is an embedding, we refer to its image as a (strictly)
convex/concave hypersurface. Clearly reversing the orientation (and therefore
the normal vector field) of a (strictly) convex hypersurface it becomes
(strictly) concave, and viceversa. An example of convex hypersurface

A classical fact is that a properly embedded strictly convex hyperurface in
$\mathbb{H}^{n+1}$ disconnects it into two connected components and that
exactly one of them is geodesically convex (the one towards which $-\nu$ is
pointing): we denote the closure of this connected component as the _convex
side_ of the hypersurface, and denote as the _concave side_ the closure of the
other one.

We need another definition before stating the next Lemma. We say that a smooth
curve $\gamma:[a,b]\to\mathbb{H}^{n}$ parameterized by arclength has _small
acceleration_ if $\|D_{\gamma^{\prime}(t)}\gamma^{\prime}(t)\|<1$ for all $t$,
where $D$ denotes the Levi-Civita connection of $\mathbb{H}^{n}$ as usual.

######  Lemma 9.3.9.

Let $\gamma:[a,b]\to\mathbb{H}^{n}$ be a smooth curve of small acceleration.
Then the image of $\gamma$ lies on the concave side of any horosphere tangent
to $\gamma$. More precisely, $\gamma$ lies in the interior of the concave side
except for the tangency point.

###### Proof.

Up to reparametrization we can assume that the tangency point is $\gamma(0)$,
and we shall prove that $\gamma(t)$ lies on the concave side of any horosphere
tangent to $\gamma^{\prime}(0)$ for every $t>0$. Recall that we are also
assuming that $\gamma$ is parameterized by arclength. We will use the upper
half-space model of $\mathbb{H}^{n}$, namely, $\mathbb{H}^{n}$ is the region
$\\{x_{n}>0\\}$ in ${\mathbb{R}}^{n}$ endowed with the metric
$(\frac{1}{x_{n}^{2}})(dx_{1}^{2}+\ldots+dx_{n}^{2})$. Up to isometry, we can
assume that $\gamma(0)=(0,\ldots,0,1)$, $\gamma^{\prime}(0)=(1,0,\ldots,0)$
and that the tangent horosphere is $\\{x_{n}=1\\}$.

Let us first show that $\gamma$ lies on the concave side of the horosphere for
small $t$, namely, denoting $\gamma(t)=(\gamma_{1}(t),\ldots,\gamma_{n}(t))$,
that $\gamma_{n}(t)<1$ for small $t$. Since $\gamma_{n}(0)=1$ and
$\gamma_{n}^{\prime}(0)=0$, it will be sufficient to check that
$\gamma_{n}^{\prime\prime}(0)<0$. Using the assumption on $\gamma^{\prime}(0)$
and a direct computation of the Christoffel symbols $\Gamma_{11}^{n}=1$, we
get

$(D_{\gamma^{\prime}}\gamma^{\prime})_{n}(0)=\gamma_{n}^{\prime\prime}(0)+1~{}.$

Since by hypothesis $\gamma$ has small acceleration, and at $\gamma(0)$ the
metric of the upper half-space model coincides with the standard metric
$dx_{1}^{2}+\ldots+dx_{n}^{2}$,
$|(D_{\gamma^{\prime}}\gamma^{\prime})_{n}(0)|<1$ and therefore
$\gamma_{n}^{\prime\prime}(0)<0$. We conclude that, for suitable $\epsilon>0$,
$\gamma_{n}(t)<1$ for all $t\in(-\epsilon,\epsilon)\setminus\\{0\\}$.

Let us now show that $\gamma(t)$ lies in the interior of the concave side of
the tangent horosphere $\\{x_{n}=1\\}$ for all $t\neq 0$, that is, that
$\gamma_{n}(t)<1$ for all $t\neq 0$. Suppose by contradiction that
$\gamma_{n}(t_{0})=1$ for some $t_{0}\geq\epsilon$. Then $\gamma_{n}$ has a
minimum point $t_{\min}$ in $(0,t_{0})$, with minimum value $m<1$. The
horosphere $\\{x_{n}=m\\}$ is then tangent to $\gamma$ at $\gamma(t_{\min})$
and $\gamma_{n}(t)\geq m$ for $t$ in a neighbourhood of $t_{\min}$. By re-
applying the argument of the previous part of the proof, this gives a
contradiction. See Figure 9.2. ∎

Figure 9.2: A schematic picture of the argument in the proof of Lemma 9.3.9.
On the left, for $t\in(-\epsilon,\epsilon)$ the image of the curve $\gamma(t)$
lies in the concave side of the horosphere tangent to $\gamma$ at $t=0$. On
the right, the same holds in fact for every $t$, for otherwise one would
obtain a contradiction with the first part of the proof at the minimum point
$t_{\min}$.

######  Remark 9.3.10.

Given an immersion $\sigma:M^{n}\to\mathbb{H}^{n+1}$ (or in general into any
Riemannian manifold), a curve $\gamma:[a,b]\to M$ is a geodesic for the first
fundamental form of $\sigma$ (in short, it is an _intrinsic_ geodesic) if and
only if $D_{(\sigma\circ\gamma)^{\prime}}(\sigma\circ\gamma)^{\prime}$ is
orthogonal to the image of $\sigma$. In this case we have indeed

$D_{(\sigma\circ\gamma)^{\prime}}(\sigma\circ\gamma)^{\prime}=\mathrm{I}\mathrm{I}(\gamma^{\prime}(t),\gamma^{\prime}(t))\nu(\gamma(t))$
(9.12)

where $\nu$ is the unit normal vector of the immersion with respect to the
chosen orientations.

By applying this remark to an intrinsic geodesic for the horosphere
$\\{x_{n}=1\\}$, which has the form $\gamma(t)=(a_{1}t,\ldots,a_{n-1}t,1)$
(here $\sigma$ is simply the inclusion), and repeating the same computation of
the proof of Lemma 9.3.9, we see that the second fundamental form of a
horosphere equals the first fundamental form. Hence the principal curvatures
of a horosphere are all identically equal to $1$ for the choice of inward
normal vector, and therefore the shape operator is the identity at every
point, a fact we have already used in Example 9.1.11.

An immediate consequence of Lemma 9.3.9 is the following:

######  Lemma 9.3.11.

Given a complete immersion $\sigma:M^{n}\to\mathbb{H}^{n+1}$ with small
principal curvatures, the image of $\sigma$ lies strictly on the concave side
of any tangent horosphere. That is, for every $p\in M$,
$\sigma(M\setminus\\{p\\})$ lies in the interior of the concave side of each
of the two horospheres tangent to $\sigma$ at $\sigma(p)$.

