# A note on hyperbolic flows in sub-Riemannian Structures

Chengbo Li  School of science, Tianjin University, Tianjin, 300072, P.R.China;
email<EMAIL_ADDRESS>
###### Abstract.

The _curvature_ and the _reduced curvature_ are basic differential invariants
of the pair (Hamiltonian system, Lagrange distribution) on the symplectic
manifold. It is shown in [4] that the negativity of the reduced curvature
implies the hyperbolicity of any compact invariant set of the Hamiltonian flow
restricted to a prescribed energy level. We consider the Hamiltonian flows of
the curve of least action of natural mechanical systems in sub-Riemannian
structures with symmetries.We give sufficient conditions for the reduced flows
(after reduction of the first integrals induced from the symmetries) to be
hyperbolic and show new examples of Anosov flows. This result is a
generalization of [8] and a partial generalization of [13] on magnetic flows.

##  1\. introduction

A prime example of Anosov flow is the geodesic flow on a compact Riemannian
manifold with negative sectional curvature ([6]). It describes inertial motion
of a point particle confined to the manifold. In this context, magnetic flows,
the flows generated by special forces, were discussed more than 30 years ago
by Anosov and Sinai [7]. They were studied recently by Gouda [8], Grognet
[12], M. and P. Paternain [9] and M. P. Wojtkowski [13]. In the last
reference, the potential and the so-called Gaussian thermostats of external
fields were also considered. .

In the present note, we focus on the hyperbolicity of the flows associated
with a natural mechanical system in a sub-Riemannian structure with
multidimensional symmetries. In this case, the sub-Riemannian structures are
reduced to a Riemannian manifold with a (vector-valued) magnetic field. We
give sufficient conditions for the reduced Hamiltonian flows (after the
reduction of the first integrals) to be hyperbolic in terms of the Riemannian
curvature tensor and the magnetic field. As a consequence, a class of Anosov
flows are also given.

In the second section, we formulate the main results of the note. We firstly
introduce the notion of a dynamical Lagrangian distribution and then discuss
the reduction after the first integrals. The key point is that we can
construct the (reduced) curvature maps (forms) for the (reduced) dynamical
Lagrangian distribution based on the work [14] and [3]. The negativity of
reduced curvature forms implies the hyperbolicity of the Hamiltonian flows
([4]). Applying this criteria we give sufficient conditions for the reduced
Hamiltonian flows to be hyperbolic, based on an expression of the reduced
curvature forms via the Riemannian curvature tensor and the magnetic field.

The last section is devoted to the proofs of the main results. We apply the
similar technique as in [10] to give the proof of the expression of the
reduced curvature forms and the sufficient conditions of hyperbolic flows then
easily follows.

##  2\. main results

###  2.1. Dynamical Lagrangian distributions

Let $M$ be an even dimensional symplectic manifold endowed with a symplectic
form $\sigma$. A _Lagrange distribution_ $\Delta\subset TM$ is a smooth vector
sub-bundle of $TM$ such that each fiber $\Delta_{x}=\Delta\cap T_{x}M,\ x\in
M$ is a Lagrangian subspace of the symplectic space $T_{x}M$. Basic examples
are cotangent bundles endowed with the standard symplectic structure and the
“vertical”distribution:

(1) $M=T^{*}N,\ \Pi_{x}=T_{x}(T^{*}_{q}N),\quad\forall x=(p,q)\in T^{*}M,p\in
T^{*}_{q}M,q\in M.$

Let $h$ be a Hamiltonian function on $M$ and denote by $\vec{h}$ the
corresponding Hamiltonian vector field: $i_{\vec{h}}\sigma=dh$. We will assume
that $\vec{h}$ is a complete vector field without loss of generality since we
will study the dynamics of the Hamiltonian systems on a compact set. The pair
$(\vec{h},\Delta)$ will be said to be _a dynamical Lagrangian distribution_ of
the symplectic manifold $(M,\sigma).$

Dynamical Lagrangian distributions appear naturally in Differential Geometry,
Calculus of Variations and Rational Mechanics. The model example can be
described as follows:

Example 1 On a manifold $M$ for a given smooth function
$L:TM\rightarrow\mathbb{R}$, which is convex on each fiber, we consider the
following standard problem of Calculus of Variation with fixed endpoints
$q_{0}$ and $q_{1}$ and fixed time $T$:

(2) $\displaystyle A(q(\cdot))=\int^{T}_{0}L(q(t),\dot{q}(t))dt\mapsto{\rm
min}$ (3) $\displaystyle q(0)=q_{0},\quad q(T)=q_{1}.$

Suppose that the Legendre transform $h:T^{*}M\rightarrow\mathbb{R}$ of the
function $L$,

(4) $h(p,q)=\max_{X\in T_{q}M}(p(X)-L(q,X)),\ q\in M,p\in T^{*}_{q}M$

is well defined and smooth on $T^{*}M$. We will say that the dynamical
Lagrangian distributions $(\vec{h},\Pi)$ is associated with the problem
(2)-(3), where $\Pi$ is as in (1). $\square$

To describe the dynamical property of a dynamical Lagrangian distribution
$(\vec{h},\Delta)$, we define the Jacobi curve (at point $x\in M$) of the pair
$(\vec{h},\Delta)$ as follows:

(5) $J_{x}(t):=e^{-t\vec{h}}_{*}\left(\Delta_{e^{t\vec{h}}x}\right),$

where $e^{t\vec{h}},\ t\in\mathbb{R}$ denotes the Hamiltonian flow generated
by the vector field $\vec{h}$.

It is clear that the Jacobi curves $J_{x}(t)$ are curves in the Lagrange
Grassmannian of the symplectic space $T_{x}^{*}M$. They are not arbitrary
curves of the Lagrangian Grassmannian but inherit special features of the pair
$(\vec{h},\mathcal{D})$. To specify these features recall that the tangent
space $T_{\Lambda}L(W)$ to the Lagrangian Grassmannian $L(W)$ of a linear
symplectic space $W$ (endowed with a symplectic form $\omega$) at the point
$\Lambda$ can be naturally identified with the space ${\rm Quad}(\Lambda)$ of
all quadratic forms on linear space $\Lambda\subset W$. Namely, given
$\mathfrak{V}\in T_{\Lambda}L(W)$ take a curve $\Lambda(t)\in L(W)$ with
$\Lambda(0)=\Lambda$ and $\dot{\Lambda}=\mathfrak{V}$. Given some vector
$l\in\Lambda$, take a curve $\ell(\cdot)$ in $W$ such that
$\ell(t)\in\Lambda(t)$ for all $t$ and $\ell(0)=l$. Define the quadratic form

(6) $Q_{\mathfrak{V}}(l)=\omega(l,\frac{d}{dt}\ell(0)).$

Using the fact that the spaces $\Lambda(t)$ are Lagrangian, it is easy to see
that $Q_{\mathfrak{V}}(l)$ does not depend on the choice of the curves
$\ell(\cdot)$ and $\Lambda(\cdot)$ with the above properties, but depends only
on $\mathfrak{V}$. So, we have the linear mapping from $T_{\Lambda}L(W)$ to
the spaces ${\rm Quad}(\Lambda)$, $\mathfrak{V}\mapsto Q_{\mathfrak{V}}$. A
simple counting of dimensions shows that this mapping is a bijection and it
defines the required identification. A curve $\Lambda(\cdot)$ in a Lagrange
Grassmannian is called _regular_ , if its velocity is a nondegenerated
quadratic form at every $\tau$. A curve $\Lambda(\cdot)$ is called _monotone_
(monotonically nondecreasing or monotonically nonincreasing) if the velocity
is sign definite (nonnegative or nonpositive) at any point. For later
convenience, a dynamical Lagrangian distribution is said to be regular
(monotone) if the associated Jacobi curves are regular (monotone).

The group of symplectomorphisms of the ambient space acts naturally on
Lagrangian distribution and Hamiltonian vector fields, therefore it acts also
on dynamical Lagrangian distributions. It turns out ([14]) that one can
construct the canonical bundle of moving frames and the complete system of
symplectic invariants for parametrized curves in Lagrange Grassmannians
satisfying very general assumptions (including monotone curves as a particular
case). The complete system of symplectic invariants (value at $t=0$) for the
Jacobi curve $J_{x}(\cdot),x\in M$ is called the curvature maps of
$(\vec{h},\Delta)$ and it is the basic differential invariants of the pair
$(\vec{h},\Delta)$ w.r.t. the action of symplectic group of $M$. In this
section, we will restrict us to the curvature maps for monotone regular
dynamical Lagrangian distribution since our goal is to obtain a sufficient
condition for hyperbolicity of the reduced Hamiltonian flows after the
reduction of first integrals, while the reduced dynamical Lagrangian
distributions are monotone regular (see Lemma 1 below). Note also that the
curvature maps for regular curves in Lagrangian Grassmannians are constructed
in earlier work [3].

More precisely, let $\mathfrak{R}_{x}(t)$ be the curvature map for the Jacobi
curve $J_{x}(t),x\in M$. Then the linear map
$\mathfrak{R}_{x}^{(\vec{h},\Delta)}:=\mathfrak{R}_{x}(0)=\mathfrak{R}_{x}(t)|_{t=0}:\Delta_{x}\rightarrow\Delta_{x}$
is said to be the _curvature map_ (at $x$) of the dynamical Lagrangian
distribution $(\vec{h},\Delta)$. It gives a symmetric bilinear forms (at $x$)

$r^{(\vec{h},\Delta)}_{x}(v,w):=\sigma(R_{x}^{(\vec{h},\Delta)}w,[\vec{h},V]),\quad
v,w\in\Delta_{x}$

where $V$ is a smooth section of the sub-bundle $\Delta$ with $V(x)=v$. The
corresponding quadratic form will be called the _curvature form_ of the
dynamical Lagrangian distribution $(\vec{h},\Delta)$.

