The Electromagnetic Field
in Gravitational Wave Interferometers
Preprint UWThPh-2021-11

1]Thomas B. Mieling
1]Piotr T. Chruściel
2]Stefan Palenta
[1]University of Vienna, Faculty of Physics and TURIS Research Platform, Vienna, Austria
[2]University of Vienna, Faculty of Physics, Vienna, Austria

We analyse the response of laser interferometric gravitational wave detectors using the full Maxwell equations in curved spacetime in the presence of weak gravitational waves.
Existence and uniqueness of solutions is ensured by setting up a suitable boundary value problem. This puts on solid ground previous approximate calculations. We find consistency with previous results obtained from eikonal expansions at the level of accuracy accessible to current gravitational wave detectors.

§ SCALAR WAVES

§ MAXWELL’S EQUATIONS

§ ACKNOWLEDGEMENTS

Research supported in part by the Austrian Science Fund (FWF), Project P34274, and by the Vienna University Research Platform TURIS.
TBM gratefully acknowledges funding via a fellowship of the Vienna Doctoral School in Physics (VDSP) and the Austrian Science Fund (FWF) through GRIPS (Grant No. P30817-N36).
PTC was further supported in part by the Polish National Center of Science (NCN) under grant 2016/21/B/ST1/00940.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

American Physical Society (APS)

titleWideband laser-interferometer graviational-radiation
journaltitlePhysical Review D

Oxford University Press (OUP)

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gravitational waves - I. Linearly polarized waves
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titleResponse of interferometric gravitational wave detectors
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Springer Science
Business Media LLC

titleResponse of Doppler spacecraft tracking to gravitational
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AIP Publishing

titleA Michelson interferometer in the field of a plane
gravitational wave
journaltitleJournal of Mathematical Physics

IOP Publishing

titleHigh-frequency corrections to the detector response and their
effect on searches for gravitational waves
journaltitleClassical and Quantum Gravity

titleOn the round-trip time for a photon propagating in the field
of a plane gravitational wave
journaltitleClassical and Quantum Gravity

American Association of Physics Teachers (AAPT)

titleThe response of laser interferometers to a gravitational
journaltitleAmerican Journal of Physics

American Physical Society (APS)

titlePhysical response of light-time gravitational wave detectors
journaltitlePhysical Review D

World Scientific

editionSecond Edition
titleFundamentals of Interferometric Gravitational Wave

IOP Publishing

titleGauge independent response of a laser interferometer to
gravitational waves
journaltitleClassical and Quantum Gravity

Walter de Gruyter GmbH

titleZum Übergang von der Wellenoptik zur geometrischen Optik in
der allgemeinen Relativitätstheorie
journaltitleZeitschrift für Naturforschung A

IOP Publishing

titleExact solution to the homogeneous Maxwell equations in the
field of a gravitational wave in linearized theory
journaltitleClassical and Quantum Gravity

titleEffect of a weak plane GW on a light beam
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American Physical Society (APS)

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journaltitlePhysical Review D

titleThe interaction between electromagnetic and gravitational
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titleLaser-interferometric detectors of gravitational waves
journaltitleClassical and Quantum Gravity

IOP Publishing

titleOn the propagation of electromagnetic radiation in the field
of a plane gravitational wave
journaltitleClassical and Quantum Gravity

IOP Publishing

titleThe response of optical fibres to gravitational waves
journaltitleClassical and Quantum Gravity

IOP Publishing

titleThe response of laser interferometric gravitational wave
detectors beyond the eikonal equation
journaltitleClassical and Quantum Gravity

titleZur Lichtfortpflanzung nach der Relativitätstheorie
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IOP Publishing

titleWeakly gravitating isotropic waveguides
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The Optical Society

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journaltitleJournal of the Optical Society of America A

Springer Science
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Springer International Publishing

booktitleAnalytic Number Theory, Modular Forms and
q-Hypergeometric Series
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University of Chicago Pr.

titleGeneral Relativity

gr-qc, physics.class-ph
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# Warped product metrics on hyperbolic and complex hyperbolic manifolds

Barry Minemyer  Department of Mathematical and Digital Sciences, Commonwealth
University - Bloomsburg campus, Bloomsburg, Pennsylvania 17815
<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

In this paper we study warped-product metrics on manifolds of the form
$X\setminus Y$, where $X$ denotes either $\mathbb{H}^{n}$ or
$\mathbb{C}\mathbb{H}^{n}$, and $Y$ is a totally geodesic submanifold with
arbitrary codimension. The main results that we prove are curvature formulas
for these metrics on $X\setminus Y$ expressed in spherical coordinates about
$Y$. We also discuss past and potential future applications of these formulas.

###### Key words and phrases:

complex hyperbolic space, hyperbolic space, totally geodesic submanifold,
warped product metric, sectional curvature

###### 2010 Mathematics Subject Classification:

Primary 53C20, 53C35; Secondary 53C56, 57R25

##  1\. Introduction

###  1.1. Main results

Let $\mathbb{H}^{n}$ denote (real) $n$-dimensional hyperbolic space and let
$\mathbb{C}\mathbb{H}^{n}$ denote (complex) $n$-dimensional complex hyperbolic
space. In this paper, $X$ will denote either $\mathbb{H}^{n}$ or
$\mathbb{C}\mathbb{H}^{n}$, and $Y$ will denote a totally geodesic submanifold
of $X$. So if $X=\mathbb{H}^{n}$ then $Y=\mathbb{H}^{k}$, and if
$X=\mathbb{C}\mathbb{H}^{n}$ then $Y$ is either $\mathbb{H}^{k}$ or
$\mathbb{C}\mathbb{H}^{k}$ for some $0\leq k\leq n-1$. Let $M$ be a Riemannian
manifold, and $N$ a totally geodesic submanifold of $M$. We say that the pair
$(M,N)$ is modeled on $(X,Y)$ if there exist lattices
$\Gamma\subset\text{Isom}(X)$ and $\Lambda\subset\text{Isom}(Y)$ such that
$M=X/\Gamma$, $N=Y/\Lambda$, and $\Lambda<\Gamma$. We also allow for the
possibility that $N$ is disconnected. That is, we allow for multiple lattices
$\Lambda<\Gamma$ which correspond to different (disjoint) copies of
$\mathbb{H}^{k}\subset\mathbb{H}^{n}$ or
$\mathbb{H}^{k},\mathbb{C}\mathbb{H}^{k}\subset\mathbb{C}\mathbb{H}^{n}$.

The purpose of this paper is to develop curvature formulas for warped-product
metrics on $X\setminus Y$ when the pair $(X,Y)$ is one of
$(\mathbb{H}^{n},\mathbb{H}^{k})$,
$(\mathbb{C}\mathbb{H}^{n},\mathbb{H}^{n})$, or
$(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{k})$. These cases are
detailed in Sections 2, 3, and 4, respectively. In each case we write the
metric on $X$ in spherical coordinates about $Y$ (Theorems 2.1, 3.1, and 4.1),
we consider the corresponding warped product metric where we allow for
variable coefficients in the metric tensor (equations (2.2), (3.3), and
(4.2)), and we compute formulas for the components of the $(4,0)$ curvature
tensor with respect to these coefficient functions (Theorems 2.2, 3.4, and
4.3). These last three Theorems should be considered the main results of this
paper.

###  1.2. Applications for these curvature formulas

Specific cases for these formulas are already known and have been used in
various applications in the literature. Some examples are as follows. The case
when $X=\mathbb{H}^{n}$ and $Y=\mathbb{H}^{n-2}$ was used by Gromov and
Thurston in [7] (discussed further below) and by Belegradek in [2]. When
$X=\mathbb{H}^{n}$ and $Y=\mathbb{H}^{0}$ is a point, this leads to the basis
for the Farrell and Jones Warping Deformation used in [6]. This process is
described by Ontaneda in [12] and used by the same author in [13]. The case
when $X=\mathbb{C}\mathbb{H}^{n}$ and $Y=\mathbb{C}\mathbb{H}^{0}$ is a point
was used by Farrell and Jones in [5], and the same $X$ but with
$Y=\mathbb{C}\mathbb{H}^{n-1}$ was considered by Belegradek in [1]. Finally,
the cases when $(X,Y)$ are either $(\mathbb{C}\mathbb{H}^{2},\mathbb{H}^{2})$
or $(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-2})$ were used by the
author in [9] and [10].

While the author believes that the curvature formulas in Theorems 2.2, 3.4,
and 4.3 will have many future uses, the primary motivation for the development
of these curvature formulas was for the following application.

In [7] Gromov and Thurston famously construct pinched negatively curved
manifolds which do not admit hyperbolic metrics. In this construction they
consider pairs $(M,N)$ modeled on $(\mathbb{H}^{n},\mathbb{H}^{n-2})$ which
satisfy a few special topological and geometric conditions. The pinched
negatively curved manifold $B$ which does not admit a hyperbolic metric is
then the $k$-fold cyclic branched cover of $M$ about $N$ (where
$k\in\mathbb{N}$ can take all but possibly finitely many values). The
difficulty in all of this is showing that $B$ exists, constructing a pinched
negatively curved metric on $B$, and proving that $B$ does not admit a
hyperbolic metric.

It is an open question as to whether or not this construction can be extended
to the locally symmetric pairs
$(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-1})$ and
$(\mathbb{C}\mathbb{H}^{2},\mathbb{H}^{2})$. In forthcoming paper ([11]) the
author shows that the $d$-fold cyclic ramified cover of $M$ about $N$ for the
case $(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-1})$ does admit an
almost negatively $\frac{1}{4}$-pinched Riemannian metric. The fact that such
a pair $(M,N)$ can be realized so that the ramified cover is a smooth manifold
for some integer $d>2$ is a result of Stover and Toledo in [14]. The
constructions of these Riemannian metrics are dependent on the curvature
formulas proved in Theorems 3.4 and 4.3 below.

One last remark about these curvature formulas. In [2], [1], and [9] it is
proved that the manifold $M\setminus N$, where $(M,N)$ is modeled on one of
$(\mathbb{H}^{n},\mathbb{H}^{n-2})$,
$(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-2})$, or
$(\mathbb{C}\mathbb{H}^{2},\mathbb{H}^{2})$, admits a complete, finite volume,
negatively curved Riemannian metric. The curvature formulas developed in this
paper generalize the curvature formulas computed and used in these three
articles.

###  1.3. Obstructions to $M\setminus N$ admitting a complete, finite volume
Riemannian metric of negative sectional curvature

Consider the finite volume manifold $M\setminus N$. The three cases where $N$
has real codimension two in $M$ are modeled on one of
$(\mathbb{H}^{n},\mathbb{H}^{n-2})$,
$(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-1})$, or
$(\mathbb{C}\mathbb{H}^{2},\mathbb{H}^{2})$. In all of these cases, the
manifold $M\setminus N$ admits a complete, finite volume Riemannian metric
whose sectional curvature is bounded above by a negative constant ([2], [1],
and [9]).

When the real codimension of $N$ is greater than two, the manifold $M\setminus
N$ should not admit a complete, finite volume, negatively curved metric
because it generally will not be aspherical. This fact should be realized in
the curvature equations in Theorems 2.2, 3.4, and 4.3. More specifically,
there should be an equation(s) which obstructs such a metric, but this (these)
curvature equations should vanish when $N$ has codimension two.

In all cases except one “exceptional case” the obstruction is a sectional
curvature equation of the form

(1.1) $\frac{1}{v^{2}}-\left(\frac{v^{\prime}}{v}\right)^{2}$

where $v:\mathbb{R}\to\mathbb{R}$ is a positive, increasing real-valued
function. In order to alter the metric on $M\setminus N$ to be complete, one
needs to define a warping function for $v$ that will make each component of
$N$ into the boundary of a cusp of the manifold. One easily checks that
equation (1.1) is nonpositive if and only if $1\leq(v^{\prime})^{2}$. But for
the Riemannian metric to have any chance of having finite volume one needs
$\lim_{r\to-\infty}v^{\prime}(r)=0$.

The one exceptional case is when $(M,N)$ is modeled on
$(\mathbb{C}\mathbb{H}^{n},\mathbb{C}\mathbb{H}^{n-2})$. Here, all curvature
equations of the form (1.1) vanish, and so this obstruction is more subtle. It
should be noted that the vanishing of (1.1) is what leads to the metric
developed in [10]. A detailed discussion about this situation is given in
Subsection 4.6.

###  1.4. Layout of this paper

In Section 2 we study manifolds of the form
$\mathbb{H}^{n}\setminus\mathbb{H}^{k}$, in Section 3 we consider
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{n}$, and in Section 4 we analyze
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{k}$. The calculations
in Section 3 and 4 become very complicated. So in Section 3 we restrict our
attention to $\mathbb{C}\mathbb{H}^{3}\setminus\mathbb{H}^{3}$ and in Section
4 we restrict to $\mathbb{C}\mathbb{H}^{5}\setminus\mathbb{C}\mathbb{H}^{2}$
to make these calculations simpler to follow. In each case, these are the
smallest choices for $n$ and $k$ which capture all of the different formulas
for the curvature tensor, up to the symmetries of the curvature tensor (and
with respect to the frames chosen in each Section). That is, from these cases
one knows all of curvature formulas for general
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{n}$ and
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{k}$. Also, notice that
we only consider $\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{n}$ in Section
3 instead of the more general
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{k}$. The reason for this is due
to simplicity: in general there are several ways that $\mathbb{H}^{k}$ can sit
inside of $\mathbb{C}\mathbb{H}^{n}$ which requires a case-by-case analysis.
But in all situations this copy of $\mathbb{H}^{k}$ is contained in a copy of
$\mathbb{H}^{n}$, and then one can apply our formulas here to
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{n}$. Section 5 is a short
Section on some known formulas that are referenced throughout the paper, and
Section 6 is devoted to computing values for Lie brackets from Section 3.

We end this Section with the following two remarks which deal with notational
differences between this paper and references [2], [1], and [9].

######  Remark 1.1.

In this paper we scale the complex hyperbolic metric to have sectional
curvatures in the interval $[-4,-1]$, whereas in the previous three references
the curvatures were scaled to $\left[-1,-\frac{1}{4}\right]$. To adjust the
formulas in [2], [1], and [9], one simply multiplies the warping functions
$h$, $v$, and $h_{r}$ by $\frac{1}{2}$. With this adjustment (and the
following Remark), one sees that the formulas in these references agree with
the codimension two versions of the formulas in Theorems 2.2, 3.4, and 4.3.