###### Proof.

Let us fix $p\in M$ and let $q\in M$, with $p\neq q$. By completeness there
exists an intrinsic geodesic $\gamma$ on $M$ joining $p$ and $q$, which we
assume to be parameterized by arclength. Applying Equation (9.12) as in Remark
9.3.10, we have

$\|D_{(\sigma\circ\gamma)^{\prime}}(\sigma\circ\gamma)^{\prime}\|=|\mathrm{I}\mathrm{I}(\gamma^{\prime}(t),\gamma^{\prime}(t))|<\mathrm{I}(\gamma^{\prime}(t),\gamma^{\prime}(t))=\|(\sigma\circ\gamma)^{\prime}(t)\|^{2}=1~{},$

hence $\sigma\circ\gamma$ has small acceleration. The conclusion follows from
Lemma 9.3.9. ∎

######  Remark 9.3.12.

Observe that any metric sphere in $\mathbb{H}^{n+1}$ is contained in the
convex side of any tangent horosphere. As a result, a hypersurface with small
principal curvatures lies in the complementary of any metric ball of
$\mathbb{H}^{n+1}$ whose boundary is tangent to the hypersurface. See Figure
9.3.

######  Remark 9.3.13.

A $r$-_cap_ in the hyperbolic space is the hypersurface at (signed) distance
$r$ from a totally geodesic plane. By a simple computation (for instance using
Equation (9.10)), $r$-caps are umbilical hypersurfaces with principal
curvatures identically equal to $-\tanh(r)$, computed with respect to the unit
normal vector pointing to the side where $r$ is increasing. Now, if
$\sigma:M\to\mathbb{H}^{n+1}$ is an immersion with principal curvatures
smaller that $\epsilon=\tanh(r){\in(0,1)}$ in absolute value, then one can
repeat wordly the proofs of Lemma 9.3.9 and Lemma 9.3.11, by replacing
horospheres with $r$-caps, and conclude that the image of $\sigma$ lies
strictly on the concave side of every tangent $r$-cap for
$r=\mathrm{arctanh}(\epsilon)$. See Figure 9.3. A similar conclusion (which is
however not interesting for the purpose of this paper) could of course be
obtained under the assumption that $\sigma$ has principal curvatures bounded
by some constant $\epsilon>1$, in terms of tangent metric spheres with
curvature greater than $\epsilon$ in absolute value.

Figure 9.3: Schematically, an immersion $\sigma$ tangent at one point to a
metric sphere (whose principal curvatures are larger than $1$), a horosphere
(equal to $1$) and a $r$-cap (smaller than $1$). The image of $\sigma$ is
contained in the concave side of the three of them.

#####  9.3.3 Injectivity results

Having established these preliminary results, let us finally discuss the
global injectivity of $\sigma$ and $G_{\sigma}$ under the hypothesis of
completeness. Before that, we relate the completeness assumption for $\sigma$
to some topological conditions.

######  Remark 9.3.14.

Let us observe that proper immersions $\sigma\colon M\to\mathbb{H}^{n+1}$ are
complete. Indeed, if $p,q\in M$ have distance at most $r$ for the first
fundamental form $\mathrm{I}$, then, by definition of distance on a Riemannian
manifold, $dist_{\mathbb{H}^{n+1}}(\sigma(p),\sigma(q))\leq r$: as a result,

$\sigma(B_{\mathrm{I}}(x,r))\subset B_{\mathbb{H}^{n+1}}(x,r).$

Assuming $\sigma$ is proper,
$\sigma^{-1}(\overline{B_{\mathbb{H}^{n+1}}(x,r)})$ is a compact subspace of
$M$ containing $B_{\mathrm{I}}(x,r)$, therefore
$\overline{B_{\mathrm{I}}(x,r)}$ is compact. We conclude that $\mathrm{I}$ is
complete by Hopf-Rinow Theorem.

A less trivial result is that Remark 9.3.14 can be reversed for immersions
with small principal curvatures: in fact, for immersions with small principal
curvatures, being properly immersed, properly embedded and complete are all
equivalent conditions

######  Proposition 9.3.15.

Let $M^{n}$ be a manifold and $\sigma:M\to\mathbb{H}^{n+1}$ be a complete
immersion with small principal curvatures. Then $\sigma$ is a proper embedding
and $M$ is diffeomorphic to ${\mathbb{R}}^{n}$.

###### Proof.

To show that $\sigma$ is injective, let us suppose by contradiction that
$\sigma(p)=\sigma(q)=y_{0}$ for $p\neq q$. Let $\gamma:[a,b]\to M$ be an
intrinsic $\mathrm{I}$-geodesic joining $p$ and $q$ parametrized by arclength,
which exists because $\mathrm{I}$ is complete. As in Lemma 9.3.11,
$\sigma\circ\gamma$ has small acceleration. Let

$r_{0}:=\max_{t\in[a,b]}d_{\mathbb{H}^{n+1}}(y_{0},\sigma\circ\gamma(t))~{}.$

Then $\sigma\circ\gamma$ is tangent at some point $\sigma\circ\gamma(t_{0})$
to the metric sphere in $\mathbb{H}^{n+1}$ centered at $y_{0}$ of radius
$r_{0}$, and contained in its convex side. By Remark 9.3.12,
$\sigma\circ\gamma$ lies in the convex side of the horosphere tangent to the
hypersurface at $\sigma\circ\gamma(t_{0})$. This contradicts Lemma 9.3.9 and
shows that $\sigma$ is an injective immersion. See Figure 9.4.

Figure 9.4: A sketch of the proof of the first part of Proposition 9.3.15,
namely the injectivity of $\sigma$. If $\sigma(p)=\sigma(q)=y_{0}$ for $p\neq
q$, then the image $\sigma\circ\gamma$ of a $\mathrm{I}$-geodesic connecting
$p$ and $q$ would be tangent to a metric ball centered at $y_{0}$, which
contradicts the assumption that $\sigma$ has small principal curvatures.

It follows that $M$ is simply connected. Indeed, let $u:\widetilde{M}\to M$ be
a universal covering. If $M$ were not simply connected, then $u$ would not be
injective, hence $\sigma\circ u$ would give a non-injective immersion in
$\mathbb{H}^{n+1}$ with small principal curvatures, contradicting the above
part of the proof. Since the first fundamental form is a complete negatively
curved Riemannian metric on $M$ (Remark 9.3.3), $M$ is diffeomorphic to
${\mathbb{R}}^{n}$ by the Cartan-Hadamard Theorem.