Example 2 (Natural mechanical system) In Example 1, let

$M=R^{n},\Pi_{(p,q)}=(R^{n},0),L(q,X)=\frac{1}{2}|X|^{2}-W(q)$

(in this case the function $A(q(t),\dot{q}(t))$ is the Action functional of
the natural mechanical system with potential energy $W(q)$). Then the
curvature forms can be written as follows:

(7)
$r^{(\vec{h},\Pi)}_{(p,q)}(\partial_{p_{i}},\partial_{p_{j}})=\frac{\partial^{2}W}{\partial
q_{i}\partial q_{j}}(q),\quad\forall 1\leq i,j\leq n.$

In other words, in this case the curvature forms are naturally identified with
the Hessian of the potential $W$. $\square$

Example 3 (Riemannian manifold) Let $(M,g)$ be a Riemannian manifold. Let
$L(q,X)=\frac{1}{2}g(X,X)$. The inner product $g(\cdot,\cdot)$ defines the
canonical isomorphism between $T_{q}M$ and $T^{*}_{q}M$. For any $q\in M$ and
$p\in T^{*}_{q}M$ we will denote by $p^{h}$ the image of $p$ under this
isomorphism, namely, the vector $p^{h}\in T_{q}M$, satisfying

(8) $p(\cdot)=g(p^{h},\cdot)$

Since the fibers $T_{q}^{*}M$ are linear spaces, one can identify
$\Pi_{\lambda}(=T_{\lambda}T^{*}_{q}M)$ with $T_{\pi(\lambda)}^{*}M$, i.e. the
operation $p^{h}$ is defined also on each $p\in\Pi_{\lambda}$ with values in
$T_{\pi(\lambda)}M$. For any given $\lambda=(p,q)\in T^{*}M,p\in M,p\in
T^{*}_{q}M$, it turns out ([3]) that

(9)
$(\mathfrak{R}^{(\vec{h},\Pi)}_{\lambda}v)^{h}=R^{\nabla}(p^{h},v^{h})p^{h},\quad
v\in\Pi_{\lambda},$

where $R^{\nabla}$ is the Riemannian curvature tensor of the metric $g$.
$\square$

Example 4 (Natural mechanical system on a Riemannian manifold) We add the
potential in the action functional in the previous example, i.e.
$L(q,X)=\frac{1}{2}g(X,X)-W(q)$ . Then the curvature maps satisfies

(10)
$(\mathfrak{R}^{(\vec{h},\Pi)}_{\lambda}v)^{h}=R^{\nabla}(p^{h},v^{h})p^{h}+\nabla_{v^{h}}(\nabla
W)(q),\quad v\in\Pi_{\lambda},$

where $\nabla W$ is the gradient of $W$ w.r.t. the Riemannian metric $g$.
$\square$

###  2.2. Reduced curvature forms and hyperbolicity

Assume that the dynamical Lagrange distribution $(\vec{h},\Delta)$ have
arbitrary $s$ first integrals $g_{1},...,g_{s}$ in involution with the
Hamiltonian $h$, i.e. $s$ functions on $M$ such that

$\\{h,g_{i}\\}=0,\quad\\{g_{i},g_{j}\\}=0,\quad\forall 1\leq i,j\leq s,$

where $\\{,\\}$ is the Poisson bracket. This problem appear naturally in the
framework of mechanical systems and variational problems with symmetries. Let
$\mathcal{G}=(g_{1},...,g_{s})$ and let

(11) $\Delta^{\mathcal{G}}_{x}=(\cap_{i=1}^{s}{\rm ker}\
d_{x}g_{i})\cap\Delta_{x}+{\rm span}\\{\vec{g}_{1}(x),...,\vec{g}_{s}(x)\\}$

Clearly, the distribution
$\Delta^{\mathcal{G}}=\\{\Delta^{\mathcal{G}}_{x},x\in M\\}$ is a Lagrangian
distribution. Hence, we get a reduced dynamical Lagrangian distribution
$(\vec{h},\Delta^{\mathcal{G}})$ after the reduction by first integrals
$\mathcal{G}$. Its curvature maps (forms) will be called _the reduced
curvature maps (forms)_ after the reduction by first integrals $\mathcal{G}$.

Example 5 Assume that we have one first integral $g$ of $h$ such that the
Hamiltonian vector field $\vec{g}$ preserves the distribution $\Delta$, i.e.
$(e^{t\vec{g}})_{*}\Delta=\Delta$. Fixing some value $c$ of $g$, one can
define (at least locally) the following quotient manifold:
$M_{g,c}=g^{-1}(c)/{\mathcal{C}}$ , where $\mathcal{C}$ is the line foliation
of the integral curves of the vector field $\vec{g}$. The manifold $M_{g,c}$
naturally inherits a symplectic form from the original symplectic structure
$(M,\sigma)$. Furthermore, if we denote by $\Phi:g^{-1}(c)\rightarrow M_{g,c}$
the canonical projection on the quotient set, the vector field
$\Phi_{*}(\vec{h})$ is well defined Hamiltonian vector field on $M_{g,c}$ due
to the fact that the vector fields $\vec{h}$ and $\vec{g}$ commute. For
simplicity, we still denote $\Phi_{*}(\vec{h})$ by $\vec{h}$. Actually we have
simply described the standard reduction of the Hamiltonian systems on the
level set of the first integrals in Mechanics (see e.g. [1]). In this way, to
any dynamical Lagrangian distribution $(\vec{h},\Delta)$ on $M$ one can
associate the dynamical Lagrangian distribution $(\vec{h},\Phi_{*}\Delta)$ on
the symplectic manifold $M_{g,c}$ of smaller dimension. $\square$

It is well known that the geodesic flows on a compact Riemannian manifold with
negative sectional curvature is Anosov ([6]). On the other hand, the reduced
curvature maps (forms) of the dynamical Lagrangian distributions associated
with the geodesic problem on a Riemannian manifold are naturally identified
with the sectional curvature tensor (See Example 6 below). Hence, we could
roughly formulate the result of Anosov as follows: negativity of the reduced
curvature forms implies the hyperbolicity of the geodesic flows. To go further
from this viewpoint, one can obtain a natural generalization ([4]) in the
framework of dynamical Lagrangian distributions. It will serve as a criteria
in the study of the hyperbolic flows in sub-Riemannian structures.

######  Definition 1.

Let $e^{tX},\ t\in\mathbb{R}$ be the flow generated by the vector field $X$ on
a manifold $P$. A compact invariant set $A\subset P$ of the flow $e^{tX}$ is
called a hyperbolic set if there exists a Riemannian structure in a
neighborhood of $A$, a positive constant $\delta$, and a splitting:
$T_{z}P=E_{z}^{+}\oplus E_{z}^{-}\oplus\mathbb{R}X(z),\ z\in A$ such that
$X(z)\neq 0$ and

  1. (1)

$e^{tX}_{*}E^{+}_{z}=E^{+}_{e^{tX}z},\ e^{tX}_{*}E^{-}_{z}=E^{-}_{e^{tX}z},$

  2. (2)

$\|e^{tX}_{*}\zeta^{+}\|\geq e^{\delta t}\|\zeta^{+}\|,\ \forall
t>0,\forall\zeta^{+}\in E^{+}_{z},$

  3. (3)

$\|e^{tX}_{*}\zeta^{-}\|\leq e^{-\delta t}\|\zeta^{-}\|,\ \forall
t>0,\forall\zeta^{-}\in E^{-}_{z}.$

If the entire manifold $P$ is a hyperbolic set, then the flow $e^{tX}$ is
called a flow of Anosov type.

######  Theorem 1.

Let $c=(c_{0},c_{1},...,c_{s})$ be constants. Let $S$ be a compact invariant
set of the flow $e^{t\vec{h}}$ contained in a fixed level of
$h^{-1}(c_{0})\cap_{i=1}^{s}g_{i}^{-1}(c_{i})$ and
$\vec{h}(x),\vec{g}_{i}(x)\notin\mathcal{D}_{x},\forall x\in S,i=1,...,s$. If
the reduced curvature form $r_{x}^{(\mathcal{G},h)}$ of the dynamical
Lagrangian distribution $(\vec{h},\mathcal{D})$ (after the reduction by first
integrals $(\mathcal{G},h$)) is negative at every point $x$ of $S$, then $S$
is a hyperbolic set of the flow
$e^{t\vec{h}}|_{h^{-1}(c_{0})\cap_{i=1}^{s}g_{i}^{-1}(c_{i})}$.

###  2.3. Descriptions of main results

We now specialize to the study of a natural mechanical system on a sub-
Riemannian manifold with symmetries.

Let $M$ be a connected smooth manifold. A distribution $\mathcal{D}$ on $M$ is
a sub-bundle of the tangent bundle $TM$. It is said to be _completely
nonholonomic_ if any local frame $\\{X_{i}:1\leq i\leq n\\}$ for
$\mathcal{D}$, together with all its iterated Lie brackets
$[X_{i},X_{j}],[X_{i},[X_{j},X_{k}]]$, … , spans the tangent bundle $TM$. A
Lipschitzian curve $\gamma:[0,T]\longrightarrow M$ is said to be admissible if
$\dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)}$ for a.e. $t\in[0,T]$. From the
Rashevskii-Chow theorem ([5]) it follows that there is an admissible curve
joining any two points of $M$. _A sub-Riemannian metric_ is a smoothly varying
positive definite inner product $\left\langle\cdot,\cdot\right\rangle$ on
$\mathcal{D}$. In particular, when $\mathcal{D}$ is equal to the tangent
bundle, $\left\langle\cdot,\cdot\right\rangle$ gives a Riemannian metric.