######  Remark 1.2.

Another major notational difference between this paper and the papers [1] and
[2] is the formula used for the curvature tensor. Let $g$ be a Riemannian
metric with Levi-Civita connection $\nabla$, and let $W,X,Y,$ and $Z$ be
vector fields. In this paper we follow [4] and use the notation

(1.2) $R(X,Y)Z=\nabla_{Y}\nabla_{X}Z-\nabla_{X}\nabla_{Y}Z+\nabla_{[X,Y]}Z$

for the curvature tensor $R$ of $g$. The negative of this formula is used in
[1] and [2]. So, in particular, the $(4,0)$-curvature tensor $\langle
R(X,Y)Z,W\rangle_{g}$ in this paper is equivalent to $\langle
R(X,Y)W,Z\rangle_{g}$ in [1] and [2].

##  2\. Curvature formulas for warped product metrics on
$\mathbb{H}^{n}\setminus\mathbb{H}^{k}$

###  2.1. Expressing the metric in $\mathbb{H}^{n}$ in spherical coordinates
about $\mathbb{H}^{k}$

Let us first note that in Subsections 2.1, 3.1, and 4.1 we closely follow the
notation and terminology used in [1].

Let ${\bf h_{n}}$ denote the hyperbolic metric on $\mathbb{H}^{n}$. Since
$\mathbb{H}^{k}$ is a complete totally geodesic submanifold of the negatively
curved manifold $\mathbb{H}^{n}$, there exists an orthogonal projection map
$\pi:\mathbb{H}^{n}\to\mathbb{H}^{k}$. This map $\pi$ is a fiber bundle whose
fibers are totally geodesic $(n-k)$-planes.

For $r>0$ let $E(r)$ denote the $r$-neighborhood of $\mathbb{H}^{k}$. Then
$E(r)$ is a real hypersurface in $\mathbb{H}^{n}$, and consequently we can
decompose ${\bf h_{n}}$ as

${\bf h_{n}}=({\bf h_{n}})_{r}+dr^{2}$

where $({\bf h_{n}})_{r}$ is the induced Riemannian metric on $E(r)$. Let
$\pi_{r}:E(r)\to\mathbb{H}^{k}$ denote the restriction of $\pi$ to $E(r)$.
Note that $\pi_{r}$ is an $\mathbb{S}^{n-k-1}$-bundle whose fiber over any
point $q\in\mathbb{H}^{k}$ is the $(n-k-1)$-sphere of radius $r$ in the
totally geodesic $(n-k)$-plane $\pi^{-1}(q)$. The tangent bundle splits as an
orthogonal sum $\mathcal{V}(r)\oplus\mathcal{H}(r)$ where $\mathcal{V}(r)$ is
tangent to the sphere $\pi_{r}^{-1}(q)$ and $\mathcal{H}(r)$ is the orthogonal
complement to $\mathcal{V}(r)$.

It is well known (see [2] or [7] when $k=n-2$ and [Ontaneda] for general $k$)
that for an appropriate identification of
$E(r)\cong\mathbb{H}^{k}\times\mathbb{S}^{n-k-1}$ the metric $({\bf
h_{n}})_{r}$ can be written as

$({\bf h_{n}})_{r}=\cosh^{2}(r){\bf h_{k}}+\sinh^{2}(r){\bf\sigma_{n-k-1}}$

where ${\bf h_{k}}$ denotes the hyperbolic metric on $\mathbb{H}^{k}$ and
${\bf\sigma_{n-k-1}}$ denotes the round metric on the unit sphere
$\mathbb{S}^{n-k-1}$. Note that $({\bf h_{n}})_{r}$ restricted to
$\mathcal{H}(r)$ is $\cosh^{2}(r){\bf h_{k}}$ and $({\bf h_{n}})_{r}$
restricted to $\mathcal{V}(r)$ is $\sinh^{2}(r){\bf\sigma_{n-k-1}}$. We
summarize this in the following Theorem.

######  Theorem 2.1.

The hyperbolic manifold $\mathbb{H}^{n}\setminus\mathbb{H}^{k}$ can be written
as $E\times(0,\infty)$ where $E\cong\mathbb{H}^{k}\times\mathbb{S}^{n-k-1}$
equipped with the metric

(2.1) ${\bf h_{n}}=\cosh^{2}(r){\bf h_{k}}+\sinh^{2}(r)\sigma^{n-k-1}+dr^{2}.$

###  2.2. The warped product metric and curvature formulas

For some positive, increasing real-valued functions
$h,v:(0,\infty)\to\mathbb{R}$ define

(2.2) ${\bf\lambda_{r}}:=h^{2}(r){\bf h_{k}}+v^{2}(r){\bf\sigma_{n-k-1}}\hskip
30.0pt\text{and}\hskip 30.0pt{\bf\lambda}:={\bf\lambda_{r}}+dr^{2}.$

Of course, ${\bf\lambda}={\bf h_{n}}$ when $h=\cosh(r)$ and $v=\sinh(r)$.

Fix $p\in E(r)$ for some $r$ and let $q=\pi(p)\in\mathbb{H}^{k}$. Let
$\\{\check{X}_{i}\\}_{i=1}^{k}$ be an orthonormal frame of $\mathbb{H}^{k}$
near $q$ which satisfies $[\check{X}_{i},\check{X}_{j}]_{q}=0$ for all $1\leq
i,j\leq k$. These vector fields can be extended to a collection of orthogonal
vector fields $\\{X_{i}\\}_{i=1}^{k}$ in a neighborhood of $p$ via the
inclusion $\mathbb{H}^{k}\to E\times(0,\infty)$. Analogously, define an
orthonormal frame $\\{\check{X}_{j}\\}_{j=k+1}^{n-1}$ of $\mathbb{S}^{n-k-1}$
near (the projection of) $p$ which satisfies
$[\check{X}_{i},\check{X}_{j}]_{p}=0$ for all $k+1\leq i,j\leq n-1$, and
extend this frame to vector fields $\\{X_{j}\\}_{j=k+1}^{n-1}$ in a
neighborhood of $p$ via the inclusion $\mathbb{S}^{n-k-1}\to
E\times(0,\infty)$. Lastly, let $X_{n}=\frac{\partial}{\partial r}$.

The orthogonal collection of vector fields $\\{X_{i}\\}_{i=1}^{n}$ satisfies
the following:

  1. (1)

$\langle X_{i},X_{i}\rangle_{\lambda}=h^{2}$ for $1\leq i\leq k$.

  2. (2)

$\langle X_{i},X_{i}\rangle_{\lambda}=v^{2}$ for $k+1\leq i\leq n-1$.

  3. (3)

$\langle X_{n},X_{n}\rangle_{\lambda}=1$.

  4. (4)

$[X_{i},X_{j}]_{p}=0$ for all $i,j$.

It should be noted that property (4) is special to the real hyperbolic case
and will not be true in Sections 3 and 4 below.

Now define the corresponding orthonormal frame near $p$ by
$Y_{i}=\frac{1}{h}X_{i}$ for $1\leq i\leq k$, $Y_{j}=\frac{1}{v}X_{j}$ for
$k+1\leq j\leq n-1$, and $Y_{n}=X_{n}$. This frame satisfies the property that
$[Y_{i},Y_{j}]_{p}=0$ for $1\leq i,j\leq n-1$. We can then apply formulas
(5.4) through (5.7) to write the (4,0) curvature tensor $R_{\lambda}$ in terms
of $R_{\lambda_{r}}$ as follows, where $1\leq a,b\leq k$ and $k+1\leq c,d\leq
n-1$.

$\displaystyle
K_{\lambda}(Y_{a},Y_{b})=K_{\lambda_{r}}(Y_{a},Y_{b})-\left(\frac{h^{\prime}}{h}\right)^{2}\hskip
30.0ptK_{\lambda}(Y_{c},Y_{d})=K_{\lambda_{r}}(Y_{c},Y_{d})-\left(\frac{v^{\prime}}{v}\right)^{2}$
$\displaystyle
K_{\lambda}(Y_{a},Y_{c})=K_{\lambda_{r}}(Y_{a},Y_{c})-\frac{h^{\prime}v^{\prime}}{hv}\hskip
20.0ptK_{\lambda}(Y_{a},Y_{n})=-\frac{h^{\prime\prime}}{h}\hskip
20.0ptK_{\lambda}(Y_{c},Y_{n})=-\frac{v^{\prime\prime}}{v}.$

In the above equations, we use the notation

$K(X,Y)=\langle R(X,Y)X,Y\rangle$

to denote the sectional curvature of the 2-plane spanned by $X$ and $Y$. The
above equations are the only terms that appear (up to the symmetries of the
curvature tensor). So, in particular, all mixed terms of $R_{\lambda}$ are
identically zero.

Now, the $(4,0)$ curvature tensor $R_{\lambda_{r}}$ is simple to calculate.
Since both $h(r)\mathbb{H}^{k}$ and $v(r)\mathbb{S}^{n-k-1}$ have constant
curvature, and $h(r)\mathbb{H}^{k}\times v(r)\mathbb{S}^{n-k-1}$ is metrically
a product, we have that for $1\leq a,b,\leq k$ and $k+1\leq c,d\leq n-1$:

$\displaystyle K_{\lambda_{r}}(Y_{a},Y_{b})=-\frac{1}{h^{2}}\hskip
30.0ptK_{\lambda_{r}}(Y_{c},Y_{d})=\frac{1}{v^{2}}\hskip
30.0ptK_{\lambda_{r}}(Y_{a},Y_{c})=0.$

Putting this all together yields the following.

######  Theorem 2.2.

Up to the symmetries of the curvature tensor, the only nonzero terms of the
$(4,0)$ curvature tensor $R_{\lambda}$ are:

$\displaystyle
K_{\lambda}(Y_{a},Y_{b})=-\frac{1}{h^{2}}-\left(\frac{h^{\prime}}{h}\right)^{2}\hskip
40.0ptK_{\lambda}(Y_{c},Y_{d})=\frac{1}{v^{2}}-\left(\frac{v^{\prime}}{v}\right)^{2}$
$\displaystyle K_{\lambda}(Y_{a},Y_{c})=-\frac{h^{\prime}v^{\prime}}{hv}\hskip
20.0ptK_{\lambda}(Y_{a},Y_{n})=-\frac{h^{\prime\prime}}{h}\hskip
20.0ptK_{\lambda}(Y_{c},Y_{n})=-\frac{v^{\prime\prime}}{v}$

where $1\leq a,b,\leq k$ and $k+1\leq c,d\leq n-1$.

One easily checks that plugging in the values $v(r)=\sinh(r)$ and
$h(r)=\cosh(r)$ gives all sectional curvatures of $-1$.

##  3\. Curvature formulas for warped product metrics on
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{n}$

As mentioned in the Introduction, for simplicity we are going to restrict
ourselves to the case when $n=3$. This is exactly the smallest dimension which
captures every nonzero component of the curvature tensor, and so nothing is
lost with this restriction (see the comments after Theorem 3.4 for more
discussion).

###  3.1. Expressing the metric in $\mathbb{C}\mathbb{H}^{3}$ in spherical
coordinates about $\mathbb{H}^{3}$

Let ${\bf c_{3}}$ denote the complex hyperbolic metric on
$\mathbb{C}\mathbb{H}^{3}$ normalized to have constant holomorphic sectional
curvature $-4$. Since $\mathbb{H}^{3}$ is a complete totally geodesic
submanifold of the negatively curved manifold $\mathbb{C}\mathbb{H}^{3}$,
there exists an orthogonal projection map
$\pi:\mathbb{C}\mathbb{H}^{3}\to\mathbb{H}^{3}$. This map $\pi$ is a fiber
bundle whose fibers are totally real totally geodesic $3$-planes, and
therefore have constant sectional curvature $-1$.

For $r>0$ let $E(r)$ denote the $r$-neighborhood of $\mathbb{H}^{3}$. Then
$E(r)$ is a real hypersurface in $\mathbb{C}\mathbb{H}^{3}$, and consequently
we can decompose ${\bf c_{3}}$ as

${\bf c_{3}}=({\bf c_{3}})_{r}+dr^{2}$

where $({\bf c_{3}})_{r}$ is the induced Riemannian metric on $E(r)$. Let
$\pi_{r}:E(r)\to\mathbb{H}^{3}$ denote the restriction of $\pi$ to $E(r)$.
Note that $\pi_{r}$ is an $\mathbb{S}^{2}$-bundle whose fiber over any point
$q\in\mathbb{H}^{3}$ is the 2-sphere of radius $r$ in the totally real totally
geodesic $3$-plane $\pi^{-1}(q)$. The tangent bundle splits as an orthogonal
sum $\mathcal{V}(r)\oplus\mathcal{H}(r)$ where $\mathcal{V}(r)$ is tangent to
the 2-sphere $\pi_{r}^{-1}(q)$ and $\mathcal{H}(r)$ is the orthogonal
complement to $\mathcal{V}(r)$.

For $r,s>0$ there exists a diffeomorphism $\phi_{sr}:E(s)\to E(r)$ induced by
the geodesic flow along the totally real totally geodesic $3$-planes
orthogonal to $\mathbb{H}^{3}$. Fix $p\in E(r)$ arbitrary, let
$q=\pi(p)\in\mathbb{H}^{3}$, and let $\gamma$ be the unit speed geodesic such
that $\gamma(0)=q$ and $\gamma(r)=p$. In what follows, all computations are
considered in the tangent space $T_{p}E(r)$.

Note that $\mathcal{V}(r)$ is tangent to both $E(r)$ and the totally real
totally geodesic $3$-plane $\pi^{-1}(q)$. Then since $\pi^{-1}(q)$ is
preserved by the geodesic flow, we have that $d\phi_{sr}$ takes
$\mathcal{V}(s)$ to $\mathcal{V}(r)$. Since
$exp_{p}^{-1}\left(\pi^{-1}(q)\right)$ is a totally real 3-plane, there exists
a suitable identification $\pi^{-1}(q)\cong\mathbb{S}^{2}\times(0,\infty)$
where the metric ${\bf c_{3}}$ restricted to $\pi^{-1}(q)$ can be written as

$\sinh^{2}(r)\sigma^{2}+dr^{2}.$

Here, $\sigma^{2}$ is the round metric on the unit 2-sphere.