Let us now show that $\sigma$ is proper, which also implies that it is a
homeomorphism onto its image and thus an embedding. As a first step, suppose
$y_{0}\in\mathbb{H}^{n+1}$ is in the closure of the image of $\sigma$. We
claim that the normal direction of $\sigma$ extends to $y_{0}$, meaning that
there exists a vector $\nu_{0}\in T^{1}_{x_{0}}\mathbb{H}^{n+1}$ such that
$[\nu(p_{n})]\to[\nu_{0}]$ for every sequence $p_{n}\in M$ satisfying
$\sigma(p_{n})\to y_{0}$, where $\nu(p)$ denotes the unit normal vector of
$\sigma$ at $p$ and $[\cdot]$ denotes the equivalence class up to
multiplication by $\pm 1$. By compactness of unit tangent spheres, if
$\sigma(p_{n})\to y_{0}$ then one can extract a subsequence $\nu(p_{n})$
converging to (say) $\nu_{0}$. Observe that by Lemma 9.3.11, the image of
$\sigma$ lies in the concave side of any horosphere orthogonal to $\nu(p_{n})$
at $\sigma(p_{n})$. By a continuity argument, it lies also on the concave side
of each of the two horospheres orthogonal to $\nu_{0}$ at $y_{0}$. The claim
follows by a standard subsequence argument once we show that there can be no
limit other than $\pm\nu_{0}$ along any subsequence.

We will assume hereafter, in the upper half-space model

$(\\{x_{n+1}>0\\},\frac{1}{x_{n+1}^{2}}(dx_{1}^{2}+\ldots+dx_{n+1}^{2}))~{},$

that $y_{0}=(0,\ldots,0,1)$ and $\nu_{0}=(0,\ldots,0,1)$. See Figure 9.5 on
the left. In this model, horospheres are either horizontal hyperplanes
$\\{x_{n+1}=c\\}$ or spheres with south pole on $\\{x_{n+1}=0\\}$. By Lemma
9.3.11, the image of $\sigma$ is contained in the concave side of both
horospheres orthogonal to $\nu_{0}$, hence it lies in the region defined by
$0<x_{n+1}\leq 1$ and
$x_{1}^{2}+\ldots+x_{n}^{2}+(x_{n+1}-\frac{1}{2})^{2}\geq\frac{1}{4}$. Now, if
$\nu_{1}\neq\pm\nu_{0}$ were a subsequential limit of $\nu(q_{n})$ for some
sequence $q_{n}$ with $\sigma(q_{n})\to y_{0}$, then the image of $\sigma$
would lie on the concave side of some sphere with south pole on
$\\{x_{n+1}=0,(x_{1},\ldots,x_{n})\neq(0,\ldots,0)\\}$. But then $\sigma$
would either enter the region $x_{n+1}>1$ or the region
$x_{1}^{2}+\ldots+x_{n}^{2}+(x_{n+1}-\frac{1}{2})^{2}<\frac{1}{4}$ in a
neighbourhood of $y_{0}$, which gives a contradiction.

Having established the convergence of the normal direction to $[\nu_{0}]$, we
can now find a neighbourhood $U$ of $y_{0}$ of the form
$B(0,\epsilon)\times(\frac{1}{2},\frac{3}{2})$, where $B(0,\epsilon)$ is the
ball of Euclidean radius $\epsilon$ centered at the origin in
$\\{x_{n+1}=0\\}$, such that if $\sigma(p)\in U$, then the vertical projection
from the tangent space of $\sigma$ at $\sigma(p)$ to $\\{x_{n+1}=0\\}$ is a
linear isomorphism. By the implicit function theorem, $\sigma(M)\cap U$ is
locally a graph over ${\mathbb{R}}^{n}$. Up to taking a smaller $\epsilon$, we
can arrange $U$ so that $\sigma(M)\cap U$ is a global graph over some open set
of $B(0,\epsilon)\subset{\mathbb{R}}^{n}$. Indeed as long as the normal vector
$\nu$ is in a small neighbourhood of $\pm\nu_{0}$, the vertical lines over
points in $B(0,\epsilon)$ may intersect the image of $\sigma$ in at most one
point as a consequence of Lemma 9.3.11. Let us denote $V\subseteq
B(0,\epsilon)$ the image of the vertical projection from $\sigma(M)\cap U$ to
${\mathbb{R}}^{n}$, so that $\sigma(M)\cap U$ is the graph of some function
$h:V\to(\frac{1}{2},\frac{3}{2})$ satisfying $h(0)=1$. Since the gradient of
$h$ converges to $0$ at $0$, up to restricting $U$ again, there is a constant
$C>0$ such that the Euclidean norm of the gradient of $h$ is bounded by $C$.

We shall now apply again the hypothesis that $\sigma$ is complete to show that
in fact $V=B(0,\epsilon)$. For this purpose, we assume that $V$ is a proper
(open) subset of $B(0,\epsilon)$ and we will derive a contradiction. Under the
assumption $V\neq B(0,\epsilon)$ we would find a Euclidean segment
$c:[0,1]\to{\mathbb{R}}^{n}$ such that $c(s)\in V$ for $s\in[0,1)$ and
$c(1)\in B(0,\epsilon)\setminus V$. The path $s\mapsto(c(s),h\circ c(s))$ is
contained in $\sigma(M)$; using $h\geq\frac{1}{2}$ and the bound on the
gradient, we obtain that its hyperbolic length is less than $2\sqrt{1+C^{2}}$
times the Euclidean length of $c$, hence is finite. This contradicts
completeness of $\sigma$. In summary, $\sigma(M)\cap U$ is the graph of a
function globally defined on $B(0,\epsilon)$, and clearly contains the point
$y_{0}$. See Figure 9.5 on the right.

We are now ready to complete the proof of the fact that $\sigma$ is proper.
Indeed, let $p_{n}\in M$ be a sequence such that $\sigma(p_{n})\to y_{0}$. We
showed above that $y_{0}$ is in the image of $\sigma$ (say
$y_{0}=\sigma(p_{0})$) and that $p_{n}$ is definitively in $\sigma^{-1}(U)$,
whose image is a graph over $B(0,\epsilon)$. Hence $p_{n}$ is at bounded
distance from $p_{0}$ for the first fundamental form of $\sigma$, and
therefore admits a subsequence $p_{n_{k}}$ converging to $p_{0}$. In
conclusion, $\sigma$ is a proper embedding. ∎

Figure 9.5: The setting of the proof that $\sigma$ is proper in Proposition
9.3.15: in the upper half-plane model, the image of $\sigma$ is contained
below the horosphere $\\{x_{n+1}=1\\}$ and in the outer side of the horosphere
$x_{1}^{2}+\ldots+x_{n}^{2}+(x_{n+1}-\frac{1}{2})^{2}=\frac{1}{4}$. On the
right, the neighbourhood $U$ of $y_{0}$, where the image of $\sigma$ is proved
to be the graph of a function $h:B(0,\epsilon)\to{\mathbb{R}}$.

