\input amstex
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\topmatter
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\author
Vladimir A. Sharafutdinov
\endauthor
\title
Some questions of integral geometry on Anosov manifolds
\endtitle
%\affil
%\endaffill
%\address
%Sobolev Institute of Mathematics, Universitetskiy pr., 4,
%Novosibirsk, 630090, Russia
%\endaddress
%\email
%sharaf\@math.nsc.ru
%endemail
\thanks
The work was started during the stay in Max-Planck-Institut f\"ur
Mathematik in Bonn,
partly supported by CRDF (Grant RM2-143).
\endthanks
\endtopmatter

\document

\head
1. Posing the problem and formulating the results
\endhead

The author is grateful to B. Kleiner for the following remark that has
been used as a start point of the present work: there is an analogy
between the modified horizontal derivative constructed in author's
works \cite{Sh1, Sh3} and asymptotic Jacobi fields first introduced by
E.~Hopf \cite{H} and then developed by many authors \cite{G, Es, Eb}.
Indeed, absence of conjugate points plays the crucial role in both the
cases, and the same Riccati equation is used. The author is also
grateful to N.~S.~Dairbekov for fruitful discussions of the subject.

Let
$ (M,g) $
be a closed (= compact without boundary) Riemannian manifold. We are
interesting in questions like the following one: to what extent is a
smooth function
$ f\in C^\infty(M) $
determined by its integrals over all closed geodesics?

Of course, the question is sensible only in the case when
$ (M,g) $
has sufficiently many closed geodesics. Therefore the question was
first investigated for symmetric Riemannian manifolds. The simplest of
such manifolds is the sphere with the standard metric. P.~Funk \cite{F}
proved that the even part of a function on the two-dimensional sphere is
determined by integrals over great circles. This work is traditionally
considered as the start point of integral geometry. Later the problem
was investigated for some other symmetric spaces \cite{Hl}.

Anosov manifolds constitute another wide class of manifolds with
sufficiently many closed geodesics. Recall that a closed Riemannian
manifold is called the {\it Anosov manifold} if its geodesic flow is of
Anosov type. For such a manifold, the set of closed geodesics is dense
in the set of all geodesics. The class of Anosov manifolds is opposite
in some sense to the class of homogeneous Riemannian manifolds because
the isometry group of an Anosov manifold is discrete \cite{Es}.

\proclaim{Theorem 1.1}
Let
$ (M,g) $
be an Anosov manifold. If a function
$ f\in C^\infty(M) $
integrates to zero over every closed geodesic then
$ f $
must itself be zero.
\endproclaim

A closed Riemannian manifold is said to have a simple length spectrum if
there do not exist two different closed geodesics such that the ratio
of their lengths is a rational number. This is a generic condition.

\proclaim{Corollary 1.2}
Let
$ (M,g) $
be an Anosov manifold with simple length spectrum, and
$ {\Delta}: C^\infty(M)\rightarrow C^\infty(M) $
be the corresponding Laplace --- Beltrami operator. If real functions
$ q_1,q_2\in C^\infty(M) $
are such that the operators
$ {\Delta}+q_1 $
and
$ {\Delta}+q_2 $
have coincident spectra, then
$ q_1\equiv q_2 $.
\endproclaim

Theorem 1.1 implies Corollary 1.2 by the same arguments as in
\cite{GK}.

The similar result is valid for 1-forms.

\proclaim{Theorem 1.3}
Let
(M,g)
be an Anosov manifold, and
$ f $
be a smooth 1-form on
$ M $.
If
$ f $
integrates to zero around every closed geodesic, then
$ f $
is an exact form.
\endproclaim