Let

(3.1) $\check{X}_{4}=\frac{\partial}{\partial\theta}\hskip
40.0pt\check{X}_{5}=\frac{1}{\sin\theta}\frac{\partial}{\partial\psi}$

be an orthonormal frame on a neighborhood of (the projection of) $p$ in
$\mathbb{S}^{2}$, and extend these to orthogonal vector fields
$\\{X_{4},X_{5}\\}$ on $\pi^{-1}(q)$ via the inclusion
$\mathbb{S}^{2}\to\pi^{-1}(q)$. Note that both $X_{4}$ and $X_{5}$ are
invariant under $d\phi_{sr}$. Let $X_{6}=\frac{\partial}{\partial r}$.

Let $J$ denote the complex structure on $\mathbb{C}\mathbb{H}^{3}$. It is well
known that $J_{p}$ preserves complex subspaces in
$T_{p}\mathbb{C}\mathbb{H}^{3}$ and maps real subspaces into their orthogonal
complement. Since $\left(X_{4},X_{5},X_{6}\right)$ spans a real $3$-plane in
$T_{p}\mathbb{C}\mathbb{H}^{3}$, its orthogonal complement
$\mathcal{H}_{p}(r)$ is spanned by $\left(JX_{4},JX_{5},JX_{6}\right)$. In
what follows we define vector fields $X_{1}$, $X_{2}$, and $X_{3}$ which are
just scaled copies of $JX_{4}$, $JX_{5}$, and $JX_{6}$, respectively.

####  3.1.1. The vector fields $X_{1}$ and $X_{2}$

First note that $\left(JX_{4},X_{6}\right)$ spans a real $2$-plane in
$T_{p}\mathbb{C}\mathbb{H}^{3}$ (since its $J$-image is contained in its
orthogonal complement). So
$P=\text{exp}\left(\text{span}\left(JX_{4},X_{6}\right)\right)$ is a totally
real totally geodesic $2$-plane in $\mathbb{C}\mathbb{H}^{3}$ which intersects
$\mathbb{H}^{3}$ orthogonally. Since this intersection is orthogonal, $P$ is
preserved by the geodesic flow $\phi$. Therefore,
$\text{span}\left(JX_{4}\right)$ is preserved by $d\phi$.

The set $P\cap\mathbb{H}^{3}$ is a (real) geodesic. Let $\alpha(s)$ denote
this geodesic parameterized with respect to arc length so that $\alpha(0)=q$.
Then define $(X_{1})_{p}=(d\pi)^{-1}_{p}\alpha^{\prime}(0)$. There exists a
positive real-valued function $a(r,s)$ so that the metric ${\bf c_{3}}$
restricted to $P$ is of the form $dr^{2}+a^{2}(r,s)ds^{2}$. But since
$\mathbb{R}$ acts by isometries on $P$ via translation along $\alpha$, the
function $a(r,s)$ is independent of $s$. Then since the curvature of a real
$2$-plane is $-1$, we have that $a(r)=\cosh(r)$.

We analogously define $X_{2}$ by replacing $X_{4}$ with $X_{5}$ in the above
description. All conclusions follow in an identical manner. Thus, we can write
the metric ${\bf c_{3}}$ restricted to $exp_{p}(X_{1},X_{2},X_{6})$ as

$\cosh^{2}(r)(dX_{1}^{2}+dX_{2}^{2})+dr^{2}.$

####  3.1.2. The vector field $X_{3}$

This is also mostly analogous to the definition of $X_{1}$. But this time note
that $\left(JX_{6},X_{6}\right)$ spans a complex line in
$T_{p}\mathbb{C}\mathbb{H}^{3}$ (since it is preserved by its $J$-image). So
$Q=\text{exp}_{p}\left(\text{span}\left(JX_{6},X_{6}\right)\right)$ is a
complex geodesic in $\mathbb{C}\mathbb{H}^{3}$ which intersects
$\mathbb{H}^{3}$ orthogonally. Since this intersection is orthogonal, $Q$ is
preserved by the geodesic flow $\phi$. Therefore,
$\text{span}\left(JX_{6}\right)$ is preserved by $d\phi$.

The set $Q\cap\mathbb{H}^{3}$ is a (real) geodesic. Let $\beta(t)$ denote this
geodesic parameterized with respect to arc length so that $\beta(0)=q$. Then
define $(X_{3})_{p}=(d\pi)^{-1}\beta^{\prime}(0)$. There exists a positive
real-valued function $b(r,t)$ so that the metric ${\bf c_{3}}$ restricted to
$Q$ is of the form $dr^{2}+b^{2}(r,t)dt^{2}$. But since $\mathbb{R}$ acts by
isometries on $Q$ via translation along $\beta$, the function $b(r,t)$ is
independent of $t$. Then since the curvature of a complex geodesic is $-4$, we
have that $b(r)=\cosh(2r)$.

####  3.1.3. Conclusion

######  Theorem 3.1.

The complex hyperbolic manifold
$\mathbb{C}\mathbb{H}^{3}\setminus\mathbb{H}^{3}$ can be written as
$E\times(0,\infty)$ where $E\cong\mathbb{H}^{3}\times\mathbb{S}^{2}$ equipped
with the metric

(3.2) ${\bf
c_{3}}=\cosh^{2}(r)(dX_{1}^{2}+dX_{2}^{2})+\cosh^{2}(2r)dX_{3}^{2}+\sinh^{2}(r)(dX_{4}^{2}+dX_{5}^{2})+dr^{2}.$

In equation (3.2), $dX_{1}$ through $dX_{5}$ denote the covector fields dual
to the vector fields $X_{1}$ through $X_{5}$, respectively. Lastly, notice
that $dX_{1}^{2}+dX_{2}^{2}$ is the hyperbolic metric with constant sectional
curvature $-1$, and $dX_{4}^{2}+dX_{5}^{2}$ is the spherical metric with
constant sectional curvature $1$.

###  3.2. The warped product metric and curvature formulas in
$\mathbb{C}\mathbb{H}^{3}\setminus\mathbb{H}^{3}$

For some positive, increasing real-valued functions
$h,h_{r},v:(0,\infty)\to\mathbb{R}$ define

${\bf\mu_{r}}:=h^{2}(r)(dX_{1}^{2}+dX_{2}^{2})+h_{r}^{2}(r)dX_{3}^{2}+v^{2}(r)(dX_{4}^{2}+dX_{5}^{2})$

and

(3.3) ${\bf\mu}:={\bf\mu_{r}}+dr^{2}.$

Of course, ${\bf\mu}={\bf c_{3}}$ when $h=\cosh(r)$, $h_{r}=\cosh(2r)$, and
$v=\sinh(r)$.

Define an orthonormal basis $\\{Y_{i}\\}_{i=1}^{6}$ with respect to $\mu$ by

(3.4) $\displaystyle Y_{1}=\frac{1}{h}X_{1}\hskip
50.0ptY_{2}=\frac{1}{h}X_{2}\hskip 50.0ptY_{3}=\frac{1}{h_{r}}X_{3}$
$\displaystyle Y_{4}=\frac{1}{v}X_{4}\hskip 50.0ptY_{5}=\frac{1}{v}X_{5}\hskip
50.0ptY_{6}=X_{6}.$

Our goal is to compute formulas for the components of the $(4,0)$ curvature
tensor $R_{\mu}$ in terms of the warping functions $h,h_{r},$ and $v$ (this is
the content of Theorem 3.4). As a first step, we need to compute the
components of the $(4,0)$ curvature tensor $R_{\bf c_{3}}$ of the complex
hyperbolic metric with respect to the orthonormal basis given above. We can do
this with the help of formula (5.1). To use this formula note that, by
construction, we have that $JY_{4}=Y_{1}$, $JY_{5}=Y_{2}$, and $JY_{6}=Y_{3}$
(again, when the metric is ${\bf c_{3}}$, that is, when $h=\cosh(r)$,
$h_{r}=\cosh(2r)$, and $v=\sinh(r)$). Lastly, we use the notation

$R^{\bf c_{3}}_{ijkl}:=\langle R_{\bf
c_{3}}(Y_{i},Y_{j})Y_{k},Y_{l}\rangle_{\bf c_{3}}.$

Then, up to the symmetries of the curvature tensor, the nonzero components of
the $(4,0)$ curvature tensor $R_{\bf c_{3}}$ are

(3.5) $\displaystyle-4=$ $\displaystyle R^{\bf c_{3}}_{1414}=R^{\bf
c_{3}}_{2525}=R^{\bf c_{3}}_{3636}$ (3.6) $\displaystyle-1=$ $\displaystyle
R^{\bf c_{3}}_{1212}=R^{\bf c_{3}}_{1313}=R^{\bf c_{3}}_{1515}=R^{\bf
c_{3}}_{1616}=R^{\bf c_{3}}_{2323}=R^{\bf c_{3}}_{2424}$ $\displaystyle=R^{\bf
c_{3}}_{2626}=R^{\bf c_{3}}_{3434}=R^{\bf c_{3}}_{3535}=R^{\bf
c_{3}}_{4545}=R^{\bf c_{3}}_{4646}=R^{\bf c_{3}}_{5656}$ (3.7)
$\displaystyle-2=$ $\displaystyle R^{\bf c_{3}}_{1425}=R^{\bf
c_{3}}_{1436}=R^{\bf c_{3}}_{2536}$ (3.8) $\displaystyle-1=$ $\displaystyle
R^{\bf c_{3}}_{1245}=R^{\bf c_{3}}_{1346}=R^{\bf c_{3}}_{2356}=R^{\bf
c_{3}}_{1524}=R^{\bf c_{3}}_{1634}=R^{\bf c_{3}}_{2635}.$

###  3.3. Lie brackets

We now need to compute the values of the Lie brackets of the orthogonal basis
$\\{X_{i}\\}_{i=1}^{6}$. A first observation is that, by construction, each of
these vector fields is invariant under the flow of $\frac{\partial}{\partial
r}$. This implies that $[X_{i},X_{6}]=0$ for all $1\leq i\leq 6$. From this we
can deduce that

$\displaystyle[Y_{1},Y_{6}]=\frac{h^{\prime}}{h}$ $\displaystyle Y_{1}\hskip
30.0pt[Y_{2},Y_{6}]=\frac{h^{\prime}}{h}Y_{2}\hskip
30.0pt[Y_{3},Y_{6}]=\frac{h_{r}^{\prime}}{h_{r}}Y_{3}$
$\displaystyle[Y_{4},Y_{6}]=\frac{v^{\prime}}{v}Y_{4}\hskip
30.0pt[Y_{5},Y_{6}]=\frac{v^{\prime}}{v}Y_{5}.$

Next, we know that each Lie bracket is tangent to the level surfaces of $r$.
Thus, for all $1\leq i,j\leq 6$, the Lie bracket $[X_{i},X_{j}]$ has no
$X_{6}$ term. For all $1\leq i,j,k\leq 5$ define structure constants
$c_{ij}^{k}$ by

(3.9) $[X_{i},X_{j}]=\sum_{k=1}^{5}c_{ij}^{k}X_{k}.$

Two quick observations about the structure constants. The first is that
$c_{ij}^{k}=-c_{ji}^{k}$ due to the anti-symmetry of the Lie bracket. The
second observation is about the values of $c_{45}^{4}$ and $c_{45}^{5}$.
Recall the definitions for $\check{X}_{4}$ and $\check{X}_{5}$ from equation
(3.1). Then

(3.10)
$\left[\check{X}_{4},\check{X}_{5}\right]=\left[\frac{\partial}{\partial\theta},\frac{1}{\sin(\theta)}\frac{\partial}{\partial\psi}\right]=\frac{-\cos(\theta)}{\sin^{2}(\theta)}\frac{\partial}{\partial\psi}=-\cot(\theta)\check{X}_{5}.$

We therefore conclude that $c_{45}^{4}=0$ and $c_{45}^{5}=-\cot(\theta)$.

The following Theorem gives almost a full description of the values of the Lie
brackets. Some quantities are only defined up to sign, but this is sufficient
to compute the curvature formulas in Theorem 3.4. The interested reader can
find the proof of Theorem 3.2 in Section 6.

######  Theorem 3.2.

The values for the Lie brackets in equation (3.9) are

$\displaystyle[X_{1},X_{2}]=\pm X_{1}\hskip
101.0pt[X_{1},X_{3}]=\mp\cot(\theta)X_{2}+X_{4}$
$\displaystyle[X_{1},X_{4}]=X_{3}\mp X_{5}\hskip
84.0pt[X_{1},X_{5}]=-\cot(\theta)X_{2}\pm X_{4}$
$\displaystyle[X_{2},X_{3}]=\pm\cot(\theta)X_{1}+X_{5}\hskip
48.0pt[X_{2},X_{4}]=0$
$\displaystyle[X_{2},X_{5}]=\cot(\theta)X_{1}+X_{3}\hskip
58.0pt[X_{3},X_{4}]=-X_{1}\pm\cot(\theta)X_{5}$
$\displaystyle[X_{3},X_{5}]=-X_{2}\mp\cot(\theta)X_{4}\hskip
50.0pt[X_{4},X_{5}]=-\cot(\theta)X_{5}$

In the above equations, all of the $\pm$ and $\mp$ signs are related. For
example, if it is the case that $[X_{1},X_{2}]=X_{1}$, then
$[X_{1},X_{4}]=X_{3}-X_{5}$ and so on.

###  3.4. The Levi-Civita connection and formulas for the (4,0) curvature
tensor $R_{\mu}$

In this Subsection we first compute the Levi-Civita connection $\nabla$
associated to the metric $\mu$ with respect to the frame $(Y_{i})_{i=1}^{6}$.
The difficult part in all of this is computing the Lie brackets in Theorem
3.2. From there it is now a simple calculation using formula (5.3) to prove
the following Theorem.