_A sub-Riemannian structure_ , denoted by the triple
$(M,\mathcal{D},\left\langle\cdot,\cdot\right\rangle)$, is a smooth
$n$-dimensional connected manifold $M$ equipped with a sub-Riemannian metric
$\left\langle\cdot,\cdot\right\rangle$ on a completely nonholonomic
distribution $\mathcal{D}$. In this case, we call the manifold $M$ _a sub-
Riemannian manifold_. In the present note we consider sub-Riemannian metrics
$\left\langle\cdot,\cdot\right\rangle$ on distribution $\mathcal{D}$ of corank
$s$, having $s$ transversal infinitesimal symmetries, i.e. $s$ vector fields
$X_{1},\ldots,X_{s}$ on $M$ such that

(12) $e^{tX_{i}}_{*}\mathcal{D}=\mathcal{D}\ ,\
(e^{tX_{i}})^{*}\left\langle\cdot,\cdot\right\rangle=\left\langle\cdot,\cdot\right\rangle,\quad
1\leq i\leq s,$

and $TM=\mathcal{D}\oplus\text{span}\\{X_{1},\ldots,X_{s}\\}$. Suppose further
that the symmetries $\\{X_{i}:\ 1\leq i\leq s\\}$ are commutative (see Remark
1 for noncommutative case), i.e.

(13) $[X_{i},X_{j}]=0,\quad\forall 1\leq i,j\leq s.$

We consider the natural mechanical system on a sub-Riemannian manifold (ASR):

(14) $\displaystyle
A(\gamma(\cdot))=\int^{T}_{0}(\frac{1}{2}\|\dot{\gamma}\|^{2}-W(\gamma))dt\mapsto{\rm
min}$ (15) $\displaystyle\gamma(\cdot)\ {\rm is\
admissible},\quad\gamma(0)=q_{0},\quad\gamma(T)=q_{1}.$

where $\|\cdot\|$ is the norm w.r.t. the metric
$\left\langle\cdot,\cdot\right\rangle$. We assume further that the potential
$W$ in (14) is constant along the integral curves of any $X_{i}$, or,
equivalently,

(16) $X_{i}(W)=0,\ i=1,...,s.$

It is more convenient to regard it as an optimal control problem and its
extremals can be described by the Pontryagin Maximum Principle of Optimal
Control Theory ([11]). There are two different types of extremals: abnormal
and normal, according to vanishing or nonvanishing of Lagrange multiplier near
the functional, respectively. The minimizers of the problem are the
projections of either normal extremals or abnormal extremals.

In the present note we will focus on normal extremals only. To describe them
let us introduce some notations. Let $T^{*}M$ be the cotangent bundle of $M$
and $\sigma$ be the canonical symplectic form on $T^{*}M$, i.e.
$\sigma=-d\varsigma$, where $\varsigma$ is the tautological (Liouville) 1-form
on $T^{*}M$. Let

(17) $h(p,q)=\max_{u\in\mathcal{D}}(p\cdot
u-\frac{1}{2}\|u\|^{2}+W(q))=\frac{1}{2}\|p|_{\mathcal{D}_{q}}\|^{2}+W(q),\
q\in M,\ p\in T^{*}_{q}M,$

where $p|_{\mathcal{D}_{q}}$ is the restriction of the linear functional $p$
to $\mathcal{D}_{q}$ and the norm $\|p|_{\mathcal{D}_{q}}\|$ is defined w.r.t.
the Euclidean structure on $\mathcal{D}_{q}.$ It is well defined and smooth in
the open set $O=T^{*}M\backslash\mathcal{D}^{\perp}$, where
$\mathcal{D}^{\perp}$ is the annihilator of $\mathcal{D}$, that is,

(18) $\mathcal{D}^{\perp}=\\{(p,q)\in T^{*}M:p(v)=0\,\,\forall
v\in\mathcal{D}_{q}\\}.$

For any vector field $X_{i}$ define the “quasiimpluses”
$u_{i}:T^{*}M\rightarrow\mathbb{R}$ by

$\quad u_{i}(p,q)=p(X_{i}(q)),\ q\in T^{*}_{q}M,q\in M,\ \forall 1\leq i\leq
s.$

Let $h$ be the sub-Riemannian Hamiltonian as in (17). Then it follows from
(12) and (16) that

(19) $\\{h,u_{i}\\}=0,\forall 1\leq i\leq s.$

and from (13) it follows that

(20) $\\{u_{i},u_{j}\\}=0,\ \forall 1\leq i,j\leq s,$

where $\\{\ ,\ \\}$ is the Poisson bracket. In other words, $u_{i}(1\leq i\leq
s)$ are first integrals in involution of the Hamiltonian system
$e^{t\vec{h}}$.

As before, let $\Pi$ be the “vertical”distribution, i.e.
$\Pi_{\lambda}=T_{\lambda}T_{\pi(\lambda)}^{*}M$, where $\pi:T^{*}M\to M$ is
the canonical projection. Now we can apply the reduction to the dynamical
Lagrangian distributions $(\vec{h},\Pi)$ after the first integrals
$u_{i}(1\leq i\leq s)$ (c.f. Example 5). For this, fix constants
$c_{0},c_{1},...,c_{s}$, where $c_{0}>0$ is sufficient large. Take a common
level set

$\mathcal{H}_{c}:=\\{h=c_{0}\\}\cap\\{u_{i}=c_{i},1\leq i\leq s\\}.$

Then

$W^{c}_{\lambda}=T_{\lambda}\mathcal{H}_{c}/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),\ 1\leq i\leq s\\}$

is a linear symplectic space with the symplectic form $\sigma^{c}$ naturally
inherited from the symplectic form $\sigma$. Moreover,

$\Pi_{\lambda}^{c}=(T_{\lambda}\mathcal{H}_{c}\cap\Pi_{\lambda})/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),\ 1\leq i\leq s\\}$

is a Lagrangian subspace in $W^{c}_{\lambda}$. Hence, we get the reduced
dynamical Lagrangian distribution $(\vec{h},\Pi^{c})$ in the linear symplectic
space $W^{c}$.

######  Remark 1.

The reduction procedure above also applies for the case that the symmetries
$\\{X_{i}:\ 1\leq i\leq s\\}$ are not commutative but still satisfy that
$g=\rm{span}_{\mathbb{R}}\\{X_{i},\ 1\leq i\leq s\\}$ is a Lie algebra and the
derived Lie algebra $g^{2}=[g,g]$ is a proper Lie subalgebra. Indeed, take a
basis of $g^{2}:(g_{1},...,g_{k})$ and complete it to a basis of
$g:(g_{1},...,g_{k},g_{k+1},...,g_{s})$. Then if we select a level set
$c=(c_{1},...,c_{s})$ such that

(21) $c_{i}=0,\ 1\leq i\leq k,$

then one can see that $g$ is commutative on this level set. Therefore, it
reduces to commutative case and then the Poission reduction can be applied.
Actually, it is equivalent to considering a sub-Riemannian structure on the
manifold obtained by reduction of the original one by $g_{1},...,g_{k}$ on
which the symmetries consist of a commutative Lie algebra $g/g^{2}$.

The (reduced) curvature maps (forms) of $(\vec{h},\Pi^{c})$ is naturally
related to the ambient sub-Riemannian structures (with symmetries), while the
later can be reduced to a Riemannian manifold equipped with a
$\mathbb{R}^{s}$-valued magnetic field. Denote by $\widetilde{M}$ the quotient
of $M$ by the leaves of the integral manifold of the involutive distribution
spanned by $X_{1},\ldots X_{s}$ and denote the factorization map by ${\rm
pr}:M\rightarrow\widetilde{M}$. Then $\widetilde{M}$ is (at least locally) a
Riemannian manifold equipped with the Riemannian metric $g$ induced from the
sub-Riemannian metric. Furthermore, let $\omega=(\omega_{i})_{1\leq i\leq s}$
be the $\mathbb{R}^{s}$-valued 1-form defined by $\omega_{i}|_{\mathcal{D}}=0$
and $\omega_{i}(X_{j})=\delta_{ij},\ \forall 1\leq i,j\leq s.$ Then
$d\omega=(d\omega_{i})_{1\leq i\leq s}$ induces a $\mathbb{R}^{s}$-valued
2-form on $\widetilde{M}$ (still denoted by $d\omega=(d\omega_{i})$) and one
can define a $\mathbb{R}^{s}$-valued tensor
$J=(J_{i}(\tilde{q}),\tilde{q}\in\widetilde{M})$ of type $(1,1)$ on
$\widetilde{M}$ satisfying

$g_{\tilde{q}}(J_{i}(\tilde{q})v,w)=d\omega_{i}(\tilde{q})(v,w),\ v,w\in
T_{\tilde{q}}\widetilde{M},\tilde{q}\in\widetilde{M},\ \forall 1\leq i\leq s.$

Let $\Xi^{c}$ be the $s$-foliation such that its leaves are integral curves of
$\\{\vec{u}_{i},1\leq i\leq s\\}$. Let ${\rm PR}^{c}:T^{*}M\to T^{*}M/\Xi^{c}$
be the canonical projection to the quotient manifold. Now we show that the
quotient manifold $N^{c}=\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}$ can be
naturally identified with $T^{*}\widetilde{M}$. Indeed, a point
$\tilde{\lambda}$ in $\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}$ can be
identified with a leaf $(\rm{PR}^{c})^{-1}(\tilde{\lambda})$ of $\Xi^{c}$
which has a form

$((e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}p,e^{\sum_{i=1}^{s}t_{i}X_{i}}q),$

where $\lambda=(p,q)\in({\rm PR}^{c})^{-1}(\tilde{\lambda})$, $q\in M$ and
$p\in T_{q}^{*}M$. On the other hand, any element in $T^{*}\widetilde{M}$ can
be identified with a one-parametric family of pairs
$((e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}(p|_{\mathcal{D}}),e^{\sum_{i=1}^{s}t_{i}X_{i}}q)$.
The mapping $I^{c}:\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}\to
T^{*}\widetilde{M}$ defined by

$I^{c}:(e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}p,e^{\sum_{i=1}^{s}t_{i}X_{i}}q)\mapsto(e^{-\sum_{i=1}^{s}t_{i}X_{i}})^{*}(p|_{\mathcal{D}}),e^{\sum_{i=1}^{s}t_{i}X_{i}}q)$

is one-to-one ($p(X_{i})=\hbox{const.}$ is already prescribed) and it defines
the required identification.