We now consider the corresponding question for symmetric tensor fields
of arbitrary degree. Let
$ \tau'_M $
be the cotangent bundle of a closed Riemannian manifold
$ M $,
and
$ C^\infty(S^m\tau'_M) $
be the space of smooth symmetric covariant tensor fields of degree
$ m $.
For such a field
$ f\in C^\infty(S^m\tau'_M) $
and a closed geodesic
$ \gamma :[a,b]\rightarrow M $,
we may consider the integral
$$
If(\gamma )=
\oint\limits_{\gamma}\langle f,{\dot\gamma}^m\rangle dt
=
\int\limits_{a}^{b}
f_{i_1\dots i_m}(\gamma (t))\dot\gamma{}^{i_1}(t)\dots
\dot\gamma{}^{i_m}(t)\,dt.
      \eqno{(1.1)} $$
The integrand on (1.1) is written with  use made by local
coordinates. Nevertheless, it is evidently invariant, i.e., independent
of the choice of coordinates. Let
$ Z(S^m\tau'_M) $
denote the subspace of
$ C^\infty(S^m\tau'_M) $
consisting of all fields such that
$ If(\gamma)=0 $
for every closed geodesic
$ \gamma  $.
For
$ m>0 $
this subspace is not zero as is seen from the following argument. Let
us introduce the first order differential operator
$$
d=\sigma {\nabla}:
C^\infty(S^{m-1}\tau'_M)\rightarrow C^\infty(S^{m}\tau'_M),
      \eqno{(1.2)} $$
where
$ \nabla  $
is the covariant derivative and
$ \sigma  $
is the symmetrization. This operator is called the {\it inner
differentiation}. A tensor field
$ f $
is called the {\it potential field} if it can be represented in the
form
$ f=dv $
for some
$ v\in C^\infty(S^{m-1}\tau'_M) $.
Let
$ P(S^{m}\tau'_M) $
denote the space of all potential fields. If
$ f=dv $,
then the integrand on (1.1) equals to
$ d(v_{i_1\dots i_{m-1}}(\gamma(t))
\dot\gamma{}^{i_1}(t)\dots \dot\gamma{}^{i_{m-1}}(t))/dt $.
Therefore there is the inclusion
$$
P(S^m\tau'_M)\subset Z(S^m\tau'_M).
      \eqno{(1.3)} $$
The principal question of integral geometry of tensor fields is
formulated as follows: for what classes of closed Riemannian manifolds
and for what values of
$ m $
is inclusion (1.3) in fact the equality? For instance, Theorems 1.1 and
1.3 say that (1.3) is the equality for an Anosov manifold in the cases
$ m=0 $
and
$ m=1 $.
The equality is also proved in the case of arbitrary
$ m $
for some symmetric spaces \cite{M1, M2}.

The following result is proved in \cite{CSh}.

\proclaim{Theorem 1.4}
For an Anosov manifold of nonpositive sectional curvature, inclusion
(1.3) is the equality for all
$ m $.
\endproclaim

In \cite{CSh} this theorem is formulated for negatively curved
manifolds. Nevertheless, only nonpositivity of the curvature and Anosov
type of the geodesic flow are used in the proof.

Let us recall some standard definitions. For an open
$ U\subset {\bold R}^n $
and
$ p\geq 1 $,
the {\it Sobolev space}
$ W^1_p(U) $
is the completion of
$ C^1(U) $
with respect to the norm
$$
\|f\|_{W^1_p}=\|f\|_{L_p}+\|{\nabla} f\|_{L_p}
=\left(\int\limits_{U}|f(x)|^p\,dx\right)^{1/p}
+
\left(\int\limits_{U}|{\nabla}f(x)|^p\,dx\right)^{1/p}.
$$

A
$ k $-dimensional {\it distribution} on a manifold
$ N $
is a section of the Grassmanian bundle
$ G_k(N) $
whose fiber at a point
$ x\in N $
is the manifold
$ G_k(T_xN) $
of all
$ k $-dimensional subspaces of
$ T_xN $.
A distribution is said to belong to the class
$ W^1_p $
if in local coordinates it is represented by functions of the class
$ W^1_{p,loc} $.

Recall that the {\it stable} and {\it unstable distributions}
participate in the definition of an Anosov flow \cite{An}. In the case
of a smooth Anosov flow, these distributions are
absolutely continuous \cite{An} and H\"older-continuous
\cite{AS} but in general they are not smooth.

In the case of an
$ n $-dimensional Anosov manifold
$ M $, the stable and unstable
distributions are defined on the manifold
$ \Omega M $
of unit tangent vectors. Note that
$ \dim (\Omega M)=2n-1 $.