######  Theorem 3.3.

The Levi-Civita connection $\nabla$ compatible with $\mu$ is determined by the
$36$ equations

$\displaystyle\bullet\nabla_{Y_{1}}Y_{1}=$
$\displaystyle\mp\frac{1}{h}Y_{2}-\frac{h^{\prime}}{h}Y_{6}\hskip
26.0pt\bullet\nabla_{Y_{3}}Y_{4}=-\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}-\frac{v}{hh_{r}}\right)Y_{1}\pm\frac{1}{h_{r}}\cot(\theta)Y_{5}$
$\displaystyle\bullet\nabla_{Y_{1}}Y_{2}=$
$\displaystyle\pm\frac{1}{h}Y_{1}\hskip
60.0pt\bullet\nabla_{Y_{3}}Y_{5}=-\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}-\frac{v}{hh_{r}}\right)Y_{2}\mp\frac{1}{h_{r}}\cot(\theta)Y_{4}$
$\displaystyle\bullet\nabla_{Y_{1}}Y_{3}=$
$\displaystyle\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{4}\hskip
44.0pt\bullet\nabla_{Y_{4}}Y_{1}=-\frac{1}{2}\left(\frac{h}{h_{r}v}+\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{3}$
$\displaystyle\bullet\nabla_{Y_{1}}Y_{4}=$
$\displaystyle-\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{3}\mp\frac{1}{h}Y_{5}\hskip
112.0pt\bullet\nabla_{Y_{4}}Y_{2}=0$
$\displaystyle\bullet\nabla_{Y_{1}}Y_{5}=$
$\displaystyle\pm\frac{1}{h}Y_{4}\hskip
132.0pt\bullet\nabla_{Y_{4}}Y_{3}=\frac{1}{2}\left(\frac{h}{h_{r}v}+\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{1}$
$\displaystyle\bullet\nabla_{Y_{2}}Y_{1}=$ $\displaystyle 0\hskip
238.0pt\bullet\nabla_{Y_{4}}Y_{4}=-\frac{v^{\prime}}{v}Y_{6}$
$\displaystyle\bullet\nabla_{Y_{2}}Y_{2}=$
$\displaystyle-\frac{h^{\prime}}{h}Y_{6}\hskip
232.0pt\bullet\nabla_{Y_{4}}Y_{5}=0$
$\displaystyle\bullet\nabla_{Y_{2}}Y_{3}=$
$\displaystyle\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{5}\hskip
117.0pt\bullet\nabla_{Y_{5}}Y_{1}=\frac{1}{v}\cot(\theta)Y_{2}$
$\displaystyle\bullet\nabla_{Y_{2}}Y_{4}=$ $\displaystyle 0\hskip
91.0pt\bullet\nabla_{Y_{5}}Y_{2}=-\frac{1}{v}\cot(\theta)Y_{1}-\frac{1}{2}\left(\frac{h}{h_{r}v}+\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{3}$
$\displaystyle\bullet\nabla_{Y_{2}}Y_{5}=$
$\displaystyle-\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{3}\hskip
40.0pt\bullet\nabla_{Y_{5}}Y_{3}=\frac{1}{2}\left(\frac{h}{h_{r}v}+\frac{h_{r}}{hv}+\frac{v}{hh_{r}}\right)Y_{2}$
$\displaystyle\bullet\nabla_{Y_{3}}Y_{1}=$
$\displaystyle\pm\frac{1}{h_{r}}\cot(\theta)Y_{2}+\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{vh}-\frac{v}{hh_{r}}\right)Y_{4}\hskip
40.0pt\bullet\nabla_{Y_{5}}Y_{4}=\frac{1}{v}\cot(\theta)Y_{5}$
$\displaystyle\bullet\nabla_{Y_{3}}Y_{2}=$
$\displaystyle\mp\frac{1}{h_{r}}\cot(\theta)Y_{1}+\frac{1}{2}\left(\frac{h}{h_{r}v}-\frac{h_{r}}{hv}-\frac{v}{hh_{r}}\right)Y_{5}$
$\displaystyle\bullet\nabla_{Y_{3}}Y_{3}=$
$\displaystyle-\frac{h_{r}^{\prime}}{h_{r}}Y_{6}\hskip
148.0pt\bullet\nabla_{Y_{5}}Y_{5}=-\frac{1}{v}\cot(\theta)Y_{4}-\frac{v^{\prime}}{v}Y_{6}$
$\displaystyle\bullet\nabla_{Y_{1}}Y_{6}=$
$\displaystyle\frac{h^{\prime}}{h}Y_{1}\hskip
3.0pt\bullet\nabla_{Y_{2}}Y_{6}=\frac{h^{\prime}}{h}Y_{2}\hskip
3.0pt\bullet\nabla_{Y_{3}}Y_{6}=\frac{h_{r}^{\prime}}{h_{r}}Y_{3}\hskip
3.0pt\bullet\nabla_{Y_{4}}Y_{6}=\frac{v^{\prime}}{v}Y_{4}\hskip
3.0pt\bullet\nabla_{Y_{5}}Y_{6}=\frac{v^{\prime}}{v}Y_{5}$
$\displaystyle\bullet\,$ $\displaystyle
0=\nabla_{Y_{6}}Y_{1}=\nabla_{Y_{6}}Y_{2}=\nabla_{Y_{6}}Y_{3}=\nabla_{Y_{6}}Y_{4}=\nabla_{Y_{6}}Y_{5}=\nabla_{Y_{6}}Y_{6}$

By combining Theorem 3.3 with equation (1.2), and remembering that
$Y_{6}=X_{6}=\frac{\partial}{\partial r}$ and
$X_{4}=\frac{\partial}{\partial\theta}$, we compute the following formulas for
the $(4,0)$ curvature tensor $R_{\mu}$. As in equations (3.5) through (3.8) we
use the notation:

$R^{\mu}_{ijkl}:=\langle R_{\mu}(Y_{i},Y_{j})Y_{k},Y_{l}\rangle_{\mu}.$

######  Theorem 3.4.

In terms of the basis given in equation (3.4), the only independent nonzero
components of the $(4,0)$ curvature tensor $R_{\mu}$ are the following:

$\displaystyle
R^{\mu}_{1212}=-\left(\frac{h^{\prime}}{h}\right)^{2}-\frac{1}{h^{2}}\hskip
30.0ptR^{\mu}_{4545}=-\left(\frac{v^{\prime}}{v}\right)^{2}+\frac{1}{v^{2}}\hskip
30.0ptR^{\mu}_{1515}=R^{\mu}_{2424}=-\frac{h^{\prime}v^{\prime}}{hv}$
$\displaystyle
R^{\mu}_{1414}=R^{\mu}_{2525}=-\frac{v^{\prime}h^{\prime}}{vh}-\left(\frac{-v^{2}}{4h^{2}h_{r}^{2}}-\frac{h^{2}}{4v^{2}h_{r}^{2}}+\frac{3h_{r}^{2}}{4v^{2}h^{2}}-\frac{1}{2v^{2}}+\frac{1}{2h^{2}}-\frac{1}{2h_{r}^{2}}\right)$
$\displaystyle
R^{\mu}_{3434}=R^{\mu}_{3535}=-\frac{v^{\prime}h_{r}^{\prime}}{vh_{r}}-\left(\frac{-v^{2}}{4h^{2}h_{r}^{2}}+\frac{3h^{2}}{4v^{2}h_{r}^{2}}-\frac{h_{r}^{2}}{4v^{2}h^{2}}-\frac{1}{2v^{2}}-\frac{1}{2h^{2}}+\frac{1}{2h_{r}^{2}}\right)$
$\displaystyle
R^{\mu}_{1313}=R^{\mu}_{2323}=-\frac{h^{\prime}h_{r}^{\prime}}{hh_{r}}-\left(\frac{3v^{2}}{4h^{2}h_{r}^{2}}-\frac{h^{2}}{4v^{2}h_{r}^{2}}-\frac{h_{r}^{2}}{4v^{2}h^{2}}+\frac{1}{2v^{2}}+\frac{1}{2h^{2}}+\frac{1}{2h_{r}^{2}}\right)$
$\displaystyle R^{\mu}_{1616}=R^{\mu}_{2626}=-\frac{h^{\prime\prime}}{h}\hskip
50.0ptR^{\mu}_{3636}=-\frac{h_{r}^{\prime\prime}}{h_{r}}\hskip
50.0ptR^{\mu}_{4646}=R^{\mu}_{5656}=-\frac{v^{\prime\prime}}{v}$
$\displaystyle
R^{\mu}_{1436}=R^{\mu}_{2536}=\frac{1}{2h_{r}}\left[\left(\frac{h}{v}\right)^{\prime}-\left(\frac{v}{h}\right)^{\prime}-\left(\frac{h_{r}^{2}}{vh}\right)^{\prime}\right]$
$\displaystyle
R^{\mu}_{1634}=R^{\mu}_{2635}=\frac{1}{2h}\left[-\left(\frac{h_{r}}{v}\right)^{\prime}+\left(\frac{v}{h_{r}}\right)^{\prime}+\left(\frac{h^{2}}{vh_{r}}\right)^{\prime}\right]$
$\displaystyle
R^{\mu}_{1346}=R^{\mu}_{2356}=\frac{-1}{2v}\left[\left(\frac{h}{h_{r}}\right)^{\prime}+\left(\frac{h_{r}}{h}\right)^{\prime}+\left(\frac{v^{2}}{hh_{r}}\right)^{\prime}\right]$
$\displaystyle
R^{\mu}_{1425}=\frac{1}{2v^{2}}-\frac{1}{2h^{2}}-\frac{h_{r}^{2}}{2h^{2}v^{2}}$
$\displaystyle
R^{\mu}_{1245}=-\frac{1}{4}\left(\frac{h^{2}}{h_{r}^{2}v^{2}}+\frac{h_{r}^{2}}{h^{2}v^{2}}+\frac{v^{2}}{h^{2}h_{r}^{2}}+\frac{2}{h^{2}}+\frac{2}{h_{r}^{2}}-\frac{2}{v^{2}}\right)$
$\displaystyle
R^{\mu}_{1524}=-\frac{1}{4}\left(\frac{-h^{2}}{h_{r}^{2}v^{2}}+\frac{h_{r}^{2}}{h^{2}v^{2}}-\frac{v^{2}}{h^{2}h_{r}^{2}}-\frac{2}{h_{r}^{2}}\right).$

It is a tedious exercise in hyperbolic trigonometric identities to check that,
when $h=\cosh(r)$, $h_{r}=\cosh(2r)$, and $v=\sinh(r)$, the above formulas
reduce to the constants in equations (3.5) through (3.8). Also, note that the
first nine equations above give the sectional curvatures of the coordinate
planes, while the last six equations are formulas for the nonzero mixed terms.

Finally, notice that the above curvature formulas contain all of the formulas
that arise in the analogous $\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{H}^{n}$
for general $n$. In general, one can write the complex hyperbolic metric ${\bf
c_{n}}$ as

${\bf c_{n}}=\cosh^{2}(r){\bf
h_{n-1}}+\cosh^{2}(2r)dX_{n}^{2}+\sinh^{2}(r){\bf\sigma_{n-1}}+dr^{2}$

and the corresponding warped-product metric as

${\bf\mu_{n}}=h^{2}(r){\bf
h_{n-1}}+h_{r}^{2}(r)dX_{n}^{2}+v^{2}(r){\bf\sigma_{n-1}}+dr^{2}$

where ${\bf\sigma_{n-1}}$ is the round metric on $\mathbb{S}^{n-1}$ and the
vector field $X_{n}$ is defined in the same manner as $X_{3}$. The curvature
formulas for the base $\mathbb{H}^{n}$ are encoded in the formulas for
$R^{\mu}_{1212}$ and $R^{\mu}_{1313}$. All curvature formulas for the
$\mathbb{S}^{n-1}$ factor are contained in the term $R^{\mu}_{4545}$. Note
that neither of these cases contain any mixed terms. Adding in the curvature
formulas above of the form $R^{\mu}_{1414}$, $R^{\mu}_{3434}$,
$R^{\mu}_{1425}$, $R^{\mu}_{1245}$, and $R^{\mu}_{1524}$ gives all curvature
formulas for $h^{2}(r){\bf
h_{n-1}}+h_{r}^{2}(r)dX_{n}^{2}+v^{2}(r){\bf\sigma_{n-1}}$ (this is where most
of the mixed terms appear). And then all of the formulas above containing a
“6” give the rest of the curvature formulas for $\mu_{n}$.

##  4\. Curvature formulas for warped product metrics on
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{k}$

As mentioned in the Introduction, for simplicity we are going to restrict
ourselves to the case when $n=5$ and $k=2$. These are the smallest choices for
$n$ and $k$ which capture every formula for the curvature tensor in the
general case, so nothing is lost with this restriction (see the comments after
Theorem 4.3 for more discussion).

###  4.1. Expressing the metric in $\mathbb{C}\mathbb{H}^{5}$ in spherical
coordinates about $\mathbb{C}\mathbb{H}^{2}$

Let ${\bf c_{5}}$ denote the complex hyperbolic metric on
$\mathbb{C}\mathbb{H}^{5}$ normalized to have constant holomorphic sectional
curvature $-4$. Since $\mathbb{C}\mathbb{H}^{2}$ is a complete totally
geodesic submanifold of the negatively curved manifold
$\mathbb{C}\mathbb{H}^{5}$, there exists an orthogonal projection map
$\pi:\mathbb{C}\mathbb{H}^{5}\to\mathbb{C}\mathbb{H}^{2}$. This map $\pi$ is a
fiber bundle whose fibers are totally geodesic $6$-planes isometric to
$\mathbb{C}\mathbb{H}^{3}$.

For $r>0$ let $E(r)$ denote the $r$-neighborhood of
$\mathbb{C}\mathbb{H}^{2}$. Then $E(r)$ is a real hypersurface in
$\mathbb{C}\mathbb{H}^{5}$, and consequently we can decompose ${\bf c_{5}}$ as

${\bf c_{5}}=({\bf c_{5}})_{r}+dr^{2}$

where $({\bf c_{5}})_{r}$ is the induced Riemannian metric on $E(r)$. Let
$\pi_{r}:E(r)\to\mathbb{C}\mathbb{H}^{2}$ denote the restriction of $\pi$ to
$E(r)$. Note that $\pi_{r}$ is an $\mathbb{S}^{5}$-bundle whose fiber over any
point $q\in\mathbb{C}\mathbb{H}^{2}$ is (topologically) the $5$-sphere of
radius $r$ in the totally geodesic $6$-plane $\pi^{-1}(q)$. The tangent bundle
splits as an orthogonal sum $\mathcal{V}(r)\oplus\mathcal{H}(r)$ where
$\mathcal{V}(r)$ is tangent to the $5$-sphere $\pi_{r}^{-1}(q)$ and
$\mathcal{H}(r)$ is the orthogonal complement to $\mathcal{V}(r)$. Note that
this copy of $\mathbb{S}^{5}$ does not have constant sectional curvature equal
to $1$, but rather it is an example of a Berger sphere. This will be discussed
further below.