By an application of Lemma 9.3.11 one can easily show that the Gauss map
$G_{\sigma}:M\to\mathbb{G}_{n+1}$ is injective as well if $\sigma$ is complete
and has small principal curvatures. However, we will prove here (Proposition
9.3.16) a stronger property of the Gauss map. Recall that the space of
oriented geodesics of $\mathbb{H}^{n+1}$ has the natural identification

$\mathbb{G}_{n+1}\cong\partial\mathbb{H}^{n+1}\times\partial\mathbb{H}^{n+1}\setminus\Delta~{},$

for $\Delta$ the diagonal, given by mapping an oriented geodesic $\ell$ to its
endpoints at infinity according to the orientation: as a consequence, the map
$G_{\sigma}$ can be seen as a pair of maps with values in the boundary of
$\mathbb{H}^{n}$. More precisely, if we denote by
$\gamma:{\mathbb{R}}\to\mathbb{H}^{n+1}$ a parameterized geodesic, then the
above identification reads:

$\gamma\mapsto\left(\lim_{t\to+\infty}\gamma(t),\lim_{t\to-\infty}\gamma(t)\right)~{}.$
(9.13)

Given an immersion of an oriented manifold $M^{n}$ into $\mathbb{H}^{n+1}$,
composing the Gauss map $G_{\sigma}:M\to\mathbb{G}_{n+1}$ with the above map
(9.13) with values in $\partial\mathbb{H}^{n+1}\times\partial\mathbb{H}^{n+1}$
and projecting on each factor, we obtain the so-called _hyperbolic Gauss maps_
$G^{\pm}_{\sigma}:M\to\partial\mathbb{H}^{n+1}$. They are explicitely
expressed by

$G^{\pm}_{\sigma}(p)=\lim_{t\to\pm\infty}\exp_{\sigma(p)}(t\nu(p))\in\partial\mathbb{H}^{n+1}~{}.$

The following proposition states their injectivity property under the small
principal curvatures assumption, which will be applied in Proposition 9.4.4,

######  Proposition 9.3.16.

Let $M^{n}$ be an oriented manifold and $\sigma:M\to\mathbb{H}^{n+1}$ be a
complete immersion with small principal curvatures. Then both hyperbolic Gauss
maps $G^{\pm}_{\sigma}:M\to\partial\mathbb{H}^{n+1}$ are diffeomorphisms onto
their images. In particular, the Gauss map $G_{\sigma}$ is an embedding.

###### Proof.

Let us first show that $G_{\sigma}^{\pm}$ are local diffeomorphisms. Recalling
the definition of $\sigma_{t}$ (Definition 9.1.4) and its expression in the
hyperboloid model of $\mathbb{H}^{n+1}$ (Equation (9.5)), $G_{\sigma}^{\pm}$
is the limit for $t\to\pm\infty$ in $\partial\mathbb{H}^{n+1}$ of

$\sigma_{t}(p)=\cosh(t)\sigma(p)\pm\sinh(t)\nu(p)~{}.$

Recalling the definition of the boundary of $\mathbb{H}^{n+1}$ as the
projectivization of the null-cone (Equation (5.1)), we will consider the
boundary at infinity of $\mathbb{H}^{n+1}$ as the slice of the null-cone
defined by $\\{x_{n+2}=1\\}$. Given $p_{0}\in M$, up to isometries we can
assume that $\sigma(p_{0})=(0,\ldots,0,1)$ and $\nu(p_{0})=(1,0,\ldots,0)$, so
that $G_{\sigma}^{\pm}(p_{0})=(\pm 1,0,\ldots,0,1)$ and the tangent spaces to
the image of $\sigma$ at $p_{0}$ and of $\partial\mathbb{H}^{n+1}$ at
$G_{\sigma}^{\pm}(p)$ are identified to the same subspace
$\\{x_{1}=x_{n+2}=0\\}$ in ${\mathbb{R}}^{n,1}$.

To compute the differential of $G_{\sigma}^{\pm}$ at $p_{0}$, we must
differentiate the maps

$p\mapsto\lim_{t\to\pm\infty}\frac{\sigma_{t}(p)}{|\langle\sigma_{t}(p),\sigma(p_{0})\rangle|}=\frac{\sigma(p)\pm\nu(p)}{|\langle\sigma(p)\pm\nu(p),\sigma(p_{0})\rangle|}$

at $p=p_{0}$. Under these identifications, a direct computation for $V\in
T_{p_{0}}M$ gives:

$dG_{\sigma}^{\pm}(V)=d\sigma\circ(\mathrm{id}\mp B)(V)~{}.$

Hence both differentials of $G_{\sigma}^{\pm}$ are invertible at $p_{0}$ if
the eigenvalues of $B$ are always different from $1$ and $-1$, as in our
hypothesis. This shows that $G_{\sigma}^{+}$ and $G_{\sigma}^{-}$ are local
diffeomorphisms.

To see that $G_{\sigma}^{\pm}$ is injective, suppose that
$G_{\sigma}^{\pm}(p)=G_{\sigma}^{\pm}(q)$. This means that $\sigma$ is
orthogonal at $p$ and $q$ to two geodesics having a common point at infinity.
Hence $\sigma$ is tangent at $p$ and $q$ to two horospheres $H_{p}$ and
$H_{q}$ having the same point at infinity. By Lemma 9.3.11 the image of
$\sigma$ must lie in the concave side of both $H_{p}$ and $H_{q}$, hence the
two horospheres must coincide. But by Lemma 9.3.11 again,
$\sigma(M\setminus\\{p\\})$ lies strictly in the concave side of $H_{p}$,
hence necessarily $p=q$. See Figure 9.6.

By the invariance of the domain, $G_{\sigma}^{\pm}$ are diffeomorphisms onto
their images. Under the identification between $\mathbb{G}_{n+1}$ and
$\partial\mathbb{H}^{n+1}\times\partial\mathbb{H}^{n+1}\setminus\Delta$ the
Gauss map $G_{\sigma}$ corresponds to $(G_{\sigma}^{+},G_{\sigma}^{-})$, and
it follows that $G_{\sigma}$ is an embedding. ∎

Figure 9.6: The proof of the injectivity of $G_{\sigma}^{+}$. Suppose two
orthogonal geodesics share the final point, hence the image of $\sigma$ is
tangent at $\sigma(p)$ and $\sigma(q)$ have the same point at infinity. As a
consequence of Lemma 9.3.11, this is only possible if the two tangent
horospheres coincide, and therefore if $p=q$. Replacing horospheres by metric
spheres, the same argument proves that the orthogonal geodesics at different
points are disjoint (See Proposition 9.4.2).

####  9.4 Nearly-Fuchsian manifolds and the space ${\mathbb{G}}_{\rho}$

Taking advantage of the results so far established in this chapter, we now
introduce nearly-Fuchsian representations and manifolds. These will appear
again in Chapter 11.

######  Definition 9.4.1.

Let $M^{n}$ be a closed orientable manifold. A representation
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$ is called _nearly-
Fuchsian_ if there exists a $\rho$-equivariant immersion
$\widetilde{\sigma}:\widetilde{M}\to\mathbb{H}^{n+1}$ with small principal
curvatures.