The main result of the present article is the following

\proclaim{Theorem 1.5}
Let
$ (M,g) $
be an
$ n $-dimensional
Anosov manifold whose stable and unstable distributions
belong to the class
$ W^1_p $
with some
$ p>2n-1 $.
Then inclusion (1.3) has a finite codimension for
every
$ m $.
\endproclaim

In the case of an Anosov surface (= two-dimensional Anosov manifold)
the stable nd unstable distributions are
$ C^1 $-smooth
\cite{KH}. We thus have

\proclaim{Corollary 1.6}
For an Anosov surface, inclusion (1.2) has a finite codimension for
every
$ m $.
\endproclaim

In view of the theorem, we pose the following

\proclaim{Conjecture 1.7}
The stable and unstable distributions of an
$ n $-dimensional Anosov manifold belong to the class
$ W^1_p $
with some
$ p>2n-1 $.
\endproclaim

The conjecture agrees with all known results on regularity of the
stable and unstable distributions of Anosov flows.

\smallpagebreak

In the case of
$ m=2 $,
the question under consideration relates to the spectral rigidity
problem. Let us recall the corresponding definitions. A smooth
one-parameter family
$ g^\tau \ (-\varepsilon <\tau <\varepsilon ) $
of metrics on a manifold
$ M $
is called the {\it deformation} of a metric
$ g $
if
$ g^0=g $.
Such a family is called the {\it isospectral deformation} if the
spectrum of the Laplace --- Beltrami operator
$ \Delta^\tau  $
of the metric
$ g^\tau  $
is independent of
$ \tau  $.
A deformation
$ g^\tau  $
is called the {\it trivial deformation} if there exists a family
$ \varphi^\tau  $
of diffeomorphisms of
$ M $
such that
$ g^\tau =(\varphi ^\tau)^*g $.
A manifold
$ (M,g) $
is called {\it spectrally rigid} if it does not admit a
nontrivial isospectral deformation.

The following result is proved in \cite{GK}.

\proclaim{Theorem 1.8}
An Anosov manifold is spectrally rigid if and only if inclusion (1.3) is
the equality for
$ m=2 $.
\endproclaim

The theorem is formulated in \cite{GK} for negatively curved
manifolds. Nevertheless, the same proof is valid for Anosov manifolds.

\smallpagebreak

Our method of proving Theorem 1.5 is a combination of methods used in
\cite{Sh2} and \cite{CSh}.
The principal idea of \cite{Sh2} is constructing a semibasic tensor
field
$ a=(a^{ij}) $
such that the curvature tensor of the modified horizontal derivative
$ \overset a \to{\nabla} $
meets the equation
$ \overset a\to R{}_{ijkl}\xi^i\xi^k=0 $.
In this case the Pestov identity does not contain the term dependent
on curvature. The stable and unstable distributions give us the
possibility to construct a similar modifying tensor field
$ a=(a^{ij}) $
in the case of an Anosov manifold. Unfortunately, this tensor field is
not smooth. Therefore we have to approximate it by a smooth field. In
the process of approximation, we should control boundedness of first
order derivatives of the smoothing field. For such the control we need
the hypothesis that the stable and unstable distributions belong to the
class
$ W^1_p $.

Theorems 1.1 and 1.3 are proved in the same way. But in this case we
need no estimate of first order derivatives of the modifying tensor
field.

\smallpagebreak

We conclude the introduction by posing the following question.

\proclaim{Problem 1.9}
Does there exist an Anosov manifold such that
inclusion (1.3) is not equality for some values of
$ m $?
\endproclaim

\head
2. Reduction of Theorem 1.5 to an estimate for the kinetic equation
\endhead

Given a Riemannian manifold
$ (M,g) $,
by
$ TM $
we denote its tangent space. Points of
$ TM $
are denoted by pairs
$ (x,\xi) $,
where
$ x\in M $
and
$ \xi \in T_xM $.
Let
$ \Omega M $
be the manifold of unit tangent vectors. By
$ G^t:TM\rightarrow TM $
we denote the geodesic flow, and by
$ H $,
the corresponding vector field on
$ TM $.
At points of
$ \Omega M $
$ H $
is tangent to
$ \Omega M $.
Therefore
$ H $
can be considered as a first order differential operator
$ H:C^\infty(\Omega M)\rightarrow C^\infty(\Omega M) $.