For $r,s>0$ there exists a diffeomorphism $\phi_{sr}:E(s)\to E(r)$ induced by
the geodesic flow along the totally geodesic $6$-planes orthogonal to
$\mathbb{C}\mathbb{H}^{2}$. Fix $p\in E(r)$ arbitrary, let
$q=\pi(p)\in\mathbb{C}\mathbb{H}^{2}$, and let $\gamma$ be the unit speed
geodesic such that $\gamma(0)=q$ and $\gamma(r)=p$. In what follows, all
computations are considered in the tangent space $T_{p}E(r)$.

Note that $\mathcal{V}(r)$ is tangent to both $E(r)$ and the totally geodesic
$6$-plane $\pi^{-1}(q)$. Then since $\pi^{-1}(q)$ is preserved by the geodesic
flow, we have that $d\phi_{sr}$ takes $\mathcal{V}(s)$ to $\mathcal{V}(r)$.
Consider the complex geodesic $P=exp(span(\frac{\partial}{\partial
r},J\frac{\partial}{\partial r}))$. P intersects $E(r)$ orthogonally, and
$P\cap E(r)$ is isometric to a circle of radius $r$. Thus, since a complex
geodesic has curvature $-4$, there exists a suitable identification
$P\cong\mathbb{S}^{1}\times(0,\infty)$ where the metric ${\bf c_{5}}$
restricted to $P$ can be written as

$\frac{1}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}$

where $d\theta^{2}$ denotes the round metric on the unit circle
$\mathbb{S}^{1}$. Note that the presence of the $``1/4"$ is to make the metric
complete when extended to the core $\mathbb{C}\mathbb{H}^{2}$.

Notice that $\frac{\partial}{\partial\theta}$ is a vector field on the five
sphere $\mathbb{S}^{5}$ mentioned above. More generally, thinking of
$\mathbb{S}^{5}$ as the unit sphere in $\mathbb{C}^{3}$ with respect to the
usual Hermitian metric, there is an obvious free action of the circle
$\mathbb{S}^{1}$ on $\mathbb{S}^{5}$. The unit tangent vector field with
respect to this action corresponds to the vector field
$\frac{\partial}{\partial\theta}$ above. This action fibers $\mathbb{S}^{5}$
over the complex projective plane $\mathbb{C}\mathbb{P}^{2}$, and the
Riemannian submersion metric on this fiber bundle is an example of a Berger
sphere (see [FarrellJones] pg. 59 for more details). Let $\alpha(t)$ be a unit
speed geodesic in $\mathbb{S}^{5}$ orthogonal to $J\frac{\partial}{\partial
r}$ such that $\alpha(0)=p$. Then $\exp_{p}(\alpha^{\prime}(0),\partial r)$
forms a totally real totally geodesic 2-plane in $\mathbb{C}\mathbb{H}^{5}$.
Thus the curvature of this $2$-plane is $-1$. Since the direction of $\alpha$
orthogonal to $J\frac{\partial}{\partial r}$ was arbitrary, we can write the
Riemannian metric $({\bf c_{5}})_{r}$ restricted to $\mathcal{V}(r)$ as

$\sinh^{2}(r){\bf p_{2}}+\frac{1}{4}\sinh^{2}(2r)d\theta^{2}$

where ${\bf p_{2}}$ denotes the complex projective metric on
$\mathbb{C}\mathbb{P}^{2}$.

Now let $\beta(t)$ be any unit speed geodesic in $\mathbb{C}\mathbb{H}^{2}$
such that $\beta(0)=q$. Then
$Q=exp(span(\beta^{\prime}(0),\gamma^{\prime}(0)))$ is a totally real totally
geodesic submanifold of $\mathbb{C}\mathbb{H}^{5}$, and thus
$K(\beta^{\prime},\gamma^{\prime})=-1$. Therefore, the metric ${\bf c_{5}}$
restricted to $Q$ can be written as $\cosh^{2}(r)dt^{2}+dr^{2}$. But since
$\gamma$ was arbitrary, we can write the metric on the 5-dimensional
submanifold determined by $\mathbb{C}\mathbb{H}^{2}$ and
$\frac{\partial}{\partial r}$ as $\cosh^{2}{\bf c_{2}}+dr^{2}$. This leads to
the following.

######  Theorem 4.1.

The complex hyperbolic manifold
$\mathbb{C}\mathbb{H}^{5}\setminus\mathbb{C}\mathbb{H}^{2}$ can be written as
$E\times(0,\infty)$ where $E\cong\mathbb{C}\mathbb{H}^{2}\times\mathbb{S}^{5}$
equipped with the metric

(4.1) ${\bf c_{5}}=\cosh^{2}(r){\bf c_{2}}+\sinh^{2}(r){\bf
p_{2}}+\frac{1}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}.$

###  4.2. The warped product metric, orthonormal basis, and curvature formulas
in $\mathbb{C}\mathbb{H}^{5}\setminus\mathbb{C}\mathbb{H}^{2}$

For some positive, increasing real-valued functions
$h,v,v_{r}:(0,\infty)\to\mathbb{R}$ define the Riemannian metrics

${\bf\gamma_{r,\theta}}=h^{2}(r){\bf c_{2}}+v^{2}(r){\bf p_{2}}$
${\bf\gamma_{r}}:={\bf\gamma_{r,\theta}}+\frac{1}{4}v_{r}^{2}(r)d\theta^{2}$

and

(4.2) ${\bf\gamma}:={\bf\gamma_{r}}+dr^{2}.$

Of course, ${\bf\gamma}={\bf c_{5}}$ when $h=\cosh(r)$, $v=\sinh(r)$, and
$v_{r}=\sinh(2r)$.

For the remainder of this Section, fix
$p=(q_{1},\bar{q},r)\in\mathbb{C}\mathbb{H}^{2}\times\mathbb{S}^{5}\times(0,\infty)\cong\mathbb{C}\mathbb{H}^{5}\setminus\mathbb{C}\mathbb{H}^{2}$,
and write $\bar{q}\in\mathbb{S}^{5}$ as $(q_{2},\theta)$ where
$q_{2}\in\mathbb{C}\mathbb{P}^{2}$ and $\theta\in\mathbb{S}^{1}$. Let
$(\check{X}_{1},\check{X}_{2},\check{X}_{3},\check{X}_{4})$ be an orthonormal
collection of vector fields near $q_{1}\in\mathbb{C}\mathbb{H}^{2}$ which
satisfies:

  1. (1)

$[\check{X}_{i},\check{X}_{j}]_{q_{1}}=0$ for all $1\leq i,j\leq 4$.

  2. (2)

$J\check{X}_{2}|_{q_{1}}=\check{X}_{1}|_{q_{1}}$ and
$J\check{X}_{4}|_{q_{1}}=\check{X}_{3}|_{q_{1}}$.

Define an analogous collection of vector fields
$(\check{X}_{5},\check{X}_{6},\check{X}_{7},\check{X}_{8})$ about
$q_{2}\in\mathbb{C}\mathbb{P}^{2}$ so that
$J\check{X}_{6}|_{q_{2}}=\check{X}_{5}|_{q_{2}}$,
$J\check{X}_{8}|_{q_{2}}=\check{X}_{7}|_{q_{2}}$, and
$[\check{X}_{i},\check{X}_{j}]_{q_{2}}=0$ for all $5\leq i,j\leq 8$. Extend
both collections to vector fields $(X_{1},\ldots,X_{8})$ near $p$. Lastly, let
$X_{9}=\frac{\partial}{\partial\theta}$ and $X_{10}=\frac{\partial}{\partial
r}$.

Define an orthonormal basis $\\{Y_{i}\\}_{i=1}^{8}$ with respect to $\gamma$
by

(4.3) $\displaystyle Y_{1}=\frac{1}{h}X_{1}\hskip
25.0ptY_{2}=\frac{1}{h}X_{2}\hskip 25.0ptY_{3}=\frac{1}{h}X_{3}\hskip
25.0ptY_{4}=\frac{1}{h}X_{4}\hskip 25.0ptY_{5}=\frac{1}{v}X_{5}$
$\displaystyle Y_{6}=\frac{1}{v}X_{6}\hskip 25.0ptY_{7}=\frac{1}{v}X_{7}\hskip
25.0ptY_{8}=\frac{1}{v}X_{8}\hskip
25.0ptY_{9}=\frac{1}{\frac{1}{2}v_{r}}X_{9}\hskip 25.0ptY_{10}=X_{10}.$

Our goal is to compute formulas for the components of the $(4,0)$ curvature
tensor $R_{\gamma}$ in terms of the warping functions $h,v,$ and $v_{r}$. As a
first step, we need to compute the components of the $(4,0)$ curvature tensor
$R_{\bf c_{5}}$ of the complex hyperbolic metric with respect to the
orthonormal basis given above. Just as in Section 3 we can do this with the
help of formula (5.1). To use this formula note that, by construction, we have
that $JY_{2}=Y_{1}$, $JY_{4}=Y_{3}$, $JY_{6}=Y_{5}$, $JY_{8}=Y_{7}$, and
$JY_{10}=Y_{9}$ at the point $p$ (and again, when the metric is ${\bf c_{5}}$.
So, when $h=\cosh(r)$, $v=\sinh(r)$, and $v_{r}=\sinh(2r)$). Lastly, we use
the notation

$R^{\bf c_{5}}_{ijkl}:=\langle R_{\bf
c_{5}}(Y_{i},Y_{j})Y_{k},Y_{l}\rangle_{\bf c_{5}}.$

Then, up to the symmetries of the curvature tensor, the nonzero components of
the $(4,0)$ curvature tensor $R_{\bf c_{5}}$ are

(4.4) $\displaystyle-4=$ $\displaystyle R^{\bf c_{5}}_{1212}=R^{\bf
c_{5}}_{3434}=R^{\bf c_{5}}_{5656}=R^{\bf c_{5}}_{7878}=R^{\bf
c_{5}}_{9,10,9,10}$ (4.5) $\displaystyle-1=$ $\displaystyle R^{\bf
c_{5}}_{ijij}\text{ where
}\\{i,j\\}\not\in\\{\\{1,2\\},\\{3,4\\},\\{5,6\\},\\{7,8\\},\\{9,10\\}\\}$
(4.6) $\displaystyle-2=$ $\displaystyle R^{\bf c_{5}}_{ijkl}\text{ where
}(i,j)\neq(k,l)\in\\{(1,2),(3,4),(5,6),(7,8),(9,10)\\}$ (4.7)
$\displaystyle-1=$ $\displaystyle R^{\bf c_{5}}_{ikjl}\text{ where
}(i,j)\neq(k,l)\in\\{(1,2),(3,4),(5,6),(7,8),(9,10)\\}$ (4.8) $\displaystyle
1=$ $\displaystyle R^{\bf c_{5}}_{iljk}\text{ where
}(i,j)\neq(k,l)\in\\{(1,2),(3,4),(5,6),(7,8),(9,10)\\}$

Let us quickly note that, since $\mathbb{C}\mathbb{P}^{2}$ is dual to
$\mathbb{C}\mathbb{H}^{2}$, we have the following curvature formulas for
$R_{\bf p_{2}}$:

$\displaystyle 4$ $\displaystyle=R^{\bf p_{2}}_{5656}=R^{\bf p_{2}}_{7878}$
$\displaystyle 1$ $\displaystyle=R^{\bf p_{2}}_{5757}=R^{\bf
p_{2}}_{5858}=R^{\bf p_{2}}_{6767}=R^{\bf p_{2}}_{6868}$ $\displaystyle 2$
$\displaystyle=R^{\bf p_{2}}_{5678}=2R^{\bf p_{2}}_{5768}=-2R^{\bf
p_{2}}_{5867}.$

In the above formulas, $R^{\bf p_{2}}_{ijkl}:=\langle R_{\bf
p_{2}}(Y_{i},Y_{j})Y_{k},Y_{l}\rangle_{\bf p_{2}}$ and with the abuse of
notation of $Y_{i}$ denoting the restriction of $Y_{i}$ to
$\mathbb{C}\mathbb{P}^{2}$.

###  4.3. Lie brackets and curvature formulas for ${\bf\gamma_{r,\theta}}$

The vector fields $\\{X_{i}\\}_{i=1}^{10}$ form an orthogonal frame near $p$
which satisfies the following properties (at $p$):

  1. (1)

$[X_{i},X_{j}]$ is tangent to the level surfaces of $r$ for $1\leq i,j\leq 9$.

  2. (2)

$[X_{i},X_{j}]$ is tangent to $\mathbb{C}\mathbb{H}^{2}\times\mathbb{S}^{1}$
for $1\leq i,j\leq 4$, where
$\mathbb{S}^{1}\cong\text{exp}_{p}(J\frac{\partial}{\partial r})$.

  3. (3)

$[X_{i},X_{j}]$ is tangent to $\mathbb{S}^{5}$ for $5\leq i,j\leq 8$.

  4. (4)

$[X_{i},X_{10}]=0$ since $X_{i}$ is invariant under the flow of
$\frac{\partial}{\partial r}$ for $1\leq i\leq 9$.

  5. (5)

$[X_{i},X_{9}]=0$ since $X_{i}$ is invariant under the flow of
$\frac{\partial}{\partial\theta}$ for $1\leq i\leq 8$.

  6. (6)

$[X_{i},X_{j}]=0$ for $i\in\\{1,2,3,4\\}$ and $j\in\\{5,6,7,8\\}$ since these
vector fields were defined via inclusion.

By the above points, and since $[\check{X}_{i},\check{X}_{j}]_{p}=0$ for all
$1\leq i,j\leq 8$, there exist structure constants $c_{ij}$ such that
$[X_{i},X_{j}]_{p}=c_{ij}X_{9}$. Note that $c_{ij}=-c_{ji}$. The following
Lemma provides the values for the structure constants.