Before the statement of the main result of the note, let us introduce some
notations. Let $D^{\bot}$ be as in (18). Denote
$\mathcal{D}_{q}^{\bot}=\mathcal{D}^{\bot}\cap T^{*}_{q}M.$ Then one has the
following series of natural identifications:

(22) $\Pi^{c}_{\lambda}\sim
T^{*}_{q}M/\mathcal{D}_{q}^{\bot}\sim\mathcal{D}^{*}_{q}\stackrel{{\scriptstyle\left\langle\cdot,\cdot\right\rangle}}{{\sim}}\mathcal{D}_{q}\sim
T_{{\rm pr}(q)}\widetilde{M}$

where $\mathcal{D}_{q}^{*}\subseteq T^{*}_{q}M$ is the dual space of
$\mathcal{D}_{q}$. Given $v\in T_{\lambda}T^{*}_{q}M$ ($\sim T^{*}_{q}M$),
where $q=\pi(\lambda)$, we can assign a unique vector $v^{h}\in T_{{\rm
pr}(q)}\widetilde{M}$ to its equivalence class in
$T_{q}^{*}M/\mathcal{D}_{q}^{\bot}$ by using the identifications (22).
Conversely, to any $X\in T_{\hbox{pr}(q)}\widetilde{M}$ one can assign an
equivalence class of $T_{\lambda}(T^{*}_{q}M)/\mathcal{D}_{q}^{\bot}$. Denote
by $X^{v}\in T_{\lambda}T^{*}_{q}M$ the unique representative of this
equivalence class such that $du_{i}(X^{v})=0,\,\forall 1\leq i\leq s$.

For simplicity, we henceforth denote: $J^{c}=\sum_{i=1}^{s}c_{i}J_{i}$.

######  Theorem 2.

The curvature forms $r_{\lambda}^{c}$ of the dynamical Lagrangian distribution
$(\vec{h},\Pi^{c})$ is expressed as follows. For any $v\in\Pi^{c}_{\lambda}$,

$\displaystyle r^{c}_{\lambda}(v)$ $\displaystyle=$ $\displaystyle
g(R^{\nabla}(p^{h},v^{h})p^{h},v^{h})+g(\nabla
J^{c}(p^{h},v^{h}),v^{h})+\frac{1}{4}g(J^{c}v^{h},J^{c}v^{h})$
$\displaystyle+$
$\displaystyle\frac{3}{8(c_{0}+W)}\left(g(J^{c}p^{h},v^{h})\right)^{2}+\frac{3}{2(c_{0}+W)}g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h})$ $\displaystyle+$
$\displaystyle\frac{3}{2(c_{0}+W)}\left(g(v^{h},\nabla W)\right)^{2}+{\rm
Hess}\ W(v^{h},v^{h}).$

It follows from relations (16),(19) and (20) that $e^{t\vec{h}}$ induces a
(reduced) Hamiltonian flow $\Phi_{t}$ on $N^{c}$, where
$N^{c}=\\{u_{i}=c_{i},1\leq i\leq s\\}/\Xi^{c}$, as before. Then following
theorem is a direct consequence Theorem 1.

######  Theorem 3.

Assume that $K^{c}\subset N^{c}$ is a compact invariant set of the flow
$\Phi_{t}$ on $N^{c}$. If the curvature form $r_{\lambda}^{c}$ is negative at
every point of $K^{c}$, then $K^{c}$ is a hyperbolic set of the flow
$\Phi_{t}$ on $N^{c}$.

As mentioned, the manifold $N^{c}$ is naturally identified $T^{*}M$. Now
denote by $S_{1}\widetilde{M}$ the unit tangent bundle. Combining the previous
theorem with Theorem 2, we get the following

######  Theorem 4.

Assume that the reduced Riemannian manifold $(\widetilde{M},g)$ is compact and
has sectional curvature bounded from above by $k_{\rm max}$. If the constants
$c_{0},c_{1},...,c_{s}$ satisfy

$\displaystyle\max_{v,w\in S_{1}\widetilde{M},v\perp w}g(v,\nabla
J^{c}(w;v))+\frac{1}{4}g(J^{c}v,J^{c}v)+\frac{3}{8(c_{0}+W)}g(w,J^{c}v)g(w,J^{c}v)$
$\displaystyle+\frac{3}{2(c_{0}+W)}g(v,\nabla
W)g(J^{c}w,v)+3\left(\frac{\|\nabla W\|}{2(c_{0}+W)}\right)^{2}+\frac{\|{\rm
Hess}\ W\|}{2(c_{0}+W)}<-k_{\rm max},$

then the flow $\Phi_{t}$ is an Anosov flow.

######  Corollary 1.

Pure potential flows, i.e. $J_{i}=0(1\leq i\leq s)$,

(23) $\max_{\tilde{q}\in\widetilde{M}}\left(3\left(\frac{\|\nabla
W\|}{2(c_{0}+W)}\right)^{2}+\frac{\|{\rm Hess}\
W\|}{2(c_{0}+W)}\right)<-k_{\rm max}.$

######  Corollary 2.

Pure magnetic flows, i.e. $W=0$,

(24) $\max_{v,w\in S_{1}\widetilde{M},v\perp w}{cg(v,\nabla
J(w;v))+c^{2}g(Jv,Jv))}<-k_{\rm{max}}.$

######  Remark 2.

The left-hand side of the inequality in Theorem 4 is always positive, because
the second term inside the max is positive and the first term can be made
nonnegative, if necessary, by changing the sign of $w$. Hence, Theorem 4 makes
sense only if $k_{\rm{max}}<0$.

The flow $\Phi_{t}$ on $N^{c}$ can be considered as a perturbation of the
Riemannian geodesic flow: the flow $\Phi_{t}$ on $N^{c}$ remains to be an
Anosov flow for sufficient small constants $c_{i}(1\leq i\leq s)$ and for
proper potential function $W$ (sufficient small norm of $W$ and its
derivatives). When $s=1$, it coincides with Theorem 4.1 (the case of Gaussian
thermostats of external fields $E=0$ there) in [13]; when $s=1$ and $W=0$, the
condition of Anosov magnetic flows (24) coincides with the main results in
[8].

##  3\. proof of the main results

The rest of the note is devoted to the proof of Theorem 2.

###  3.1. Reduced curvature maps

As before, fix constants $c_{0},c_{1},...,c_{s}$, where $c_{0}>0$ is
sufficient large. Let $J^{c}_{\lambda}(t)$ be the Jacobi curves associated
with the reduced dynamical Lagrangian distribution $(\vec{h},\Pi^{c})$,
namely,

(25) $J^{c}_{\lambda}(t):=e^{-t\vec{h}}_{*}\Pi^{c}_{e^{t\vec{h}}\lambda}.$

######  Lemma 1.

The reduced Jacobi curve $J^{c}_{\lambda}(\cdot)$ is a regular monotone
nondecreasing curve in Lagrange Grassmannian $L(W^{c}_{\lambda})$.

###### Proof.

First note that if $\bar{\lambda}=e^{\bar{t}\vec{h}}\lambda$ and
$\phi:W^{c}_{\lambda}\rightarrow W^{c}_{\bar{\lambda}}$ is a symplectic
transformation induced in the natural way by a linear mapping
$e^{t\vec{h}}_{*}:T_{\lambda}\mathcal{H}_{c}\rightarrow
T_{\bar{\lambda}}\mathcal{H}_{c}$, where, as before,

$\mathcal{H}_{c}:=\\{h=c_{0}\\}\cap\\{u_{i}=c_{i},1\leq i\leq s\\}.$

Then by (25) we have

(26)
$J^{c}_{\bar{\lambda}}(t)=\phi\bigl{(}J^{c}_{\lambda}(t-\bar{t})\bigr{)}.$

Further, it turns out (see, for example, [2, Proposition 1]) that the velocity
of the Jacobi curve $J_{\lambda}^{c}(\cdot)$ at $t=0$ is equal to the
restriction of the Hessian of $h$ to the tangent space to $\Pi_{\lambda}^{c}$
at the point $\lambda$. This together with the relation (26) and the
construction of $\Pi_{\lambda}^{c}$ implies easily that $J^{c}_{\lambda}(t)$
is a regular monotone nondecreasing curve. ∎

######  Theorem 5.

Let $\Lambda(\cdot)$ be a regular curve in the Lagrange Grassmannian $L(G)$ of
a $2n$-dimensional linear symplectic space $G$. Then there exists a moving
Darboux frame $(E(t),F(t))$ of $G$:

$E(t)=(e_{1}(t),...,e_{n}(t)),\ F(t)=(f_{1}(t),...,f_{n}(t))$

such that $\Lambda(t)=\rm{span}\\{E(t)\\}$ and there exists a one-parametric
family of linear self-adjoint operators
$\mathfrak{R}(t):\Lambda(t)\rightarrow\Lambda(t)$ satisfying

(27) $\begin{cases}E^{\prime}(t)=F(t),\\\ F^{\prime}(t)=-R(t)E(t).\end{cases}$

The moving frame $(E(t),F(t))$ is a called _a normal moving frame_ of
$\Lambda(t)$ and the linear operator $\mathfrak{R}(t)$ is called _the
curvature map_ of $\Lambda(t)$. A moving frame
$(\widetilde{E}(t),\widetilde{F}(t))$ is a normal moving frame of $\Lambda(t)$
if and only there exists a constant orthogonal matrix $U$ of size $n\times n$
such that

(28) $\widetilde{E}(t)=E(t)U,\ \widetilde{F}(t)=F(t)U.$

######  Remark 3.

Note that from (27) it follows that if
$\bigl{(}\widetilde{E}(t),\widetilde{F}(t)\bigr{)}$ is a Darboux moving frame
such that $\widetilde{E}(t)$ is an orthonormal frame of $\Lambda(t)$ and ${\rm
span}\,\\{\widetilde{F}(t)\\}=\Lambda^{\rm trans}(t)$. Then there exists a
curve of antisymmetric matrices $B(t)$ such that

(29)
$\left\\{\begin{array}[]{l}\widetilde{E}^{\prime}(t)=\widetilde{E}(t)B(t)+\widetilde{F}_{a}(t)\\\
\widetilde{F}^{\prime}(t)=-\widetilde{E}(t)\widetilde{\mathcal{R}}(t)+\widetilde{F}(t)B(t),\end{array}\right.$

where $\widetilde{\mathcal{R}}(t)$ is the matrix of the curvature map
$\mathfrak{R}(t)$ on $\Lambda(t)$ w.r.t. the basis $\widetilde{E}(t)$.