The operator
$$ -\delta :C^\infty(S^m\tau'_M)\rightarrow C^\infty(S^{m-1}\tau'_M) $$
is defined as the formal dual of operator (1.2). We call
$ \delta  $
the {\it divergence operator}.

Here we will show that Theorem 1.5 is a corollary of the following

\proclaim{Lemma 2.1}
Let
$ (M,g) $
be an
$ n $-dimensional
Anosov manifold whose stable and unstable distributions
belong to the class
$ W^1_p $
with some
$ p>2n-1 $.
If a function
$ u\in C^\infty(\Omega M) $
and tensor field
$ f\in C^\infty(S^m\tau'_M) $
are connected by the kinetic equation
$$
Hu(x,\xi)=f_{i_1\dots i_m}(x)\xi^{i_1}\dots\xi^{i_m},
      \eqno{(2.1)} $$
then the estimate
$$
\|u\|^2_{H^1(\Omega M)}\leq
C\left(\|u\|^2_{L_q(\Omega M)}+
\|\delta f\|_{L_2(S^{m-1}\tau'_M)}\cdot\|u\|_{L_2(\Omega M)}\right)
      \eqno{(2.2)} $$
holds with
$ q=2p/(p-2) $
and a constant
$ C $
independent of
$ u $
and
$ f $.
\endproclaim

\demo{Proof of Theorem 1.5}
Let
$ N=2n-1=\dim(\Omega M) $.
The inequality
$ p>N $
implies that
$ q<2N/(N-2) $,
and there is the compact imbedding
$ H^1(\Omega M)\subset L_q(\Omega M) $.

If the theorem is not true, there exists an infinite sequence of tensor
fields
$ z_k\in Z(S^m\tau'_M)\ (k=1,2,\dots) $
which is linearly independent
$ \text{mod}\,(P(S^m\tau'_M)) $.
Applying Theorem 2.2 of \cite{CSh}, we decompose every field
$ z_k $
into potential and solenoidal parts
$$
z_k=y_k+dv_k,\quad \quad \delta y_k=0.
      \eqno{(2.3)} $$
Then the sequence
$ y_k\in Z(S^m\tau'_M) $
is linearly independent. Applying the smooth version of the Liv\v cic
theorem \cite{LMM}, we find functions
$ w_k\in C^\infty(\Omega M) $
satisfying the kinetic equation
$$
Hw_k=\langle y_k(x),\xi^m\rangle .
      \eqno{(2.4)} $$
The sequence
$ w_k $
is linearly independent since in the other case (2.4) would imply
linear dependence of the sequence
$ y_k $.
Applying the Riesz theorem on an almost perpendicular \cite{Y, p.~124},
we construct a new sequence of functions
$ u_k\in C^\infty(\Omega M) $
such that
$$
\|u_k\|_{L_q(\Omega M)}=1,\quad  \quad
\|u_k-u_l\|_{L_q(\Omega M)}>1/2\quad \text{for}\quad k\neq l,
      \eqno{(2.5)} $$
and every
$ u_k $
is a linear combination of
$ w_1,\dots,w_k $.
Equation (2.4) implies that
$$
Hu_k = \langle f_k(x),\xi ^m\rangle ,
$$
where
$ f_k\in C^\infty(S^m\tau'_M) $
is a linear combination of
$ y_1,\dots,y_k $.
Therefore (2.3) implies that
$$
\delta f_k=0.
      \eqno{(2.6)} $$

By equalities (2.5) and (2.6),
estimate (2.2) has the following form for the pair
$ u_k, f_k $:
$$
\|u_k\|_{H^1(\Omega M)}\leq C\|u_k\|_{L_q(\Omega M)}=C.
$$
In other words, the sequence
$ u_k $
is bounded in
$ H^1(\Omega M) $.
Since the imbedding
$ H^1(\Omega M)\subset L_q(\Omega M) $
is compact, the sequence
$ u_k $
contains a subsequence converging in
$ L_q(\Omega M) $.
But this contradicts to inequality (2.5).
\enddemo

\head
3. The modifying tensor fields
$ \overset s\to a $
and
$ \overset u\to a $
\endhead

Given a manifold
$ M $
and open set
$ U\subset TM $,
by
$ C(\beta^r_sM;U) $
we denote the space of continuous semibasic tensor fields of degree
$ (r,s) $
over
$ U $.
The definition of a semibasic tensor field is given in Chapter 3 of
\cite{Sh1}, see also \cite{PSh}.