We recall that an immersion
$\widetilde{\sigma}\colon\widetilde{M}\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$
is _$\rho$ -equivariant_ if

$\widetilde{\sigma}\circ\alpha=\rho(\alpha)\circ\widetilde{\sigma}~{}.$ (9.14)

for all $\alpha\in\pi_{1}(M)$. Let us show that the action of nearly-Fuchsian
representations is “good” on $\mathbb{H}^{n+1}$ (Proposition 9.4.2) and also
on a region in $\partial\mathbb{H}^{n+1}$ which is the disjoint union of two
topological discs (Proposition 9.4.4).

######  Proposition 9.4.2.

Let $M^{n}$ be a closed orientable manifold and
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$ be a nearly-Fuchsian
representation. Then $\rho$ gives a free and properly discontinous action of
$\pi_{1}(M)$ on $\mathbb{H}^{n+1}$. Moreover $\rho$ is convex cocompact,
namely there exists a $\rho$-invariant geodesically convex subset
$\mathcal{C}\subset\mathbb{H}^{n+1}$ such that the quotient
$\faktor{\mathcal{C}}{\rho(\pi_{1}(M))}$ is compact.

###### Proof.

Let $\widetilde{\sigma}$ be an equivariant immersion as in Definition 9.4.1.
We claim that the family of geodesics orthogonal to
$\widetilde{\sigma}(\widetilde{M})$ gives a foliation of $\mathbb{H}^{n+1}$.
Observing that the action of $\pi_{1}(M)$ on $\widetilde{M}$ is free and
properly discontinous, this immediately implies that the action of
$\pi_{1}(M)$ on $\mathbb{H}^{n+1}$ induced by $\rho$ is free and properly
discontinous.

By repeating the same argument that shows, in the proof of Proposition 9.3.16,
the injectivity of $G_{\widetilde{\sigma}}^{\pm}$, replacing horospheres with
metric spheres of $\mathbb{H}^{n+1}$ and using Remark 9.3.12, one can prove
that two geodesics orthogonal to $\widetilde{\sigma}(\widetilde{M})$ at
different points are disjoint.

To show that the orthogonal geodesics give a foliation of $\mathbb{H}^{n+1}$,
it remains to show that every point $x\in\mathbb{H}^{n+1}$ is contained in a
geodesic of this family (which is necessarily unique). Of course we can assume
$x\notin\widetilde{\sigma}(\widetilde{M})$. By cocompactness,
$\widetilde{\sigma}$ is complete, hence it is a proper embedding by
Proposition 9.3.15. Then the map that associates to each element of
$\widetilde{\sigma}(\widetilde{M})$ its distance from $x$ attains its minimum:
this implies that there exists $r>0$ such that the metric sphere of radius $r$
centered at $x$ is tangent to $\widetilde{\sigma}(\widetilde{M})$ at some
point $p$. Hence $x$ is on the geodesic through $p$. See Figure 9.8.

Let us now prove that $\rho$ is also convex-cocompact. To show this, we claim
that there exists $t_{+},t_{-}\in{\mathbb{R}}$ such that
$\widetilde{\sigma}_{t_{+}}$ is a convex embedding, and
$\widetilde{\sigma}_{t_{-}}$ a concave one. Indeed in the proof of Lemma 9.3.5
we showed that the principal curvatures of the normal evolution
$\widetilde{\sigma}_{t}$ are equal to $\tanh(\mu_{i}-t)$, where $\mu_{i}$ is
the hyperbolic arctangent of the corresponding principal curvature of
$\widetilde{\sigma}$. Hence taking $t\ll 0$ (resp. $t\gg 0$) one can make sure
that the principal curvatures of $\widetilde{\sigma}_{t}$ are all negative
(resp. positive), hence $\widetilde{\sigma}_{t}$ is convex (resp. concave).
The region bounded by the images of $\widetilde{\sigma}_{t_{+}}$ and
$\widetilde{\sigma}_{t_{-}}$ is then geodesically convex and diffeomorphic to
$\widetilde{M}\times[t_{-},t_{+}]$. Under this diffeomorphism, the action of
$\pi_{1}(M)$ corresponds to the action by deck transformations on
$\widetilde{M}$ and the trivial action on the second factor. Hence its
quotient is compact, being diffeomorphic to $M\times[t_{-},t_{+}]$. ∎

This implies that in dimension three, nearly-Fuchsian manifolds are quasi-
Fuchsian.

######  Remark 9.4.3.

There is another important consequence of Proposition 9.4.2. Given
$\widetilde{\sigma}$ an equivariant immersion as in Definition 9.4.1, it
follows from the cocompactness of the action of $\rho$ on the geodesically
convex region $\mathcal{C}$ that $\widetilde{\sigma}$ is a quasi-isometric
embedding in the sense of metric spaces. By cocompactness and Remark 9.3.3,
$\widetilde{M}$ is a complete simply connected Riemannian manifold of negative
sectional curvature, hence its visual boundary $\partial\widetilde{M}$ in the
sense of Gromov is homeomorphic to $S^{n-1}$. By [[, Proposition
6.3,]]zbMATH01496599, $\widetilde{\sigma}$ extends to a continuous injective
map $\partial\widetilde{\sigma}$ from the visual boundary
$\partial\widetilde{M}$ of $\widetilde{M}$ to $\partial\mathbb{H}^{n+1}$. By
compactness of $\partial\widetilde{M}$, the extension of $\widetilde{\sigma}$
is a homeomorphism onto its image.

Since any two $\rho$-equivariant embeddings
$\widetilde{\sigma}_{1},\widetilde{\sigma}_{2}:\widetilde{M}\to\mathbb{H}^{n+1}$
are at bounded distance from each other by cocompactness, the extension
$\partial\widetilde{\sigma}$ does not depend on $\widetilde{\sigma}$, but only
on the representation $\rho$. In conclusion, the image of
$\partial\widetilde{\sigma}$ is a topological $(n-1)$-sphere $\Lambda_{\rho}$
in $\partial\mathbb{H}^{n+1}$, called the _limit set_ of the representation
$\rho$. See Figure 9.7.

Figure 9.7: A picture of the limit set $\Lambda_{\rho}$, which is a
topological $(n-1)$-sphere and disconnects $\partial\mathbb{H}^{n+1}$ in two
connected components $\Omega_{+}$ and $\Omega_{-}$, which are homeomorphic to
$n$-discs.

######  Proposition 9.4.4.

Let $M^{n}$ be a closed orientable manifold,
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$ be a nearly-Fuchsian
representation and $\Lambda_{\rho}$ be its limit set. Then the action of
$\rho$ extends to a free and properly discontinous action on
$\partial\mathbb{H}^{n+1}\setminus\Lambda_{\rho}$, which is the disjoint union
of two topological $n$-discs.