As a matter of fact, normal moving frames define a principal $O(n)$-bundle of
symplectic frame in $G$ endowed with a canonical connection. Also, relations
(28) imply that the following $n$-dimensional subspaces

(30) $\Lambda^{\rm trans}(t)={\rm span}\\{F(t)\\}$

of $G$ does not depend on the choice of the normal moving frame. It is called
the _canonical complement_ of $\Lambda(t)$ in $G$. Moreover, the subspaces
$\Lambda(t)$ and $\Lambda^{\rm trans}(t)$ are endowed with the _canonical
Euclidean structure_ such that the tuple of vectors $E(t)$ and $F(t)$
constitute an orthonormal frame w.r.t. to it, respectively.

Finally, the linear map from $\Lambda(t)$ to $\Lambda(t)$ with the matrix
$R(t)$ from (27) in the basis $\\{E(t)\\}$, is independent of the choice of
normal moving frames. It will be denoted by $\mathfrak{R}(t)$ and it is called
the _curvature map_ of the curve $\Lambda(t)$.

Now we apply the above results for curves in Lagrange Grassmannians to sub-
Riemannian structures. Since $\mathfrak{J}^{c}_{\lambda}(0)$ and
$\Pi^{c}_{\lambda}$ can be naturally identified, there is a canonical
splitting of $W^{c}_{\lambda}:$

(31)
$W^{c}_{\lambda}=\Pi^{c}_{\lambda}\oplus\widetilde{\mathfrak{J}}^{c}(\lambda),$

where $\widetilde{\mathfrak{J}}^{c}(\lambda)={\rm span}(F^{\lambda}(0))$ is
the canonical complement. In other words,
$\widetilde{\mathfrak{J}}^{c}(\lambda)$ is actually a (nonlinear) Ehresmann
connection of $\Pi^{c}_{\lambda}$ in $W^{c}_{\lambda}$. It also follows that
the subspaces $\Pi_{\lambda}^{c}$ and $\widetilde{\mathfrak{J}}^{c}(\lambda)$
are equipped with a canonical Euclidean structure. Moreover, one can define
the curvature map of the dynamical Lagrangian distribution
$(\vec{h},\Pi^{c})$, i.e.
$\mathfrak{R}^{c}_{\lambda}:\Pi^{c}_{\lambda}\rightarrow\Pi^{c}_{\lambda}$
such that $\mathfrak{R}^{c}_{\lambda}=\mathfrak{R}_{\lambda}(0)$ of the
curvature maps of the Jacobi curve $\mathfrak{J}^{c}_{\lambda}(\cdot)$ at
$t=0$. This curvature maps are intrinsically related to the sub-Riemannian
structure and will be called _the reduced curvature map_ of the sub-Riemannian
structure.

Let $\lambda\in T^{*}M$ and let $\lambda(t)=e^{t\vec{h}}\lambda$. Assume that
$(E^{\lambda}(t),F^{\lambda}(t))$ is a normal moving frame of the Jacobi curve
$\mathfrak{J}^{c}_{\lambda}(t)$ attached at point $\lambda$. Let
$\mathfrak{E}$ be the Euler field on $T^{*}M$, i.e. the infinitesimal
generator of the homotheties on its fibers. Clearly
$T_{\lambda}(T^{*}M)=T_{\lambda}\mathcal{H}_{c}\oplus\mathbb{R}\mathfrak{E}(\lambda)\oplus{\rm
span}\\{\partial_{u_{i}}(\lambda),1\leq i\leq s\\}$. The flow $e^{t\vec{h}}$
on $T^{*}M$ induces the push-forward maps $e^{t\vec{h}}_{*}$ between the
corresponding tangent spaces $T_{\lambda}T^{*}M$ and $T_{\lambda(t)}T^{*}M$,
which in turn induce naturally the maps between the spaces
$T_{\lambda}(T^{*}M)/{\rm span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),1\leq
i\leq s\\}$ and $T_{\lambda(t)}T^{*}M/{\rm
span}\\{\vec{h}(\lambda(t)),\vec{u}_{i}(\lambda(t)),1\leq i\leq s\\}$. The map
$\mathcal{K}^{t}$ between $T_{\lambda}(T^{*}M)/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),1\leq i\leq s\\}$ and
$T_{\lambda(t)}T^{*}M/{\rm
span}\\{\vec{h}(\lambda(t)),\vec{u}_{i}(\lambda(t)),1\leq i\leq s\\}$, sending
$E^{\lambda}(0)$ to $e^{t\vec{h}}_{*}E^{\lambda}(t)$, $F^{\lambda}(0)$ to
$e^{t\vec{h}}_{*}F^{\lambda}(t)$, and the equivalence class of
$\mathfrak{E}(\lambda),\partial_{u_{i}}(\lambda)(1\leq i\leq s)$ to the
equivalence class of
$\mathfrak{E}(e^{t\vec{h}}\lambda),\partial_{u_{i}}(\lambda(t))(1\leq i\leq
s)$, is independent of the choice of normal moving frames. The map
$\mathcal{K}^{t}$ is called _the parallel transport_ along the extremal
$e^{t\vec{h}}\lambda$ at time $t$. For any $v\in T_{\lambda}(T^{*}M)/{\rm
span}\\{\vec{h}(\lambda),\vec{u}_{i}(\lambda),1\leq i\leq s\\}$, its image
$v(t)=\mathcal{K}^{t}(v)$ is called _the parallel transport of $v$ at time
$t$_. Note that from the definition of the reduced Jacobi curves and the
construction of normal moving frames it follows that the restriction of the
parallel transport $\mathcal{K}_{t}$ to the vertical subspace
$T_{\lambda}(T_{\pi(\lambda)}^{*}M)$ of $T_{\lambda}(T^{*}M)$ can be
considered as a map onto the vertical subspace
$T_{\lambda(t)}(T_{\pi(\lambda(t))}^{*}M)$ of $T_{\lambda(t)}(T^{*}M)$. A
vertical vector field $V$ is called _parallel_ if
$V(e^{t\vec{h}}\lambda)=\mathcal{K}^{t}\bigl{(}V(\lambda)\bigr{)}$.

Example 6 (Riemannian geodesic flow) In this case, $\mathcal{D}=TM,W=0$ and
there is no symmetries at all ($s=0$). In [3] the reduced curvature map was
expressed by the Riemannian curvature tensor. If we adopt the notations from
Example 3 and take the constants $c_{0}=\frac{1}{2},c_{i}=0(1\leq i\leq s)$.
Then

(32)
$\mathfrak{R}_{\lambda}^{c}(v)=R^{\nabla}(p^{h},v^{h})p^{h},\quad\forall\lambda=(q,p)\in\mathcal{H}_{c},q\in
M,p\in T^{*}_{q}M,v\in\Pi^{c}_{\lambda}.$

Given a vector $X\in T_{q}M$ denote by $\nabla_{X}$ its lift to the Levi-
Civita connection, considered as an Ehresmann connection on $T^{*}M$. Then by
constructions the Hamiltonian vector field $\vec{h}$ is horizontal and
satisfies $\vec{h}=\nabla_{p^{h}}$. Take any $v,w\in\Pi_{\lambda}^{c}$ and let
$V$ be a vertical vector field such that $V(\lambda)=v$. From (32) , structure
equation (27), and the fact that the Levi-Civita connection (as an Eheresmann
connection on $T^{*}M$) is a Lagrangian distribution (c.f. [3]) it follows
that the Riemannian curvature tensor satisfies the following identity:

(33) $\langle
R^{\nabla}(p^{h},v^{h})p^{h},w^{h}\rangle=-\sigma\left([\nabla_{p^{h}},\nabla_{V^{h}}](\lambda),\nabla_{w^{h}}\right).\quad\square$

###  3.2. Proof of Theorem 2

We first express the canonical complement in terms of the Levi-Civita
connection of the Riemannian metric and the tensor $J_{i}^{c}$ and then we can
give the proof of Theorem 2 using some calclus formulae which is developped in
[10].

####  3.2.1. The canonical complement $\widetilde{\mathfrak{J}}^{c}(\lambda)$

The restriction of the parallel transport $\mathcal{K}^{t}$ to
$\Pi^{c}_{\lambda}$ is characterized by the following two properties:

  1. (1)

$\mathcal{K}^{t}$ is an orthogonal transformation of spaces
$\Pi_{\lambda}^{c}$ and $\Pi_{e^{t\vec{h}}\lambda}^{c}$;

  2. (2)

The space ${\rm
span}\\{\frac{d}{dt}\bigl{(}(e^{-t\vec{h}})_{*}(\mathcal{K}^{t}v)\bigr{)}|_{{}_{t=0}}:v\in\Pi_{\lambda}^{c}\\}$
is isotropic.