\proclaim{Lemma 3.1}
Let
$ (M,g) $
be an
$ n $-dimensional
Anosov manifold. There exist continuous semibasic tensor fields
$ \overset s\to a = ( \overset s\to a{}_{ij}(x,\xi))
\in C(\beta^0_2M;T^0M) $
and
$ \overset u\to a = ( \overset u\to a{}_{ij}(x,\xi))
\in C(\beta^0_2M;T^0M) $
defined on
$ T^0M=\{(x,\xi)\in TM\mid\xi \neq 0\} $
such that

(1) the fields are symmetric
$$
\overset s\to a{}_{ij}=\overset s\to a{}_{ji},\quad
\overset u\to a{}_{ij}=\overset u\to a{}_{ji}
$$
and orthogonal to the vector
$ \xi  $
$$
\xi^i\overset s\to a{}_{ij}(x,\xi)=0,\quad
\xi ^i\overset u\to a{}_{ij}(x,\xi)=0;
$$

(2) they are positively homogeneous of degree 1 in
$ \xi  $
$$
\overset s\to a(x,t\xi)=t\overset s\to a(x,\xi),\quad
\overset u\to a(x,t\xi)=t\overset u\to a(x,\xi)\quad
\text{for}\quad t>0;
$$

(3) the rank of the matrix
$ (\overset s\to a{}_{ij}-\overset u\to a{}_{ij}) $
equals
$ n-1 $
at every point
$ (x,\xi )\in T^0M $;

(4) along every geodesic
$ \gamma :{\bold R}\rightarrow M $,
the fields
$ \overset s\to a{}^i_j(t)=
(g^{ik}\overset s\to a{}_{kj})(\gamma(t),\dot\gamma(t)) $
and
$ \overset u\to a{}^i_j(t)=
(g^{ik}\overset u\to a{}_{kj})(\gamma(t),\dot\gamma(t)) $
are smooth and satisfy the Riccati equation
$$
a'+a^2+R=0,
      \eqno{(3.1)} $$
where the prime denotes the covariant derivative, and
$ R=R(t) $
is the curvature operator,
$ R^i_j=R^i_{\,kjl}\dot\gamma{}^k\dot\gamma{}^l $;

(5) if the stable and unstable distributions belong to the class
$ W^1_p $,
then the fields
$ \overset s\to a $
and
$ \overset u\to a $
belong also to
$ W^1_p $.
\endproclaim

Before proving the lemma we recall some notions concerning the geodesic
 flow and Jacobi fields.

Let
$ \tau _M=(TM,\pi,M) $
be the tangent bundle of a Riemannian manifold
$ (M,g) $.
Given a point
$ (x,\xi )\in TM $,
the canonical isomorphism
$$
T_{(x,\xi)}(TM)\cong T_xM\oplus T_xM,\quad
v\mapsto (d\pi (v),Kv)
$$
is defined, where
$ K:TTM\rightarrow TM $
is the {\it connection mapping}. The subspaces of
$ T_{(x,\xi)}(TM) $
corresponding to the summands of the right-hand side are called the
{\it horizontal} and {\it vertical spaces} respectively. This
isomorphism defines the {\it Sasaki metric} on
$ TM $
$$
\langle v,w\rangle =\langle d\pi (v),d\pi (w)\rangle +
\langle Kv,Kw\rangle .
$$

The metric
$ g $
determines the isomorphism of the tangent and cotangent bundles. The
standard symplectic structure of the cotangent bundle, being shifted to
$ TM $
with the help of the isomorphism, is defined by the 2-form
$$
\omega (v,w)=\langle d\pi (v),Kw\rangle -
\langle d\pi (w),Kv\rangle .
$$