######  Lemma 4.2.

The values for the structure constants are $c_{12}=c_{34}=c_{56}=c_{78}=2$,
and all other (independent) structure constants are equal to zero.

A quick note is that by “independent” structure constants we just mean that,
obviously, $c_{21}=c_{43}=c_{65}=c_{87}=-2\neq 0$.

###### Proof.

All of the structure constants can be found by combining formula (5.7) with
the curvature formulas (4.6) through (4.8). To see that $c_{12}=2$, we combine
equation (4.6) with (5.7) to obtain

$\displaystyle 4=2R^{\bf c_{5}}_{10,9,1,2}$
$\displaystyle=0+0+\langle[Y_{1},Y_{2}],Y_{9}\rangle_{\bf
c_{5}}\left(\ln\left[\frac{\frac{1}{4}\sinh^{2}(2r)}{\cosh^{2}(r)}\right]\right)^{\prime}$
$\displaystyle=\frac{c_{12}\sinh(2r)}{\cosh^{2}(r)}\left(\frac{\cosh(r)}{\sinh(r)}\right)=2c_{12}.$

An analogous argument shows that $c_{34}=c_{56}=c_{78}=2$. To see that
$c_{13}=c_{14}=c_{23}=c_{24}=c_{57}=c_{58}=c_{67}=c_{68}=0$ we use the same
equations as above, but note that the left hand side is now $0$ instead of
$4$.

Lastly, to see that $c_{15}=0$, note that

$\displaystyle 0=R^{\bf c_{5}}_{10,1,5,9}$
$\displaystyle=0+\langle[Y_{5},Y_{1}],Y_{9}\rangle_{\bf
c_{5}}\left(\ln\left[\frac{\frac{1}{2}\sinh(2r)}{\sinh(r)}\right]\right)^{\prime}+0$
$\displaystyle=\frac{-\frac{1}{2}c_{15}\sinh(2r)}{\sinh(r)\cosh(r)}(\ln
2\cosh(r))^{\prime}=-c_{15}\tanh(r).$

The argument that the remaining structure constants are $0$ is identical to
the argument above. ∎

We now, for some fixed $r$ and $\theta$, compute the components of the $(4,0)$
curvature tensor $R_{\bf\lambda_{r,\theta}}$ with respect to the orthonormal
frame $\\{Y_{i}\\}_{i=1}^{8}$. Since $[X_{i},X_{j}]=0$ for $i\in\\{1,2,3,4\\}$
and $j\in\\{5,6,7,8\\}$, the metric ${\bf\gamma_{r,\theta}}$ is a product
metric. Then since the $(4,0)$ curvature tensor scales like the metric, up to
the symmetries of the curvature tensor the only nonzero components of
$R_{\bf\gamma_{r,\theta}}$ are

$\displaystyle
R^{\bf\gamma_{r,\theta}}_{1212}=R^{\bf\gamma_{r,\theta}}_{3434}=-\frac{4}{h^{2}}\hskip
68.0ptR^{\bf\gamma_{r,\theta}}_{1313}=R^{\bf\gamma_{r,\theta}}_{1414}=R^{\bf\gamma_{r,\theta}}_{2323}=R^{\bf\gamma_{r,\theta}}_{2424}=-\frac{1}{h^{2}}$
$\displaystyle
R^{\bf\gamma_{r,\theta}}_{5656}=R^{\bf\gamma_{r,\theta}}_{7878}=\frac{4}{v^{2}}\hskip
84.0ptR^{\bf\gamma_{r,\theta}}_{5757}=R^{\bf\gamma_{r,\theta}}_{5858}=R^{\bf\gamma_{r,\theta}}_{6767}=R^{\bf\gamma_{r,\theta}}_{6868}=\frac{1}{v^{2}}$
$\displaystyle
R^{\bf\gamma_{r,\theta}}_{1234}=2R^{\bf\gamma_{r,\theta}}_{1324}=-2R^{\bf\gamma_{r,\theta}}_{1423}=-\frac{2}{h^{2}}\hskip
40.0ptR^{\bf\gamma_{r,\theta}}_{5678}=2R^{\bf\gamma_{r,\theta}}_{5768}=-2R^{\bf\gamma_{r,\theta}}_{5867}=\frac{2}{v^{2}}.$

In particular, note that mixed terms of the form
$R^{\bf\gamma_{r,\theta}}_{1256}$ are $0$.

###  4.4. Curvature formulas for ${\bf\gamma_{r}}$

Formulas (5.4) through (5.7) allow us to compute the $(4,0)$ curvature tensor
$R_{\gamma}$ in terms of $R_{\gamma_{r}}$. We use a very different approach
from Section 3 to compute the nonzero components of $R_{\gamma_{r}}$. The
background for our current computations can be found in [3] pg. 235-242. The
metric $\gamma_{r}$ is a Riemannian submersion metric with (horizontal) base
$\gamma_{r,\theta}$ and (vertical) fiber $\frac{1}{4}v_{r}^{2}d\theta^{2}$. So
our approach is to compute the A and T tensors of $\gamma_{r}$, and to then
use Theorem 9.28 in [3] to compute the components of $R_{\gamma_{r}}$.

First, the T tensor is identically zero since the vertical
$\mathbb{S}^{1}$-fibers are totally geodesic. The argument for this is
identical to that in Section 6 of [1].

We now compute the $A$ tensor associated with $\gamma_{r}$. By ([3] Prop.
9.24) we have that

$A_{X_{1}}X_{2}=\frac{1}{2}\mathscr{V}[X_{1},X_{2}]=X_{9}.$

Analogously, $A_{X_{3}}X_{4}=A_{X_{5}}X_{6}=A_{X_{7}}X_{8}=X_{9}$ and
$A_{X_{i}}X_{j}=0$ if
$\\{i,j\\}\not\in\\{\\{1,2\\},\\{3,4\\},\\{5,6\\},\\{7,8\\}\\}$. Also, by ([3]
eqn. 9.21a) we have $A_{X_{9}}X_{i}=0$ for $1\leq i\leq 8$.

Now, by ([3] eqn. 9.21d) we see that

$\langle A_{X_{1}}X_{9},X_{2}\rangle_{\gamma_{r}}=-\langle
A_{X_{1}}X_{2},X_{9}\rangle_{\gamma_{r}}=-\frac{1}{4}v_{r}^{2}.$

By this same equation we know that there are no other nonzero components of
$A_{X_{1}}X_{9}$. Therefore,
$A_{X_{1}}X_{9}=-\frac{1}{4}\frac{v_{r}^{2}}{h^{2}}X_{2}$. Analogously, we
have that

$\displaystyle A_{X_{2}}X_{9}=\frac{1}{4}\frac{v_{r}^{2}}{h^{2}}X_{1}\hskip
25.0ptA_{X_{3}}X_{9}=-\frac{1}{4}\frac{v_{r}^{2}}{h^{2}}X_{4}\hskip
25.0ptA_{X_{4}}X_{9}=\frac{1}{4}\frac{v_{r}^{2}}{h^{2}}X_{3}$ $\displaystyle
A_{X_{5}}X_{9}=$ $\displaystyle-\frac{1}{4}\frac{v_{r}^{2}}{v^{2}}X_{6}\hskip
15.0ptA_{X_{6}}X_{9}=\frac{1}{4}\frac{v_{r}^{2}}{v^{2}}X_{5}\hskip
15.0ptA_{X_{7}}X_{9}=-\frac{1}{4}\frac{v_{r}^{2}}{v^{2}}X_{8}\hskip
15.0ptA_{X_{8}}X_{9}=\frac{1}{4}\frac{v_{r}^{2}}{v^{2}}X_{7}$

We are now ready to use Theorem 9.28 from [3] to compute the nonzero
components of $R_{\gamma_{r}}$. By (9.28c) we have that

$\langle R_{\gamma_{r}}(X_{1},X_{9})X_{1},X_{9}\rangle_{\gamma_{r}}=\langle
A_{X_{1}}X_{9},A_{X_{1}}X_{9}\rangle_{\gamma_{r}}=\frac{1}{16}\frac{v_{r}^{4}}{h^{2}}$

and thus

(4.9) $R^{\bf\gamma_{r}}_{1919}=\frac{4}{h^{2}v_{r}^{2}}\langle
R_{\gamma_{r}}(X_{1},X_{9})X_{1},X_{9}\rangle_{\gamma_{r}}=\frac{v_{r}^{2}}{4h^{4}}.$

Identically,
$R^{\bf\gamma_{r}}_{2929}=R^{\bf\gamma_{r}}_{3939}=R^{\bf\gamma_{r}}_{4949}=\frac{v_{r}^{2}}{4h^{4}}$.
Also, a completely analogous computation shows that

(4.10)
$R^{\bf\gamma_{r}}_{5959}=R^{\bf\gamma_{r}}_{6969}=R^{\bf\gamma_{r}}_{7979}=R^{\bf\gamma_{r}}_{8989}=\frac{v_{r}^{2}}{4v^{4}}.$

By (9.28f) we have that

$\displaystyle\langle
R_{\gamma_{r}}(X_{1},X_{2})X_{1},X_{2}\rangle_{\gamma_{r}}$
$\displaystyle=\langle
R_{\gamma_{r,\theta}}(X_{1},X_{2})X_{1},X_{2}\rangle_{\gamma_{r,\theta}}-3\langle
A_{X_{1}}X_{2},A_{X_{1}}X_{2}\rangle_{\gamma_{r}}$ $\displaystyle=\langle
R_{\gamma_{r,\theta}}(X_{1},X_{2})X_{1},X_{2}\rangle_{\gamma_{r,\theta}}-\frac{3}{4}v_{r}^{2}$

and thus

(4.11)
$R^{\gamma_{r}}_{1212}=R^{\gamma_{r,\theta}}_{1212}-\frac{3v_{r}^{2}}{4h^{4}}=-\frac{4}{h^{2}}-\frac{3v_{r}^{2}}{4h^{4}}=R^{\gamma_{r}}_{3434}.$

An identical argument shows that

(4.12) $\displaystyle
R^{\gamma_{r}}_{5656}=R^{\gamma_{r}}_{7878}=\frac{4}{v^{2}}-\frac{3v_{r}^{2}}{4v^{4}}.$

Since $A_{X_{i}}X_{j}=0$ if
$\\{i,j\\}\not\in\\{\\{1,2\\},\\{3,4\\},\\{5,6\\},\\{7,8\\}\\}$ the above
argument also provides

(4.13) $R^{\gamma_{r}}_{ijij}=R^{\gamma_{r,\theta}}_{ijij}\hskip 30.0pt\text{
if }\hskip
10.0pt\\{i,j\\}\not\in\\{\\{1,2\\},\\{3,4\\},\\{5,6\\},\\{7,8\\}\\}.$

We now compute the mixed terms of $R_{\gamma_{r}}$.

$\displaystyle\langle
R_{\gamma_{r}}(X_{1},X_{2})X_{3},X_{4}\rangle_{\gamma_{r}}$
$\displaystyle=\langle
R_{\gamma_{r,\theta}}(X_{1},X_{2})X_{3},X_{4}\rangle_{\gamma_{r,\theta}}-2\langle
A_{X_{1}}X_{2},A_{X_{3}}X_{4}\rangle_{\gamma_{r}}$ $\displaystyle=\langle
R_{\gamma_{r,\theta}}(X_{1},X_{2})X_{3},X_{4}\rangle_{\gamma_{r,\theta}}-\frac{1}{2}v_{r}^{2}$

and therefore

(4.14)
$R^{\gamma_{r}}_{1234}=R^{\gamma_{r,\theta}}_{1234}-\frac{v_{r}^{2}}{2h^{4}}=-\frac{2}{h^{2}}-\frac{v_{r}^{2}}{2h^{4}}=2R^{\gamma_{r}}_{1324}=-2R^{\gamma_{r}}_{1423}.$

Identically

(4.15)
$R^{\gamma_{r}}_{5678}=R^{\gamma_{r,\theta}}_{5678}-\frac{v_{r}^{2}}{2v^{4}}=\frac{2}{v^{2}}-\frac{v_{r}^{2}}{2v^{4}}=2R^{\gamma_{r}}_{5768}=-2R^{\gamma_{r}}_{5867}.$

Now

$\displaystyle\langle
R_{\gamma_{r}}(X_{1},X_{2})X_{5},X_{6}\rangle_{\gamma_{r}}$
$\displaystyle=\langle
R_{\gamma_{r,\theta}}(X_{1},X_{2})X_{5},X_{6}\rangle_{\gamma_{r,\theta}}-2\langle
A_{X_{1}}X_{2},A_{X_{5}}X_{6}\rangle_{\gamma_{r}}=-\frac{1}{2}v_{r}^{2}$

and hence

(4.16)
$R^{\gamma_{r}}_{1256}=-\frac{v_{r}^{2}}{2h^{2}v^{2}}=R^{\gamma_{r}}_{1278}=R^{\gamma_{r}}_{3456}=R^{\gamma_{r}}_{3478}.$

Finally, the same argument yields

(4.17) $\displaystyle
R^{\gamma_{r}}_{1526}=R^{\gamma_{r}}_{1728}=R^{\gamma_{r}}_{3546}=R^{\gamma_{r}}_{3748}=-\frac{v_{r}^{2}}{4h^{4}v^{4}}$
(4.18) $\displaystyle
R^{\gamma_{r}}_{1625}=R^{\gamma_{r}}_{1827}=R^{\gamma_{r}}_{3645}=R^{\gamma_{r}}_{3847}=\frac{v_{r}^{2}}{4h^{4}v^{4}}.$

###  4.5. Curvature formulas for $\gamma$

Combining equations (4.9) through (4.18) with formulas (5.4) through (5.7)
proves the following Theorem.

######  Theorem 4.3.

In terms of the basis given in equation (4.3), the only independent nonzero
components of the $(4,0)$ curvature tensor $R_{\gamma}$ are given by the
following formulas, where $i\in\\{1,2,3,4\\}$, $k\in\\{5,6,7,8\\}$,
$(i,j)\in\\{(1,2),(3,4)\\}$, and $(k,l)\in\\{(5,6),(7,8)\\}$.