Then $\widetilde{\mathfrak{J}}^{c}(\lambda)={\rm
span}\\{\frac{d}{dt}\bigl{(}(e^{-t\vec{h}})_{*}(\mathcal{K}^{t}v)\bigr{)}|_{{}_{t=0}}:v\in\Pi_{\lambda}^{c}\\}$.

To express $\widetilde{\mathfrak{J}}^{c}(\lambda)$ in terms of the Riemannian
manifold and the magnetic field, we show the decomposition of the symplectic
form $\sigma$ (the standard symplectic form on $T^{*}\widetilde{M}$) and the
Hamiltonian field $\vec{h}$. One can see that the diffeomorphism $I^{c}$,
defined as before, are not in general symplectic. Indeed, each level set
inherits a symplectic structure depending on the choice of the level
$\\{c_{i}:1\leq i\leq s\\}$.

By the construction of the map $I^{c}$, for any vector field $X$ on
$T^{*}\widetilde{M}$, we can assign the vector field $\underline{X}$ on
$T^{*}M$ s.t. $PR^{c}_{*}\underline{X}=((I^{c})^{-1})_{*}X$ and
$\pi_{*}\underline{X}\in\mathcal{D}$. In the following, denote by
$\nabla_{p^{h}}$ the lift of $p^{h}$ to $T^{*}\widetilde{M}$ with respect to
the Levi-Civita connection and denote
$\Omega^{c}=\sum_{i=1}^{s}c_{i}d\omega_{i}$ and
$\bar{\sigma}=(I^{c}\circ\rm{PR}^{c})^{*}\sigma$. We will denote by
$\tilde{\sigma}$ the standard symplectic form on $T^{*}\widetilde{M}$. The
proof of the following lemma is complete similar to that of Lemma 3.1-3.3 in
[10] and thus is omitted.

######  Lemma 2.

The following decomposition formulae hold.

  1. (1)

On the level set
$\\{u_{i}=c_{i},i=1,...,s\\},\quad\sigma=\bar{\sigma}-(\pi\circ{\rm
pr})^{*}(\Omega^{c})$;

  2. (2)

For any vectors $X,V\in T_{\lambda}T^{*}M$ with $\pi_{*}V=0$ we have
$\sigma(X,v)=g(\pi_{*}X,V^{h});$

  3. (3)

$\vec{h}(p,q)=\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v}+\overrightarrow{W}$.

Comparing with the sub-Riemannian geodesic problem, we will develop some
additional calculus formulae about the potential $W$.

######  Lemma 3.

Let $V_{1},V_{2}$ be the vector fields on $T^{*}M$ with
$\pi_{*}V_{1}=\pi_{*}V_{2}=0$. Then

  1. (1)

$\overrightarrow{W}=-(\nabla W)^{v};$

  2. (2)

$\bar{\sigma}([\overrightarrow{W},\underline{\nabla_{V_{1}^{h}}}],\underline{\nabla_{V_{2}^{h}}}]=-{\rm
Hess}\ W(V_{1}^{h},V_{2}^{h});$

  3. (3)

$\overrightarrow{W}\left(g(V_{1}^{h},V_{2}^{h})\right)-g\left(([\overrightarrow{W},V_{1}])^{h},V_{2}^{h}\right)-g\left(V_{1}^{h},([\overrightarrow{W},V_{2}])^{h}\right)=0.$

###### Proof.

(1) From the second item of the last lemma it follows that

$\sigma(\overrightarrow{W},V_{1}^{h})=V_{1}^{h}(W)=dW(V_{1}^{h})=g(\nabla
W,V_{1}^{h})=-\sigma((\nabla W)^{v},V_{1}^{h}).$

Taking into account that $\pi_{*}\overrightarrow{W}=0$, we get

$\overrightarrow{W}=-(\nabla W)^{v},\quad{\rm
span}\\{\partial_{u_{i}},i=1,...s\\}.$

On the other hand, it follows from (50) and item (1) of the present lemma that

$\sigma(\vec{u}_{i},\overrightarrow{W})=-\vec{u}_{i}(W)=-X_{i}(W)=0,\
i=1,...,s.$

Thus, we get the required identity $\overrightarrow{W}=-(\nabla W)^{v}$.

(2) Both sides of the required identity are linear w.r.t. $V_{1},V_{2}$,
respectively, thus it is sufficient to prove it for the case that
$V_{1}^{h},V_{2}^{h}$ are both vector fields on $T^{*}\widetilde{M}$. But for
this case the required identity is a direct consequence of the definition of
the Hessian.

(3) Left-hand side is linear w.r.t. $V_{1},V_{2}$, respectively, thus it is
sufficient to prove it for the case that $V_{1}^{h},V_{2}^{h}$ are both vector
fields on $T^{*}\widetilde{M}$. In this case, the vector fields
$(V_{1}^{h})^{v},(V_{2}^{h})^{v}$, together with $(\nabla W)^{v}$ are all
constants on the fibers of $T^{*}M$ and then the required identity become
trivial. ∎

Given any $X\in\Pi_{\lambda}^{c}$ denote by $\widetilde{\nabla}_{X^{h}}$ the
lift of $X$ to $\widetilde{\mathfrak{J}}^{c}(\lambda)$: the unique vector
$\widetilde{\nabla}_{X^{h}}\in\widetilde{\mathfrak{J}}^{c}(\lambda)$ such that
$({\rm pr}\circ\pi)_{*}\widetilde{\nabla}_{X^{h}}=X^{h}$. Then there exist the
unique $B\in{\rm End}(\Pi^{c}_{\lambda})$ and
$\tilde{A}\in(\Pi_{\lambda}^{c})^{*}$ such that

(34) $\widetilde{\nabla}_{v^{h}}=\underline{\nabla_{v^{h}}}+Bv,\quad\forall
v\in\Pi_{\lambda}^{c},$

where $\nabla$ stands for the lifts to the Levi-Civita connection on
$T^{*}\widetilde{M}$, as before.

######  Lemma 4.

The linear operator $B$ is antisymmetric w.r.t. the canonical Euclidean
structure in $\Pi_{\lambda}^{c}$.

###### Proof.

Fix a point $\bar{\lambda}\in T^{*}M$ and consider a small neighborhood $U$ of
$\bar{\lambda}$. Let $\mathcal{E}=\\{\mathcal{E}^{i}\\}_{i=1}^{m-1}$ be a
frame of $\Pi_{\lambda}^{c}$ (i.e. $\Pi_{\lambda}^{c}={\rm
span}\,\mathcal{E}(\lambda)$) for any $\lambda\in U$ such that the following
four conditions hold

  1. (1)

$\mathcal{E}$ is orthogonal w.r.t. the canonical Euclidean structure on
$\Pi_{\lambda}^{c}$;

  2. (2)

Each vector field $\mathcal{E}^{i}$ is parallel w.r.t the canonical parallel
transport $\mathcal{K}_{t}$, i.e.
$\mathcal{E}^{i}(e^{t}\vec{h}\lambda)=\mathcal{K}^{t}\mathcal{E}^{i}(\lambda)$
for any $\lambda$ and $t$ such that $\lambda,e^{t\vec{h}}\lambda\in U$;

  3. (3)

The vector fields $(J^{c}p^{h})^{v}$ and $\mathcal{E}^{i}$ commute on $U\cap
T_{\pi(\bar{\lambda})}^{*}M$;

  4. (4)

The vector fields $\vec{u}_{i},\ \forall 1\leq i\leq s$ and $\mathcal{E}^{i}$
commute on $U\cap T_{\pi(\bar{\lambda})}^{*}M$.

Note that the frame $\mathcal{E}$ with properties above exists, because the
Hamiltonian vector field $\vec{h}$ is transversal to the fibers of $T^{*}M$
and it commutes with $\vec{u}_{i},\ \forall 1\leq i\leq s$.

From the property (2) of the parallel transport $\mathcal{K}^{t}$ in this
subsection it follows that

(35) $\widetilde{\nabla}_{(\mathcal{E}^{i})^{h}}=-{\rm
ad}\vec{h}\,\mathcal{E}^{i}$

Using the above defined identification $I^{c}:N^{c}\to T^{*}\widetilde{M}$,
one can look on the restriction of the tuple of vector fields $\mathcal{E}$ to
the submanifold $\\{u_{i}=c_{i},i=1,...,s\\}$ as on the tuple of the vertical
vector fields of $T^{*}\widetilde{M}$ (which actually span the tangent to the
intersection of the fiber of $T^{*}\widetilde{M}$ with the level to the
corresponding Riemannian Hamiltonian). Then first the tuple $\mathcal{E}$ is
the tuple of orthonormal vector fields (w.r.t. the canonical Euclidean
structure on the fibers of $T^{*}\widetilde{M}$, induced by the Riemannian
metric $g$). Further, by the equations (29) the Levi-Civita connection of $g$
is characterized by the fact that there exists a field of antisymmetric
operators $\widetilde{B}\in{\rm End}(\Pi^{c}_{\lambda})$ such that

(36)
$[\nabla_{p^{h}},\widetilde{\mathcal{E}}^{i}(\lambda)]=-\nabla_{\bigl{(}\widetilde{\mathcal{E}}^{i}(\lambda)\bigr{)}^{h}}-\widetilde{B}\widetilde{\mathcal{E}}^{i}(\lambda)$

On the other hand, from (35),(36), using the second item of Lemma 2 and the
property (3) of $\mathcal{E}^{i}$, one has

(37) $\begin{split}\widetilde{\nabla}_{(\mathcal{E}^{i})^{h}}=-{\rm
ad}\vec{h}\,\mathcal{E}^{i}=-\bigl{[}\underline{\nabla_{p^{h}}}-\sum_{i=1}^{s}c_{i}(J_{i}p^{h})^{v}+\overrightarrow{W},\mathcal{E}^{i}]=\underline{\nabla_{\bigl{(}\mathcal{E}^{i}(\lambda)\bigr{)}^{h}}}+\widetilde{B}\,\mathcal{E}^{i}(\lambda)+[\overrightarrow{W},\mathcal{E}^{i}(\lambda)].\end{split}$