###### Proof.

Since the action of $\pi_{1}(M)$ on $\widetilde{M}$ is free and properly
discontinuous, and $G_{\widetilde{\sigma}}^{\pm}$ are diffeomorphisms onto
their image by Proposition 9.3.16, it follows that the action of
$\rho(\pi_{1}(M))$ is free and properly discontinuous on
$G_{\widetilde{\sigma}}^{+}(\widetilde{M})$ and
$G_{\widetilde{\sigma}}^{-}(\widetilde{M})$, which are topological discs in
$\mathbb{H}^{n+1}$ since $\widetilde{M}$ is diffeomorphic to
${\mathbb{R}}^{n}$. We claim that

$G_{\widetilde{\sigma}}^{+}(\widetilde{M})\cup
G_{\widetilde{\sigma}}^{-}(\widetilde{M})=\partial\mathbb{H}^{n+1}\setminus\Lambda_{\rho}~{}.$

Observe that, by the Jordan-Brouwer separation Theorem, the complement of
$\Lambda_{\rho}$ has two connected components, hence the claim will also imply
that $G_{\widetilde{\sigma}}^{+}(\widetilde{M})$ and
$G_{\widetilde{\sigma}}^{-}(\widetilde{M})$ are disjoint because they are both
connected.

In order to show that
$\partial\mathbb{H}^{n+1}\setminus\Lambda_{\rho}\subseteq
G_{\widetilde{\sigma}}^{+}(\widetilde{M})\cup
G_{\widetilde{\sigma}}^{-}(\widetilde{M})$, one can repeat the same argument
as Proposition 9.4.2, now using horospheres, to see that every $x$ in the
complement of $\Lambda_{\rho}$ is the endpoint of some geodesic orthogonal to
${\widetilde{\sigma}}(\widetilde{M})$. See Figure 9.8.

It only remains to show the other inclusion. By continuity, it suffices to
show that every $x\in\Lambda_{\rho}$ is not on the image of
$G_{\widetilde{\sigma}}^{\pm}$. Observe that by cocompactness the principal
curvatures of $\widetilde{\sigma}$ are bounded by some constant $\epsilon<1$
in absolute value. Now, if $x\in\partial\mathbb{H}^{n+1}$ is the endpoint of
an orthogonal line $\ell$, then, for all $r$, one would be able to construct a
$r$-cap tangent to $\ell\cap\widetilde{\sigma}(\widetilde{M})$ such that $x$
lies in the convex side of the $r$-cap: since, by Remark 9.3.13, for some $r$
the image of $\widetilde{\sigma}$ lies in the concave side of the $r$-cap, $x$
cannot lie in $\partial\widetilde{\sigma}(\widetilde{M})=\Lambda_{\rho}$. See
Figure 9.8 again. ∎

Figure 9.8: The arguments in the proof of Proposition 9.4.4. On the left,
since $\widetilde{\sigma}$ is proper and extends to $\Lambda_{\rho}$ in
$\partial\mathbb{H}^{n+1}$, from every point $x\notin\Lambda_{\rho}$ one can
find a horosphere with point at infinity $x$ tangent to the image of
$\widetilde{\sigma}$. The same argument works for an interior point $x$, using
metric balls, which is the argument of Proposition 9.4.2. On the right, a
$r$-cap orthogonal to a geodesic $\ell$ with endpoint $x$. Since
$\widetilde{\sigma}$ lies on the concave side of tangent $r$-caps for large
$r$, $x$ cannot be in the image of $G_{\widetilde{\sigma}}^{\pm}$. The same
argument is used in Lemma 11.3.1, under the convexity assumption, in which
case it suffices to use tangent hyperplanes instead of $r$-caps.

As a consequence of Proposition 9.4.2, if
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$ is a nearly-Fuchsian
representation, then the quotient
$\faktor{\mathbb{H}^{n+1}}{\rho(\pi_{1}(M))}$ is a complete hyperbolic
manifold diffeomorphic to $M\times{\mathbb{R}}$. This motivates the following
definition.

######  Definition 9.4.5.

A hyperbolic manifold of dimension $n+1$ is _nearly-Fuchsian_ if it is
isometric to the quotient $\faktor{\mathbb{H}^{n+1}}{\rho(\pi_{1}(M))}$, for
$M$ a closed orientable $n$-manifold and
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$ a nearly-Fuchsian
representation.

######  Remark 9.4.6.

If $\widetilde{\sigma}:\widetilde{M}\to\mathbb{H}^{n+1}$ is a
$\rho$-equivariant embedding with small principal curvatures, then
$\widetilde{\sigma}$ descends to the quotient defining a smooth injective map
$\sigma\colon M\to\faktor{\mathbb{H}^{n+1}}{\rho(\pi_{1}(M))}$. Moreover,
since $\widetilde{\sigma}$ is a $\rho$-equivariant homeomorphism with its
image, $\sigma$ is a homeomorphism with its image as well hence its image is
an embedded hypersurface.

We conclude this section with a final definition which appears in the
statement of Theorem 11.1.4. As a preliminary remark, recall from Propositions
9.3.16 and 9.4.4 that if $\widetilde{\sigma}$ is a $\rho$-equivariant
embedding with small principal curvatures, then each of the Gauss maps
$G_{\widetilde{\sigma}}^{\pm}$ of ${\widetilde{\sigma}}$ is a diffeomorphism
between $\widetilde{M}$ and a connected component of
$\partial\mathbb{H}^{n+1}\setminus\Lambda_{\rho}$. Let us denote these
connected components by $\Omega_{\pm}$ as in Figure 9.7, so that:

$\partial\mathbb{H}^{n+1}\setminus\Lambda_{\rho}=\Omega_{+}\sqcup\Omega_{-}\qquad
G_{\widetilde{\sigma}}^{+}(\widetilde{M})=\Omega_{+}\qquad
G_{\widetilde{\sigma}}^{-}(\widetilde{M})=\Omega_{-}~{}.$

with the representation $\rho$ inducing an action of $\pi_{1}(M)$ on both
$\Omega_{+}$ and $\Omega_{-}$. Recalling the identification

$\mathbb{G}_{n+1}\cong\partial\mathbb{H}^{n+1}\times\partial\mathbb{H}^{n+1}\setminus\Delta~{},$

given by

$\gamma\mapsto\left(\lim_{t\to+\infty}\gamma(t),\lim_{t\to-\infty}\gamma(t)\right)~{},$

the following definition is well-posed.

######  Definition 9.4.7.

Given a closed oriented $n$-manifold $M$ and a nearly-Fuchsian representation
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$, we define
${\mathbb{G}}_{\rho}$ as the quotient:

$\faktor{\\{\gamma\in\mathbb{G}_{n+1}\,|\,\lim_{t\to+\infty}\gamma(t)\in\Omega_{+}\text{
or }\lim_{t\to-\infty}\gamma(t)\in\Omega_{-}\\}}{\rho(\pi_{1}(M))}~{}.$

Observe that, since the action of $\rho(\pi_{1}(M))$ on
$\partial\mathbb{H}^{n+1}$ is free and properly discontinuous on both
$\Omega_{+}$ and $\Omega_{-}$, it is also free and properly discontinuous on
the region of $\mathbb{G}_{n+1}$ consisting of geodesics having either final
point in $\Omega_{+}$ or initial point in $\Omega_{-}$. Moreover, such region
is simply connected, because it is the union of
$\Omega_{+}\times\partial\mathbb{H}^{n+1}\setminus\Delta$ and
$\partial\mathbb{H}^{n+1}\times\Omega_{-}\setminus\Delta$, both of which are
simply connected (they retract on $\Omega_{+}\times\\{\star\\}$ and
$\\{\star\\}\times\Omega_{-}$ respectively) and whose intersection
$\Omega_{+}\times\Omega_{-}$ is again simply connected. We conclude that
${\mathbb{G}}_{\rho}$ is a $2n$-manifold whose fundamental group is isomorphic
to $\pi_{1}(M)$, and is endowed with a natural para-Kähler structure induced
from that of $\mathbb{G}_{n+1}$ (which is preserved by the action of
$\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$).