$\displaystyle
R^{\gamma}_{1212}=R^{\gamma}_{3434}=-\left(\frac{h^{\prime}}{h}\right)^{2}-\frac{4}{h^{2}}-\frac{3v_{r}^{2}}{4h^{4}}\hskip
20.0ptR^{\gamma}_{5656}=R^{\gamma}_{7878}=-\left(\frac{v^{\prime}}{v}\right)^{2}+\frac{4}{v^{2}}-\frac{3v_{r}^{2}}{4v^{4}}$
$\displaystyle
R^{\gamma}_{i9i9}=-\frac{h^{\prime}v_{r}^{\prime}}{hv_{r}}+\frac{v_{r}^{2}}{4h^{4}}\hskip
40.0ptR^{\gamma}_{k9k9}=-\frac{v^{\prime}v_{r}^{\prime}}{vv_{r}}+\frac{v_{r}^{2}}{4v^{4}}\hskip
40.0ptR^{\gamma}_{ikik}=-\frac{h^{\prime}v^{\prime}}{hv}$ $\displaystyle
R^{\gamma}_{1313}=R^{\gamma}_{1414}=R^{\gamma}_{2323}=R^{\gamma}_{2424}=-\left(\frac{h^{\prime}}{h}\right)^{2}-\frac{1}{h^{2}}\hskip
40.0pt\text{(continued on next page)}$ $\displaystyle
R^{\gamma}_{5757}=R^{\gamma}_{5858}=R^{\gamma}_{6767}=R^{\gamma}_{6868}=-\left(\frac{v^{\prime}}{v}\right)^{2}+\frac{1}{v^{2}}$
$\displaystyle R^{\gamma}_{i,10,i,10}=-\frac{h^{\prime\prime}}{h}\hskip
60.0ptR^{\gamma}_{k,10,k,10}=-\frac{v^{\prime\prime}}{v}\hskip
60.0ptR^{\gamma}_{9,10,9,10}=-\frac{v_{r}^{\prime\prime}}{v_{r}}$
$\displaystyle
R^{\gamma}_{1234}=2R^{\gamma}_{1324}=-2R^{\gamma}_{1423}=-\frac{2}{h^{2}}-\frac{v_{r}^{2}}{2h^{4}}$
$\displaystyle
R^{\gamma}_{5678}=2R^{\gamma}_{5768}=-2R^{\gamma}_{5867}=\frac{2}{v^{2}}-\frac{v_{r}^{2}}{2v^{4}}$
$\displaystyle
R^{\gamma}_{ijkl}=2R^{\gamma}_{ikjl}=-2R^{\gamma}_{iljk}=-\frac{v_{r}^{2}}{2h^{2}v^{2}}$
$\displaystyle
R^{\gamma}_{i,j,9,10}=2R^{\gamma}_{i,9,j,10}=-2R^{\gamma}_{i,10,j,9}=-\frac{v_{r}}{h^{2}}\left(\ln\frac{v_{r}}{h}\right)^{\prime}$
$\displaystyle
R^{\gamma}_{k,l,9,10}=2R^{\gamma}_{k,9,l,10}=-2R^{\gamma}_{k,10,l,9}=-\frac{v_{r}}{v^{2}}\left(\ln\frac{v_{r}}{v}\right)^{\prime}.$

Unlike Section 3, this time is is a much simpler exercise in hyperbolic
trigonometric identities to check that, when $h=\cosh(r)$, $v=\sinh(r)$, and
$v_{r}=\sinh(2r)$, the above formulas reduce to the constants in equations
(4.4) through (4.8).

Finally, notice that the above curvature formulas contain all of the formulas
that arise in the analogous
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{k}$ for general $n$
and $k$. In general, one can write the complex hyperbolic metric ${\bf c_{n}}$
as

${\bf c_{n}}=\cosh^{2}(r){\bf c_{k}}+\sinh^{2}(r){\bf
p_{n-k-1}}+\frac{1}{4}\sinh^{2}(2r)d\theta^{2}+dr^{2}$

and the corresponding warped-product metric as

${\bf\gamma_{n}}=h^{2}(r){\bf c_{k}}+v^{2}(r){\bf
p_{n-k-1}}+\frac{1}{4}v_{r}^{2}(r)d\theta^{2}+dr^{2}$

where ${\bf p_{n-k-1}}$ is the complex projective metric on
$\mathbb{C}\mathbb{P}^{n-k-1}$. The curvature formulas for the base
$\mathbb{C}\mathbb{H}^{k}$ are encoded in the formulas for
$R^{\gamma}_{1212},R^{\gamma}_{1313},$ and $R^{\gamma}_{1234}$. The analogous
formulas for $\mathbb{C}\mathbb{P}^{n-k-1}$ are contained in
$R^{\gamma}_{5656},R^{\gamma}_{5757},$ and $R^{\gamma}_{5678}$. Adding in the
curvature formulas above of the form $R^{\gamma}_{ijij}$ and
$R^{\gamma}_{ijkl}$ gives all curvature formulas for $h^{2}(r){\bf
c_{k}}+v^{2}(r){\bf p_{n-k-1}}$. And then all of the formulas above containing
either a “9” or a “10” give the rest of the curvature formulas for
${\bf\gamma_{n}}$.

###  4.6. The exceptional case
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{n-2}$

Notice that, when $k=n-2$, there are no sectional curvatures of $\gamma$ of
the form

$-\left(\frac{v^{\prime}}{v}\right)^{2}+\frac{1}{v}^{2}.$

That is because we can write
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{n-2}\cong\mathbb{C}\mathbb{H}^{n-2}\times\mathbb{S}^{3}\times(0,\infty)$,
and $\mathbb{C}\mathbb{P}^{1}$ (the base of the Hopf fibration) has constant
holomorphic curvature $4$. So the purpose of this subsection is to prove the
following:

######  Lemma 4.4.

There do not exist functions $h,v,$ and $v_{r}$ that, when inserted into
equation (4.2), yield a complete finite volume Riemannian metric on
$\mathbb{C}\mathbb{H}^{n}\setminus\mathbb{C}\mathbb{H}^{n-2}$ with nonpositive
sectional curvature.

###### Proof.

Since we are removing a copy of $\mathbb{C}\mathbb{H}^{n-2}$ from
$\mathbb{C}\mathbb{H}^{n}$, for the metric to be complete we need to define
$h,v,$ and $v_{r}$ on $(-\infty,\infty)$. These functions need to be positive
for the metric to be Riemannian, and they need to be non-decreasing for there
to be any chance of nonpositive curvature. For any hope of finite volume, we
need all of the following limits to hold:

$\lim_{r\to-\infty}h,h^{\prime},v,v^{\prime},v_{r},v_{r}^{\prime}=0.$

Now, from the formula for the $R_{5656}$ term in Theorem 4.3 we must have that

(4.19)
$\displaystyle\frac{4-(v^{\prime})^{2}}{v^{2}}-\frac{3v_{r}^{2}}{4v^{4}}$
$\displaystyle\leq 0$ $\displaystyle\iff\hskip
30.0pt\frac{16-4(v^{\prime})^{2}}{3}$
$\displaystyle\leq\left(\frac{v_{r}}{v}\right)^{2}.$

In particular, we see that a necessary requirement for nonpositive curvature
is that

(4.20) $\lim_{r\to-\infty}\frac{v_{r}}{v}>1.$

From the formula for the $R_{k9k9}$ term in Theorem 4.3 we must have that

$\displaystyle-\frac{v^{\prime}v_{r}^{\prime}}{vv_{r}}+\frac{v_{r}^{2}}{4v^{4}}$
$\displaystyle\leq 0$ (4.21) $\displaystyle\Longrightarrow\hskip
30.0pt\frac{3v_{r}^{2}}{4v^{4}}$
$\displaystyle\leq\frac{3v^{\prime}v_{r}^{\prime}}{vv_{r}}.$

Comparing equations (4.19) and (4.21), we see that

$\frac{4-(v^{\prime})^{2}}{v^{2}}\leq\frac{3v^{\prime}v_{r}^{\prime}}{vv_{r}}\hskip
20.0pt\Longrightarrow\hskip
20.0pt4-(v^{\prime})^{2}\leq(3v^{\prime}v_{r}^{\prime})\cdot\frac{v}{v_{r}}$

is also a necessary requirement for nonpositive curvature. But as
$r\to-\infty$, we know that $4-(v^{\prime})^{2}\to 4$ and
$3v^{\prime}v_{r}^{\prime}\to 0$. Thus, we must have that

(4.22) $\lim_{r\to-\infty}\frac{v}{v_{r}}=\infty\hskip
20.0pt\Longrightarrow\hskip 20.0pt\lim_{r\to-\infty}\frac{v_{r}}{v}=0.$

Equations (4.20) and (4.22) provide a contradiction, proving the Lemma.

∎

##  5\. Preliminaries

###  5.1. Formula for the curvature tensor of $\mathbb{C}\mathbb{H}^{n}$ in
terms of the complex structure $J$

The components of the (4,0) curvature tensor of the complex hyperbolic metric
$g$ can be expressed in terms of $g$ and the complex structure $J$. The
following formula can be found in [8] or in Section 5 of [1] (recall Remark
1.2 from the Introduction). In this formula $X,Y,Z$, and $W$ are arbitrary
vector fields.

(5.1) $\displaystyle\langle R_{g}$ $\displaystyle(X,Y)Z,W\rangle_{g}=\langle
X,W\rangle_{g}\langle Y,Z\rangle_{g}-\langle X,Z\rangle_{g}\langle
Y,W\rangle_{g}$ $\displaystyle+\langle X,JW\rangle_{g}\langle
Y,JZ\rangle_{g}-\langle X,JZ\rangle_{g}\langle Y,JW\rangle_{g}+2\langle
X,JY\rangle_{g}\langle W,JZ\rangle_{g}.$

###  5.2. Koszul’s formula for the Levi-Civita connection

Let $X,Y,$ and $Z$ denote vector fields on a Riemannian manifold $(M,g)$. The
following is the well-known “Koszul formula” for the values of the Levi-Civita
connection $\nabla$ (which can be found on pg. 55 of [4])

(5.2) $\displaystyle\langle\nabla_{Y}X,Z\rangle_{g}=$
$\displaystyle\,\frac{1}{2}(X\langle Y,Z\rangle_{g}+Y\langle
Z,X\rangle_{g}-Z\langle X,Y\rangle_{g}$
$\displaystyle-\langle[X,Z],Y\rangle_{g}-\langle[Y,Z],X\rangle_{g}-\langle[X,Y],Z\rangle_{g}).$

In this paper we will usually be considering an orthonormal frame $(Y_{i})$.
In this setting we know that $\langle Y_{i},Y_{j}\rangle_{g}=\delta_{ij}$,
where $\delta_{ij}$ denotes Kronecker’s delta. Therefore the first three terms
on the right hand side of formula (5.2) are all zero. Thus, in an orthonormal
frame, formula (5.2) reduces to

(5.3)
$\langle\nabla_{Y}X,Z\rangle_{g}=-\frac{1}{2}\left(\langle[X,Z],Y\rangle_{g}+\langle[Y,Z],X\rangle_{g}+\langle[X,Y],Z\rangle_{g}\right).$

###  5.3. General curvature formulas for warped product metrics

The curvature formulas below, which were worked out by Belegradek in [2] and
stated in Appendix B of [1], apply to metrics of the form $g=g_{r}+dr^{2}$ on
manifolds of the form $E\times I$ where $I$ is an open interval and $E$ is a
manifold. The formulas are true provided that for each point $q\in E$ there
exists a local frame $\\{X_{i}\\}$ on a neighborhood $U_{q}$ in $E$ which is
$g_{r}$-orthogonal for each $r$. Such a family of metrics $(E,g_{r})$ is
called simultaneously diagonalizable. Let

$h_{i}(r):=\sqrt{g_{r}(X_{i},X_{i})}.$

Then the local frame $\\{Y_{i}\\}$ defined by

$Y_{i}=\frac{1}{h_{i}}X_{i}$

is a $g_{r}$-orthonormal frame on $U_{q}$ for any value of $r$. We then have
the following formulas for the (4,0) curvature tensor $R_{g}$ in terms of the
(4,0) curvature tensor $R_{g_{r}}$, the collection $\\{h_{i}\\}$, and the Lie
brackets $[Y_{i},Y_{j}]$. Note that $\langle,\rangle$ is used to denote the
metric $g$ and $\partial r=\frac{\partial}{\partial r}$.

(5.4) $\displaystyle\langle R_{g}(Y_{i},Y_{j})Y_{i},Y_{j}\rangle$
$\displaystyle=\langle
R_{g_{r}}(Y_{i},Y_{j})Y_{i},Y_{j}\rangle-\frac{h_{i}^{\prime}h_{j}^{\prime}}{h_{i}h_{j}}$
(5.5) $\displaystyle\langle R_{g}(Y_{i},Y_{j})Y_{k},Y_{l}\rangle$
$\displaystyle=\langle R_{g_{r}}(Y_{i},Y_{j})Y_{k},Y_{l}\rangle\hskip
10.0pt\text{if }\\{i,j\\}\neq\\{k,l\\}$ (5.6) $\displaystyle\langle
R_{g}(Y_{i},\partial r)Y_{i},\partial r\rangle$
$\displaystyle=-\frac{h_{i}^{\prime\prime}}{h_{i}}\hskip 40.0pt\langle
R_{g}(Y_{i},\partial r)Y_{j},\partial r\rangle=0\hskip 10.0pt\text{if }i\neq
j$ (5.7) $\displaystyle 2\langle R\left(\partial
r,Y_{i}\right)Y_{j},Y_{k}\rangle$
$\displaystyle=\langle[Y_{i},Y_{k}],Y_{j}\rangle\left(\ln\frac{h_{j}}{h_{k}}\right)^{\prime}+\langle[Y_{j},Y_{i}],Y_{k}\rangle\left(\ln\frac{h_{k}}{h_{j}}\right)^{\prime}$
$\displaystyle+\langle[Y_{j},Y_{k}],Y_{i}\rangle\left(\ln\frac{h_{i}^{2}}{h_{j}h_{k}}\right)^{\prime}.$

###  5.4. The Nijenhuis Tensor

In Sections 3 and 4 we will be explicitly dealing with
$\mathbb{C}\mathbb{H}^{n}$. Since the almost complex structure on
$\mathbb{C}\mathbb{H}^{n}$ is integrable, we have that the Nijenhuis Tensor is
identically equal to zero. Explicitly, for any vector fields $X$ and $Y$ on
$\mathbb{C}\mathbb{H}^{n}$, we have that

(5.8) $0=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]$

where $J$ denotes the complex structure on $\mathbb{C}\mathbb{H}^{n}$.