From item (3) of Lemma 3 and property (1) of $\mathcal{E}^{i}$ it follows

(38)
$g(([\overrightarrow{W},\mathcal{E}^{i}])^{h},(\mathcal{E}^{j})^{h})+g((\mathcal{E}^{i})^{h},([\overrightarrow{W},\mathcal{E}^{j}])^{h})=\overrightarrow{W}\left(g((\mathcal{E}^{i})^{h},(\mathcal{E}^{j})^{h})\right)=0.$

Therefore, from (37) and (38) we conclude that $B$ is antisymmetric. ∎

######  Lemma 5.

The operator $B$ satisfies

(39) $(Bv)^{h}=-\frac{1}{2}J^{c}v^{h}\quad{\rm mod}\ p^{h},\quad\forall
v\in\Pi^{c}_{\lambda}.$

###### Proof.

Since $\widetilde{\mathfrak{J}}^{c}(\lambda)$ is an isotropic subspace, we
have

$\sigma(\widetilde{\nabla}_{v_{1}^{h}},\widetilde{\nabla}_{v_{2}^{h}})=0,\quad\forall\,v_{1},v_{2}\in\Pi^{c}_{\lambda}.$

On the other hand, using Proposition 2, the fact that the Levi-Civita
connection (as an Ehresmann connection) is a Lagrangian distribution in
$T^{*}\widetilde{M}$ and Lemma 2, we get

$\displaystyle
0=\sigma(\widetilde{\nabla}_{v_{1}^{h}},\widetilde{\nabla}^{c}_{v_{2}^{h}})$
$\displaystyle=$
$\displaystyle\Bigl{(}(I^{c}\circ\hbox{PR}^{c})^{*}\tilde{\sigma}-({\rm
pr}\circ\pi)^{*}\Omega^{c}\Bigr{)}\Bigl{(}\underline{\nabla_{v^{h}_{1}}}+Bv_{1},\underline{\nabla_{v^{h}_{2}}}+Bv_{2}\Bigr{)}$
$\displaystyle=$
$\displaystyle-\Omega^{c}(v_{1}^{h},v_{2}^{h})-g\big{(}(Bv_{1})^{h},v_{2}^{h}\big{)}+g\big{(}(Bv_{2})^{h},v_{1}^{h})$
$\displaystyle=$
$\displaystyle-g(J^{c}v_{1}^{h},v_{2}^{h})-g\big{(}(Bv_{1})^{h},v_{2}^{h}\big{)}+g\big{(}(B^{*}v_{1})^{h},v_{2}^{h}).$

where $B^{*}$ is the dual of $B$ w.r.t. the Euclidean structure in
$\Pi_{\lambda}^{c}$. Taking into account that $B$ is antisymmetric, we get

(40) $(Bv)^{h}=-\frac{1}{2}J^{c}v^{h}\quad{\rm mod}\ p^{h}.$

∎

######  Corollary 3.

The canonical complement $\widetilde{\mathfrak{J}}^{c}(\lambda)$ can be
expressed as follows:

$\widetilde{\mathfrak{J}}^{c}(\lambda)=\\{\underline{\nabla_{v^{h}}}-\frac{1}{2}(J^{c}v^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g\left(v^{h},J^{c}p^{h}+2\nabla W\right)(p^{h})^{v},\
v\in\Pi^{c}_{\lambda}\\}.$

###### Proof.

It follows from (34) and (39) that there exist $A\in(\Pi^{c}_{\lambda})^{*}$
such that

$\widetilde{\nabla}_{v^{h}}=\underline{\nabla_{v^{h}}}-\frac{1}{2}(J^{c}v^{h})^{v}+A(v)(p^{h})^{v}.$

Note that $\sigma(\vec{h},(p^{h})^{v})=g(p^{h},p^{h})=\|p^{h}\|^{2}$. Hence,
from the fact that $\widetilde{\nabla}_{v^{h}}$ is tangent to the Hamiltonian
vector field $\vec{h}$, we get easily that

$A(v)=-\frac{1}{2\|p^{h}\|^{2}}\cdot g\left(v^{h},J^{c}p^{h}+2\nabla
W\right),$

which completes the proof of the corollary. ∎

####  3.2.2. The reduced curvature map

As a direct consequence of structure equation (27), we get the following
preliminary descriptions of the reduced curvature map:

######  Proposition 1.

Let $v\in\Pi^{c}_{\lambda}$. Let $V$ be a parallel vector field such that
$V(\lambda)=v$. Then the curvature maps satisfy the following identities:

(41) $\displaystyle
g\big{(}(\mathfrak{R}_{\lambda}^{c}v)^{h},v^{h}\big{)}=-\sigma(\hbox{ad}\vec{h}\
(\widetilde{\nabla}_{V^{h}}),\widetilde{\nabla}_{v^{h}}).$

It follows that in order to calculate the reduced curvature map it is
sufficient to know how to express the Lie bracket of vector fields on the
cotangent bundle $T^{*}M$ via the covariant derivatives of Levi-Civita
connection on $T^{*}\widetilde{M}$.

######  Proposition 2.

For any tensors $A,B$ of type $(1,1)$ on $\widetilde{M}$, the following
identity holds:

  1. (1)

$[(Ap^{h})^{v},(Bp^{h})^{v}]=(B(Ap^{h}))^{v}-(A(Bp^{h}))^{v},$

  2. (2)

$[\underline{\nabla_{p^{h}}},(Ap^{h})^{v}]=-\underline{\nabla_{Ap^{h}}}+((\nabla_{p^{h}}A)p^{h})^{v}.$

For simplicity, denote $\bar{\sigma}=(I^{c}\circ\rm{PR}^{c})^{*}\sigma$ and
$\Omega^{c}=\sum_{i=1}^{s}c_{i}d\omega_{i},$ as before. As in the proof of
Lemma 4, we can take a parallel vector field $V$ such that $V(\lambda)=v$ and

(42) $[(J^{c}p^{h})^{v},V](\bar{\lambda})=0,\quad\bar{\lambda}\in U\cap
T_{q}^{*}M,$

where $U$ is a neighborhood of $\lambda.$ Similar to Proposition 4 of [10], we
have

######  Lemma 6.

Let $V,V_{1},V_{2}$ be vector fields on $T^{*}M$ with
$\pi_{*}V=\pi_{*}V_{1}=\pi_{*}V_{2}=0$. Then

  1. (1)

$([(J^{c}p^{h})^{v},(J^{c}V^{h})^{v}])^{h}=J^{c}([(J^{c}p^{h})^{v},(V^{h})^{v}])^{h}$,

  2. (2)

$\bar{\sigma}([(J^{c}p^{h})^{v},\underline{\nabla_{V_{1}^{h}}}],\underline{\nabla_{V_{2}^{h}}})=g(\nabla
J^{c}(p^{h},V_{1}^{h}),V_{2}^{h}),$

  3. (3)

$({\rm
pr}\circ\pi)_{*}([(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}])=J^{c}V^{h},$

  4. (4)

$({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}])=\frac{1}{2}J^{c}V^{h}-\frac{1}{2\|p^{h}\|^{2}}g(J^{c}V^{h},p^{h})p^{h}$.

Let us simplify the right-hand side of the identity (41). First, from the last
line of the structural equations (27) it follows that

(43) $({\rm
pr}\circ\pi)_{*}(\hbox{ad}\vec{h}(\widetilde{\nabla}_{V^{h}}))\in\mathbb{R}p^{h}.$

Besides,

$\sigma\left(\vec{h},\frac{1}{2}(J^{c}v^{h})^{v}+\frac{1}{2\|p^{h}\|^{2}}g\left(v^{h},J^{c}p^{h}\right)(p^{h})^{v}\right)=\frac{1}{2}g(p^{h},J^{c}v^{h})+\frac{1}{2\|p^{h}\|^{2}}g(v^{h},J^{c}p^{h})g(p^{h},p^{h})=0.$

Hence from Lemma 2 and Corollary 3 it follows that

$\displaystyle\sigma(\hbox{ad}\vec{h}(\nabla_{V^{h}}),\nabla_{v^{h}})=\sigma\left(\hbox{ad}\vec{h}(\nabla_{V^{h}}),\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}-\frac{1}{2}(J^{c}V^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h})(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)$
$\displaystyle+$
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}-\frac{1}{2}(J^{c}V^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h})(p^{h})^{v}],-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle+$
$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v},-\frac{1}{\|p^{h}\|^{2}}\cdot
g(V^{h},\nabla
W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle+$
$\displaystyle\sigma\left([\overrightarrow{W},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle+$
$\displaystyle\sigma\left([\overrightarrow{W},-\frac{1}{2}(J^{c}V^{h})^{v}-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h}+2\nabla W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)$
$\displaystyle=:$ $\displaystyle T_{1}+T_{2}+T_{3}+T_{4}+T_{5}$

We will deal with the terms $T_{i}(1\leq i\leq 5)$ in steps.