It is important to stress once more that $\Omega_{+}$ and $\Omega_{-}$ only
depend on $\rho$ and not on ${\widetilde{\sigma}}$. We made here a choice in
the labelling of $\Omega_{+}$ and $\Omega_{-}$ which only depends on the
orientation of $M$. The other choice of labelling would result in a “twin”
region, which differs from ${\mathbb{G}}_{\rho}$ by switching the roles of
initial and final endpoints.

A consequence of this construction, which is implicit in the statement of
Theorem 11.1.4, is the following:

######  Corollary 9.4.8.

Let $M^{n}$ be a closed orientable manifold,
$\rho:\pi_{1}(M)\to\mathrm{Isom}_{0}(\mathbb{H}^{n+1})$ be a nearly-Fuchsian
representation and ${\widetilde{\sigma}}:\widetilde{M}\to\mathbb{H}^{n+1}$ be
a $\rho$-equivariant embedding of small principal curvatures. For a suitable
choice of an orientation on $M$, the Gauss map of $\widetilde{\sigma}$ takes
values in $\Omega_{+}\times\Omega_{-}$, and induces an embedding of $M$ in
${\mathbb{G}}_{\rho}$

###### Proof.

The only part of the statement which is left to prove is that the induced
immersion of $M$ in ${\mathbb{G}}_{\rho}$ is an embedding, but the proof
follows with the same argument as in Remark 9.4.6. ∎

###  Chapter 10 Local, global and equivariant integrability of immersions into
$\mathbb{G}_{n+1}$

In this chapter we introduce a connection on the principal
${\mathbb{R}}$-bundle $\mathrm{p}$ and relate its curvature with the
symplectic geometry of the space of geodesics.

As a consequence, we characterize, in terms of the Lagrangian condition, the
immersions in the space of geodesics which can be locally seen as Gauss maps
of immersions into $\mathbb{H}^{n+1}$. After that, we focus on the problem of
global integrability for immersions into $\mathbb{G}_{n+1}$, looking at the
obstructions and then focusing on Riemannian immersions. The final sections of
this chapter are devoted to characterizing equivariantly integrable Riemannian
immersions into $\mathbb{G}_{n+1}$ in terms of their extrinsic geometry,
namely in terms of their Maslov class.

####  10.1 Connection on the bundle $T^{1}\mathbb{H}^{n}\to\mathbb{G}_{n}$

Recall that a _connection_ on a principal $G$-bundle $P$ is a
$\mathfrak{g}$-valued 1-form $\omega$ on $P$ such that:

  * •

$\mathrm{Ad}_{g}(R_{g}^{*}\omega)=\omega$;

  * •

for every $\xi\in G$, $\omega(X_{\xi})=\xi$ where $X_{\xi}$ is the vector
field associated to $\xi$ by differentiating the action of $G$.

In the special case $G={\mathbb{R}}$ which we consider in this paper, $\omega$
is a real-valued 1-form and the first property simply means that $\omega$ is
invariant under the action of ${\mathbb{R}}$.

Let us now introduce the connection that we will use in the concrete.

######  Definition 10.1.1.

We define the connection form on
$\mathrm{p}:T^{1}\mathbb{H}^{n}\to\mathbb{G}_{n}$ as

$\omega(X)={\overline{Sas}_{n}}(X,\chi_{(x,v)})$

for $X\in T_{(x,v)}T^{1}\mathbb{H}^{n}$, where $\chi$ is the infinitesimal
generator of the geodesic flow.

The 1-form $\omega$ indeed satisfies the two properties of a connection: the
invariance under the ${\mathbb{R}}$-action follows immediately from the
invariance of ${\overline{Sas}_{n}}$ (Lemma 5.1.3) and of $\chi$ (Equation
(5.13)); the second property follows from Equation (5.9), namely
$\omega(\chi_{(x,v)})={\overline{Sas}_{n}}(\chi_{(x,v)},\chi_{(x,v)})=1$.

The connection $\omega$ is defined in such a way that the associated
_Ehresmann connection_ , which we recall being a

distribution of $T^{1}\mathbb{H}^{n}$ in direct sum with the tangent space of
the fibers of $\mathrm{p}$, is simply the distribution orthogonal to $\chi$.
Indeed the Ehresmann connection associated to $\omega$ is the kernel of
$\omega$. The subspaces in the Ehresmann connections defined by the kernel of
$\omega$ are usually called _horizontal_ ; we will avoid this term here since
it might be confused with the horizontal distribution $\mathcal{H}$ with
respect to the other bundle structure of $T^{1}\mathbb{H}^{n}$, namely the
unit tangent bundle $\pi:T^{1}\mathbb{H}^{n}\to\mathbb{H}^{n}$.

Now, a connection on a principal $G$-bundle is _flat_ if the Ehresmann
distribution is integrable, namely, every point admits a local section tangent
to the kernel of $\omega$. We will simply refer to such a section as a _flat_
section. The bundle is a _trivial flat_ principal $G$-bundle if it has a
global flat section.

Having introduced the necessary language, the following statement is a direct
reformulation of Proposition 9.1.2.

######  Proposition 10.1.2.

Given an oriented manifold $M^{n}$ and an immersion
$\sigma:M\to\mathbb{H}^{n+1}$, the $G_{\sigma}$-pull-back of
$\mathrm{p}:T^{1}\mathbb{H}^{n+1}\to\mathbb{G}_{n+1}$ is a trivial flat
bundle.

###### Proof.

The lift $\zeta_{\sigma}:M\to T^{1}\mathbb{H}^{n+1}$ is orthogonal to $\chi$
by Proposition 9.1.2 and therefore induces a global flat section of the pull-
back bundle via $G_{\sigma}=\mathrm{p}\circ\zeta_{\sigma}$. ∎

####  10.2 Curvature of the connection

The purpose of this section is to compute the curvature of the connection
$\omega$, which is simply the ${\mathbb{R}}$-valued 2-form $d\omega$. (The
general formula for the curvature of a connection on a principal $G$-bundle is
$d\omega+\frac{1}{2}\omega\wedge\omega$, but the last term vanishes in our
case $G={\mathbb{R}}$.)