##  6\. Computations for the Lie brackets for
$\mathbb{C}\mathbb{H}^{3}\setminus\mathbb{H}^{3}$

The whole purpose of this Section is to prove Theorem 3.2.

###### Proof of Theorem 3.2.

There are $5\times 10=50$ structure constants to compute from equation (3.9).
From equation (3.10) we know that $c_{45}^{4}=0$ and
$c_{45}^{5}=-\cot(\theta)$, leaving 48 unknown structure constants.

We can combine formula (5.7) with equations (3.5) through (3.8) to compute
many of the constants. As a first example, note that

$\displaystyle 0$ $\displaystyle=2R^{\bf
c_{3}}_{6131}=0+2\langle[Y_{3},Y_{1}],Y_{1}\rangle_{\bf
c_{3}}\left(\ln\frac{h}{h_{r}}\right)^{\prime}$
$\displaystyle=-\frac{2c_{13}^{1}}{\cosh(2r)}\left(\ln\frac{\cosh(r)}{\cosh(2r)}\right)^{\prime}$

and thus $c_{13}^{1}=0$. We can analogously show

$\displaystyle 0$
$\displaystyle=c_{14}^{1}=c_{15}^{1}=c_{23}^{2}=c_{24}^{2}=c_{25}^{2}=c_{13}^{3}=c_{23}^{3}$
$\displaystyle=c_{34}^{3}=c_{35}^{3}=c_{14}^{4}=c_{24}^{4}=c_{34}^{4}=c_{15}^{5}=c_{25}^{5}=c_{35}^{5}.$

This narrows us down to $32$ unknown constants.

Continuing with the same formula and equations, we have that

$\displaystyle 0$ $\displaystyle=2R^{\bf
c_{3}}_{6145}=0+0+\langle[Y_{4},Y_{5}],Y_{1}\rangle_{\bf
c_{3}}\left(\ln\frac{\cosh^{2}(r)}{\sinh^{2}(r)}\right)^{\prime}$
$\displaystyle=\frac{c_{45}^{1}h}{v^{2}}\left(\ln\frac{\cosh^{2}(r)}{\sinh^{2}(r)}\right)^{\prime}$

and therefore $c_{45}^{1}=0$. Analogously,
$c_{45}^{2}=c_{45}^{3}=c_{12}^{3}=c_{12}^{4}=c_{12}^{5}=0$. This reduces us to
$26$ unknowns. But we can also use the same curvature formulas here, but with
the indices permuted, to derive some simple equations relating some of the
constants. For example,

$\displaystyle 0$ $\displaystyle=R^{\bf
c_{3}}_{6415}=0+\langle[Y_{1},Y_{4}],Y_{5}\rangle_{\bf
c_{3}}\left(\ln\frac{\sinh(r)}{\cosh(r)}\right)^{\prime}+\langle[Y_{1},Y_{5}],Y_{4}\rangle_{\bf
c_{3}}\left(\ln\frac{\sinh(r)}{\cosh(r)}\right)^{\prime}$
$\displaystyle=\frac{1}{h}(c_{14}^{5}+c_{15}^{4})\left(\ln\frac{\sinh(r)}{\cosh(r)}\right)^{\prime}$

and thus $c_{14}^{5}=-c_{15}^{4}$. Analogously, we have the identities

$\displaystyle c_{24}^{5}=-c_{25}^{4}$ $\displaystyle\hskip
40.0ptc_{34}^{5}=-c_{35}^{4}\hskip 40.0ptc_{13}^{2}=-c_{23}^{1}$
$\displaystyle c_{14}^{2}=-c_{24}^{1}\hskip 40.0ptc_{15}^{2}=-c_{25}^{1}.$

Combining formula (5.7) with the fact that $2R^{\bf c_{3}}_{6413}=2$ gives
that

$\displaystyle 2$ $\displaystyle=\langle[Y_{4},Y_{3}],Y_{1}\rangle_{\bf
c_{3}}\left(\ln\frac{h}{h_{r}}\right)^{\prime}+\langle[Y_{1},Y_{4}],Y_{3}\rangle_{\bf
c_{3}}\left(\ln\frac{h_{r}}{h}\right)^{\prime}+\langle[Y_{1},Y_{3}],Y_{4}\rangle_{\bf
c_{3}}\left(\ln\frac{v^{2}}{hh_{r}}\right)^{\prime}$
$\displaystyle=-\frac{c_{34}^{1}h}{h_{r}v}\left(\ln\frac{h}{h_{r}}\right)^{\prime}+\frac{c_{14}^{3}h_{r}}{hv}\left(\ln\frac{h_{r}}{h}\right)^{\prime}+\frac{c_{13}^{4}v}{hh_{r}}\left(\ln\frac{v^{2}}{hh_{r}}\right)^{\prime}$
$\displaystyle=\left(\frac{-c_{34}^{1}\cosh(r)}{\cosh(2r)\sinh(r)}-\frac{c_{14}^{3}\cosh(2r)}{\cosh(r)\sinh(r)}\right)\left(\tanh(r)-2\tanh(2r)\right)$
$\displaystyle+\frac{c_{13}^{4}\sinh(r)}{\cosh(r)\cosh(2r)}\left(2\coth(r)-\tanh(r)-2\tanh(2r)\right).$

Now one consults equation (5.9) in [9] to see that the solutions to this
equation are

$c_{13}^{4}=1\hskip 40.0ptc_{14}^{3}=1\hskip 40.0ptc_{34}^{1}=-1.$

In exactly the same manner we can use $R^{\bf c_{3}}_{6253}$ with equation
(5.9) in [9] to compute

$c_{23}^{5}=1\hskip 40.0ptc_{25}^{3}=1\hskip 40.0ptc_{35}^{2}=-1.$

This leaves $20$ unknowns together with the $6$ identities listed above. Now,
using $R^{\bf c_{3}}_{6135}$ we have that

$\displaystyle 0$ $\displaystyle=\langle[Y_{1},Y_{5}],Y_{3}\rangle_{\bf
c_{3}}\left(\ln\frac{h_{r}}{v}\right)^{\prime}+\langle[Y_{3},Y_{1}],Y_{5}\rangle_{\bf
c_{3}}\left(\ln\frac{v}{h_{r}}\right)^{\prime}+\langle[Y_{3},Y_{5}],Y_{1}\rangle_{\bf
c_{3}}\left(\ln\frac{h^{2}}{h_{r}v}\right)^{\prime}$
$\displaystyle=\left(\frac{c_{15}^{3}\cosh(2r)}{\cosh(r)\sinh(r)}+\frac{c_{13}^{5}\sinh(r)}{\cosh(r)\cosh(2r)}\right)\left(2\tanh(2r)-\coth(r)\right)$
$\displaystyle+\frac{c_{35}^{1}\cosh(r)}{\cosh(2r)\sinh(r)}\left(2\tanh(r)-2\tanh(2r)-\coth(r)\right).$

One can check that the only solution to this equation (which holds for all
values of $r$) is $c_{13}^{5}=c_{15}^{3}=c_{35}^{1}=0$. Analogously, we can
use $\mathbb{R}^{\bf c_{3}}_{6234}$ to show that
$c_{23}^{4}=c_{24}^{3}=c_{34}^{2}=0$. These equations reduce us to $14$
unknowns.

This is as much information as we can gain from formula (5.7). So we next turn
to the Nijenhuis Tensor (5.8). First applying this to $(Y_{1},Y_{2})$, we have

$\displaystyle 0$
$\displaystyle=[Y_{1},Y_{2}]-J[Y_{4},Y_{2}]-J[Y_{1},Y_{5}]-[Y_{4},Y_{5}]$
$\displaystyle=\frac{1}{h}(c_{12}^{1}Y_{1}+c_{12}^{2}Y_{2})+J\left(\frac{c_{24}^{1}}{v}Y_{1}+\frac{c_{24}^{5}}{h}Y_{5}\right)-J\left(\frac{c_{15}^{2}}{v}Y_{2}+\frac{c_{15}^{4}}{h}Y_{4}\right)+\frac{1}{v}\cot(\theta)Y_{5}$
$\displaystyle=\frac{1}{h}\left(c_{12}^{1}-c_{15}^{4}\right)Y_{1}+\frac{1}{h}(c_{12}^{2}+c_{24}^{5})Y_{2}-\frac{c_{24}^{1}}{v}Y_{4}+\frac{1}{v}(c_{15}^{2}+\cot(\theta))Y_{5}.$

Therefore, we have that

$c_{24}^{1}=0=-c_{14}^{2}\hskip
25.0ptc_{15}^{2}=-\cot(\theta)=-c_{25}^{1}\hskip
25.0ptc_{12}^{1}=c_{15}^{4}=-c_{14}^{5}\hskip 25.0ptc_{12}^{2}=-c_{24}^{5}.$

We can also apply the Nijenhuis tensor to the pairs $(Y_{1},Y_{3})$ and
$(Y_{2},Y_{3})$, but these are much less productive. These applications only
give us the pair of identities

$c_{13}^{2}=-c_{34}^{5}\hskip 50.0ptc_{23}^{1}=-c_{35}^{4}$

the former of which comes from the pair $(Y_{1},Y_{3})$, and the latter from
the pair $(Y_{2},Y_{3})$.

At this stage, we have reduced our $10$ Lie brackets as follows:

$\displaystyle[X_{1},X_{2}]=c_{12}^{1}X_{1}+c_{12}^{2}X_{2}\hskip
64.0pt[X_{1},X_{3}]=c_{13}^{2}X_{2}+X_{4}$
$\displaystyle[X_{1},X_{4}]=X_{3}-c_{12}^{1}X_{5}\hskip
77.0pt[X_{1},X_{5}]=-\cot(\theta)X_{2}+c_{12}^{1}X_{4}$
$\displaystyle[X_{2},X_{3}]=-c_{13}^{2}X_{1}+X_{5}\hskip
70.0pt[X_{2},X_{4}]=-c_{12}^{2}X_{5}$
$\displaystyle[X_{2},X_{5}]=\cot(\theta)X_{1}+X_{3}+c_{12}^{2}X_{4}\hskip
26.0pt[X_{3},X_{4}]=-X_{1}-c_{13}^{2}X_{5}$
$\displaystyle[X_{3},X_{5}]=-X_{2}+c_{13}^{2}X_{4}\hskip
70.0pt[X_{4},X_{5}]=-\cot(\theta)X_{5}.$

Notice that, using the known identities, we can reduce the system to three
unknowns: $c_{12}^{1},c_{12}^{2}$, and $c_{13}^{2}$. All that is left is to
show that $c_{12}^{1}=\pm 1$, $c_{12}^{2}=0$, and
$c_{13}^{2}=\mp\cot(\theta)$.

At this point we have exhausted all of our “easy” options. The only way to
obtain new relationships between the structure constants is to compute new
components of $R_{\bf c_{3}}$. To do this, one needs to first use equation
(5.2) with the values for the Lie brackets given above to compute the Levi-
Civita connection $\nabla$ compatible with ${\bf c_{3}}$. Of course, these
formulas will contain the constants $c_{12}^{1},c_{12}^{2}$, and $c_{13}^{2}$.
But when the correct values for these constants are inserted, these formulas
will reduce to those of Theorem 3.3. Then once one has computed $\nabla$, they
can use those values to compute the components of $R_{\bf c_{3}}$.

The first component that will be useful is $R^{\bf c_{3}}_{1212}$:

$\displaystyle-1=$ $\displaystyle R^{\bf
c_{3}}_{1212}=\langle\nabla_{Y_{2}}\nabla_{Y_{1}}Y_{1}-\nabla_{Y_{1}}\nabla_{Y_{2}}Y_{1}+\nabla_{[Y_{1},Y_{2}]}Y_{1},Y_{2}\rangle_{\bf
c_{3}}$ $\displaystyle=$
$\displaystyle\langle\nabla_{Y_{2}}\left(\frac{-c_{12}^{1}}{\cosh(r)}Y_{2}-\tanh(r)Y_{6}\right)-\nabla_{Y_{1}}\left(\frac{-c_{12}^{2}}{\cosh(r)}Y_{2}\right)$
$\displaystyle+\frac{c_{12}^{1}}{\cosh(r)}\nabla_{Y_{1}}Y_{1}+\frac{c_{12}^{2}}{\cosh(r)}\nabla_{Y_{2}}Y_{1},Y_{1}\rangle_{\bf
c_{3}}$ $\displaystyle=$
$\displaystyle-\frac{\sinh^{2}(r)}{\cosh^{2}(r)}-\frac{((c_{12}^{1})^{2}+(c_{12}^{2})^{2})}{\cosh^{2}(r)}$
$\displaystyle\Longrightarrow\hskip
30.0pt(c_{12}^{1})^{2}+(c_{12}^{2})^{2}=1.$

The next component that we use is $R^{\bf c_{3}}_{1512}$. We will skip the
details and just note that

$0=R^{\bf c_{3}}_{1512}=\frac{c_{12}^{2}}{\sinh(r)\cosh(r)}\cdot\cot(\theta)$

which implies that $c_{12}^{2}=0$. Combining this with the first equation
shows that $c_{12}^{1}=\pm 1$. Finally, to compute $c_{13}^{2}$ we use $R^{\bf
c_{3}}_{1412}$:

$0=R^{\bf
c_{3}}_{1412}=\frac{-c_{13}^{2}}{\sinh(r)\cosh(r)}-\frac{c_{12}^{1}}{\sinh(r)\cosh(r)}\cdot\cot(\theta).$

Therefore

$c_{13}^{2}=-(\pm 1)\cot(\theta)=\mp\cot(\theta).$

∎

### Acknowledgements

The author would like to thank J.F. Lafont for both suggesting this problem
and for many helpful discussions about the work contained in this paper. The
author would also like to thank I. Belegradek, J. Meyer, and B. Tshishiku for
various comments which aided in this research.

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