Step 1 It follows identity (33) that

(44)
$\bar{\sigma}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}})=-g(R^{\nabla}(p^{h},v^{h})p^{h},v^{h}).$

Also it follows from item (3) of Lemma 6 and item (4) of Lemma 6 that

$\displaystyle\Omega^{c}(({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}]),v^{h})+\frac{1}{2}\bar{\sigma}([\underline{\nabla_{p^{h}}},(J^{c}V^{h})^{v}],\underline{\nabla_{v^{h}}})$
$\displaystyle=$ $\displaystyle-g\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}]),J^{c}v^{h}\bigr{)}+\frac{1}{2}\bar{\sigma}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}],(J^{c}v^{h})^{v})$
$\displaystyle=$ $\displaystyle-\frac{1}{2}g\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}]),J^{c}v^{h}\bigr{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\|J^{c}v^{h}\|^{2}+\frac{1}{4\|p^{h}\|^{2}}\left(g(J^{c}v^{h},p^{h})\right)^{2}$

Also it follows from straightforward computations that

(45) $\Omega^{c}\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},(J^{c}v^{h})^{v}]),v^{h}\bigr{)}=-\Omega^{c}\bigl{(}J^{c}v^{h},v^{h}\bigr{)}=\|J^{c}v^{h}\|^{2}$

Also it follows from item (2) of Proposition 2 that

(46)
$\bar{\sigma}([\underline{\nabla_{p^{h}}},(p^{h})^{v}],\underline{\nabla_{v^{h}}})=\bar{\sigma}(-\underline{\nabla_{p^{h}}},\underline{\nabla_{v^{h}}})=0.$

and

(47) $\Omega^{c}\bigl{(}({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}},(p^{h})^{v}]),v^{h}\bigr{)}=-\Omega^{c}(p^{h},v^{h}\bigr{)}=g(p^{h},J^{c}v^{h}).$

It follows from item (2) of Lemma 6 that

(48)
$\bar{\sigma}([(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}})=g(\nabla
J^{c}(p^{h},v^{h}),v^{h}).$

Applying item (3) of Lemma 6, we get

(49) $\Omega^{c}(({\rm
pr}\circ\pi)_{*}([(J^{c}p^{h})^{v},\underline{\nabla_{V^{h}}}]),v^{h})=\Omega^{c}(J^{c}v^{h},v^{h})=-\|J^{c}v^{h}\|^{2}.$

And it follows from (42) and item (1) of Lemma 6 that

(50) $[(J^{c}p^{h})^{v},(J^{c}V^{h})^{v}]=0.$

And it follows from item (1) of Proposition 2 that

(51) $[(J^{c}p^{h})^{v},(p^{h})^{v}]=0.$

Then it follows that

$\sigma\left([-(J^{c}p^{h})^{v},-\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h}+2\nabla W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=0$

Summarizing all the calculations above, we have

(52) $T_{1}=-g(R^{\nabla}(p^{h},v^{h})p^{h},v^{h})-g(\nabla
J^{c}(p^{h},v^{h}),v^{h})-\frac{1}{4}\|J^{c}v^{h}\|^{2}-\frac{3}{4\|p^{h}\|^{2}}\left(g(p^{h},J^{c}v^{h})\right)^{2}$

Step 2 Again, it follows from item (4) of Lemma 6 that

(53)
$\sigma([\underline{\nabla_{p^{h}}},\underline{\nabla_{V^{h}}}],(p^{h})^{v})=-\frac{1}{2}g(J^{c}v^{h},p^{h}).$

And it follows from straightforward computations that

(54)
$\sigma\left([\underline{\nabla_{p^{h}}},\frac{1}{2}(J^{c}V^{h})^{v}+\frac{1}{2\|p^{h}\|^{2}}\cdot
g(V^{h},J^{c}p^{h})(p^{h})^{v}],(p^{h})^{v}\right)=0.$

And it follows from item (3) of Lemma 6 that

(55)
$\sigma\left([(J^{c}V^{h})^{v},\underline{\nabla_{V^{h}}}],(p^{h})^{v}\right)=g(J^{c}v^{h},p^{h})).$

Hence, we have

(56) $T_{2}=-\frac{1}{\|p^{h}\|^{2}}g(J^{c}p^{h},v^{h})g(\nabla W,v^{h}).$

Step 3 It follows from item (2) of Proposition 2 that
$[\underline{\nabla_{p^{h}}},(p^{h})^{v}]=-\underline{\nabla_{p^{h}}}$, hence

$\displaystyle\sigma\left([\underline{\nabla_{p^{h}}},-\frac{1}{\|p^{h}\|^{2}}\cdot
g(V^{h},\nabla
W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)\sigma\left([\underline{\nabla_{p^{h}}},(p^{h})^{v}],\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)\sigma\left(-\underline{\nabla_{p^{h}}},\underline{\nabla_{v^{h}}}-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(p^{h})^{v}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h})-\frac{1}{\|p^{h}\|^{2}}\cdot\left(g(V^{h},\nabla
W)\right)^{2}.$

And it follows from (51) that

(57) $\sigma\left([(J^{c}p^{h})^{v},-\frac{1}{\|p^{h}\|^{2}}\cdot
g(V^{h},\nabla W)(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=0.$

Hence,

(58) $T_{3}=-\frac{1}{\|p^{h}\|^{2}}\cdot g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h})-\frac{1}{\|p^{h}\|^{2}}\cdot\left(g(V^{h},\nabla
W)\right)^{2}$

Step 4 From item (2) of Lemma 3, we have

(59)
$\bar{\sigma}\left([\overrightarrow{W},\underline{\nabla_{V^{h}}}],\underline{\nabla_{v^{h}}}\right)=-{\rm
Hess}\ W(v^{h},v^{h}).$

######  Lemma 7.

The following identity holds.

$({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}])=\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)p^{h}.$

###### Proof.

First of all, from (43),item (3) of Lemma 2 and Lemma 3 it follows

$({\rm
pr}\circ\pi)_{*}([\underline{\nabla_{p^{h}}}-(J^{c}p^{h})^{v}+\overrightarrow{W},\underline{\nabla_{V^{h}}}-\frac{1}{2}(J^{c}V^{h})^{v}])=0,\quad{\rm
mod}\ p^{h}.$

Then together with item (3)-(4) of Lemma 6 and item (2) of Proposition 2 we
have

$({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}])=0,\quad{\rm
mod}\ p^{h}.$

Furthermore, from the classical Cartan’s formula we have

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
d\sigma(\overrightarrow{W},\underline{\nabla_{V^{h}}},(p^{h})^{v})=\overrightarrow{W}(\sigma(\underline{\nabla_{V^{h}}},(p^{h})^{v}))-\underline{\nabla_{V^{h}}}(\sigma(\overrightarrow{W},(p^{h})^{v}))+(p^{h})^{v}(\sigma(\overrightarrow{W},\underline{\nabla_{V^{h}}}))$
$\displaystyle-$
$\displaystyle\sigma([\overrightarrow{W},\underline{\nabla_{V^{h}}}],(p^{h})^{v})+\sigma([\overrightarrow{W},(p^{h})^{v}],\underline{\nabla_{V^{h}}})-\sigma([\underline{\nabla_{V^{h}}},(p^{h})^{v}],\overrightarrow{W}).$

Since $\overrightarrow{W}$ is constant on the fiber of $T^{*}M$, one can
easily show

$(p^{h})^{v}(\sigma(\overrightarrow{W},\underline{\nabla_{V^{h}}}))-\sigma([\underline{\nabla_{V^{h}}},(p^{h})^{v}],\overrightarrow{W})=0.$

Then,

$\sigma([\overrightarrow{W},\underline{\nabla_{V^{h}}}],(p^{h})^{v})=\sigma([\overrightarrow{W},(p^{h})^{v}],\underline{\nabla_{V^{h}}})=g(\nabla
W,v^{h}).$

Hence, the required identity follows and the lemma is proved. ∎

As a direct consequence of the last lemma, we have

$\displaystyle\Omega^{c}\bigl{(}({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}]),v^{h}\bigr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\|p^{h}\|^{2}}g(J^{c}p^{h},v^{h})g(v^{h},\nabla W),$
$\displaystyle\sigma\left([\overrightarrow{W},\underline{\nabla_{V^{h}}}],(p^{h})^{v}\right)$
$\displaystyle=$ $\displaystyle g(v^{h},\nabla W).$

As a result of above calculations, we get

$T_{4}=-{\rm Hess}\
W(v^{h},v^{h})-\frac{1}{\|p^{h}\|^{2}}g(J^{c}p^{h},v^{h})g(v^{h},\nabla
W)-\frac{1}{\|p^{h}\|^{2}}(g(v^{h},\nabla W))^{2}.$

Step 5 First of all, we show the following

######  Lemma 8.

The following identity holds.

$[\overrightarrow{W},(J^{c}V^{h})^{v}]=\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)(Jp^{h})^{v}.$

###### Proof.

As $\overrightarrow{W}=-(\nabla W)^{v}$ is constant on the fiber of $T^{*}M$,
we can proceed with the following calculations

(61) $([\overrightarrow{W},(J^{c}V^{h})^{v}])^{h}=({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{J^{c}V^{h}}}])=J^{c}\left(({\rm
pr}\circ\pi)_{*}([\overrightarrow{W},\underline{\nabla_{V^{h}}}])\right).$

Substituting the identity of Lemma 7 into the last identity, we get

$([\overrightarrow{W},(J^{c}V^{h})^{v}])^{h}=\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)J^{c}p^{h},$

and then the required identity follows. ∎

As a direct consequence, we have

$\sigma\left([\overrightarrow{W},(J^{c}V^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=-\frac{1}{\|p^{h}\|^{2}}g(v^{h},\nabla
W)g(J^{c}p^{h},v^{h}).$

Furthermore, since
$[\overrightarrow{W},(p^{h})^{v}]=\overrightarrow{W}=-(\nabla W)^{v}$,then

$\sigma\left([\overrightarrow{W},(p^{h})^{v}],\underline{\nabla_{v^{h}}}\right)=g(\nabla
W,v^{h}).$

Therefore,

$T_{5}=-\frac{1}{\|p^{h}\|^{2}}(g(v^{h},\nabla W))^{2}.$

Combining the results of Step 1-5 and using the fact
$\|p^{h}\|^{2}=2(c_{0}+W)$, we get the expression of the reduced curvature
maps, as shown in Theorem 2.

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