######  Remark 10.2.1.

It is known that the curvature of $\omega$ is the obstruction to the existence
of local flat sections. In particular in the next proposition we will use
extensively that, given $X,Y\in\chi_{(x,v)}^{\perp}\subset
T_{(x,v)}T^{1}\mathbb{H}^{n+1}$, if there exists an embedding $\zeta:M\to
T^{1}\mathbb{H}^{n+1}$ such that $\zeta(p)=(x,v)$, that $X,Y\in
d\zeta(T_{p}M)$ and that $d\zeta(T_{p}M)\subset\chi^{\perp}$, then
$d\omega(X,Y)=0$.

This can be easily seen by a direct argument: if we now denote by $X$ and $Y$
some extensions tangential to the image of $\zeta$, one has

$d\omega(X,Y)=\partial_{X}(\omega(Y))-\partial_{Y}(\omega(X))-\omega([X,Y])=0~{},$

since $\omega(X)=\omega(Y)=0$ by the hypothesis that $X$ and $Y$ are
orthogonal to $\chi$, whence $\omega(X)={\overline{Sas}_{n}}(X,\chi)=0$, and
moreover $\omega([X,Y])=0$ since $[X,Y]$ remains tangential to the image of
$\sigma$.

The argument can in fact be reversed to see that $d\omega$ is exactly the
obstruction to the existence of a flat section, by the Frobenius theorem.

The following proposition represents an essential step to relate the curvature
of $\omega$ and the symplectic geometry of the space of geodesics.

######  Proposition 10.2.2.

The following identity holds for the connection form $\omega$ on the principal
${\mathbb{R}}$-bundle $\mathrm{p}:T^{1}\mathbb{H}^{n+1}\to\mathbb{G}_{n+1}$
and the symplectic form $\Omega$ of $\mathbb{G}_{n+1}$:

$d\omega=\mathrm{p}^{*}\Omega~{}.$

###### Proof.

We shall divide the proof in several steps.

First, let us show that $d\omega$ is the pull-back of some 2-form on
$\mathbb{G}_{n+1}$. Since $d\omega$ is obviously invariant under the geodesic
flow, we only need to show that $d\omega(X,\chi_{(x,v)})=0$ for all $X\in
T_{(x,v)}T^{1}\mathbb{H}^{n+1}$. Clearly, it suffices to check this for
$X\in\chi^{\perp}$. To apply the exterior derivative formula for $d\omega$, we
consider $\chi$ as a globally defined vector field and we shall extend $X$
around $(x,v)$. For this purpose, take a curve $c:(-\epsilon,\epsilon)\to
T^{1}\mathbb{H}^{n+1}$ such that $c^{\prime}(0)=X$ and $c^{\prime}(s)$ is
tangent to $\chi^{\perp}$ for every $s$. Then define the map
$f(s,t)=\varphi_{t}(c(s))$ and observe that $\chi=\partial_{t}f$. We thus set
$X=\partial_{s}f$, which is the desired extension along a two-dimensional
submanifold. Then we have:

$d\omega(X,\chi)=\partial_{X}(\omega(\chi))-\partial_{\chi}(\omega(X))-\omega[X,\chi]=0~{},$

where we have used that $\omega(\chi)\equiv 1$, that $\omega(X)=0$ along the
curves $t\mapsto\varphi_{t}(x,v)$ (since the curve $c$ is taken to be in the
distribution $\chi^{\perp}$ and the geodesic flow preserves both $\chi$ and
its orthogonal complement), and finally that $[X,\chi]=0$ since
$\chi=\partial_{t}f$ and $X=\partial_{s}f$ are coordinate vector fields for a
submanifold in neighborhood of $(x,v)$.

Having proved this, it is now sufficient to show that $d\omega$ and
$\mathrm{p}^{*}\Omega$ agree when restricted to $\chi^{\perp}$. Recall that
$\Omega$ is defined as ${\mathbbm{g}}(\cdot,\mathbbm{J}\cdot)$, where
${\mathbbm{g}}$ and $\mathbbm{J}$ are the push-forward to
$T_{\mathrm{p}(x,v)}\mathbb{G}_{n+1}$, by means of the differential of
$\mathrm{p}$, of the metric ${\overline{Sas}_{n+1}}$ and of the para-complex
structure $\mathrm{J}$ on $\chi^{\perp}$. Thus we must equivalently show that

$d\omega(X,Y)={\overline{Sas}_{n+1}}(X,\mathrm{J}Y)\qquad\text{for all
}X,Y\in\chi_{(x,v)}^{\perp}~{}.$ (10.1)

To see this, take an orthonormal basis $\\{w_{1},\ldots,w_{n}\\}$ for
$v^{\perp}\subset T_{x}\mathbb{H}^{n+1}$, and observe that
$\\{w_{1}^{\mathcal{H}},\ldots,w_{n}^{\mathcal{H}},w_{1}^{\mathcal{V}},\ldots,w_{n}^{\mathcal{V}}\\}$
is a ${\overline{Sas}_{n+1}}$-orthonormal basis of $\chi^{\perp}$. It is
sufficient to check (10.1) for $X,Y$ distinct elements of this basis. We
distinguish several cases.

  * •

First, consider the case $X=w_{i}^{\mathcal{H}}$ and $Y=w_{j}^{\mathcal{H}}$,
for $i\neq j$. Then
${\overline{Sas}_{n+1}}(X,{\mathrm{J}}Y)={\overline{Sas}_{n+1}}(w_{i}^{\mathcal{H}},w_{j}^{\mathcal{V}})=0$
by Definition 5.1.2. On the other hand, by Example 9.1.8, if $\sigma$ is the
inclusion of the totally geodesic hyperplane orthogonal to $v$ at $x$, then
its lift $\zeta_{\sigma}$ is a submanifold in $T^{1}\mathbb{H}^{n+1}$
orthogonal to $\chi$ at every point and tangent to $X$ and $Y$. Then
$d\omega(X,Y)=0$ by Remark 10.2.1.

  * •

Second, consider $X=w_{i}^{\mathcal{V}}$ and $Y=w_{j}^{\mathcal{V}}$, for
$i\neq j$. Then again
${\overline{Sas}_{n+1}}(X,{\mathrm{J}}Y)={\overline{Sas}_{n+1}}(w_{i}^{\mathcal{V}},w_{j}^{\mathcal{H}})=0$.
Here we can apply Example 9.1.9 instead, and see that there is a
$n$-dimensional sphere in $\mathcal{V}^{0}_{(x,v)}$ orthogonal to the fibers
of $\mathrm{p}$ and tangent to $X$ and $Y$, whence $d\omega(X,Y)=0$ by Remark
10.2.1.

  * •

Third, consider $X=w_{i}^{\mathcal{H}}$ and $Y=w_{j}^{\mathcal{V}}$, for
$i\neq j$. Then