The tangent space of the manifold
$ \Omega M $
of unit tangent vectors is defined at a point
$ (x,\xi )\in \Omega M $
by the equality
$$
T_{(x,\xi)}(\Omega M)=\{v\in T_{(x,\xi)}(TM)\mid
\langle Kv,\xi \rangle =0\}.
      \eqno{(3.2)} $$

Let
$ H $
be the geodesic vector field on
$ TM $
generating the geodesic flow
$ G^t:TM\rightarrow TM $.
Note that
$ H $
is horizontal,
$ KH=0 $.
Given a vector
$ v\in T_{(x,\xi)}(TM) $,
the vector field
$ Y_v(t)=d\pi\circ dG^t(v) $
is a Jacobi vector field along the geodesic
$ \gamma (t)=\exp_xt\xi  $
with the covariant derivative
$ Y'_v(t)=K\circ dG^t(v) $.

Let now
$ (M,g) $
be an Anosov manifold. Given
$ (x,\xi)\in \Omega M $,
two
$ (n-1) $-dimensional subspaces
$ X_s(x,\xi) $
and
$ X_u(x,\xi) $
of
$ T_{(x,\xi)}(\Omega M) $
are defined which are called the {\it stable} and {\it unstable spaces}
respectively. We will use the following properties of these spaces
which follow from Proposition 1.7 and Theorem 3.2 of \cite{Eb}.

(i) The distributions
$ (x,\xi )\mapsto X_s(x,\xi) $
and
$ (x,\xi)\mapsto X_u(x,\xi) $
are continuous and invariant with respect to the geodesic flow, i.e.,
$$
dG^t(X_s(x,\xi))=X_s(G^t(x,\xi)),\quad
dG^t(X_u(x,\xi))=X_u(G^t(x,\xi)).
$$

(ii) Each of the spaces
$ X_s(x,\xi) $
and
$ X_u(x,\xi) $
is orthogonal to the vector
$ H(x,\xi) $.
The space
$ T_{(x,\xi)}(\Omega M) $
splits to the (not orthogonal) direct sum of tree subspaces
$$
T_{(x,\xi)}(\Omega M)=
X_s(x,\xi)\oplus X_u(x,\xi)\oplus \{H(x,\xi)\}.
$$

(iii) The restriction of the mapping
$ d\pi  $
to each of the spaces
$ X_s(x,\xi) $
and
$ X_u(x,\xi) $
is an isomorphism onto
$ N_{(x,\xi)}=\{\eta \in T_xM\mid\langle \xi ,\eta \rangle =0\} $.

(iv)
$ X_s(x,\xi) $
and
$ X_u(x,\xi) $
are Lagrangian spaces, i.e.,
$ \omega (v,w)=0\quad \text{for}\quad v,w\in X_s(x,\xi)\ (X_u(x,\xi))
$.

\demo{Proof of Lemma 3.1}
Given
$ (x,\xi)\in \Omega M $,
we define two endomorfisms
$ b_s(x,\xi) $
and
$ b_u(x,\xi) $
of the space
$ N_{(x,\xi)} $
as compositions of the following mappings:
$$
b_s(x,\xi):N_{(x,\xi)}{\overset{(d\pi)^{-1}}\to
{\longrightarrow}}
X_s(x,\xi){\overset K\to\longrightarrow} N_{(x,\xi)},
$$
$$
b_u(x,\xi):N_{(x,\xi)}{\overset{(d\pi)^{-1}}\to
{\longrightarrow}}
X_u(x,\xi){\overset K\to\longrightarrow} N_{(x,\xi)}.
$$
Note that
$ Kv\in N_{(x,\xi)} $
for every
$ v\in T_{(x,\xi)}(\Omega M) $
because of equality (3.2). This definition is equivalent to the
following rule that is more comfortable for using: two vectors
$ \eta,\zeta \in N_{(x,\xi)} $
are connected by the equality
$ b_s(x,\xi)\eta =\zeta  $
if and only if there exists
$ v\in X_s(x,\xi) $
such that
$ d\pi(v)=\eta  $
and
$ Kv=\zeta  $.
The similar rule is valid for the operator
$ b_u(x,\xi) $.

We establish the following properties of the operators
$ b_s(x,\xi) $
and
$ b_u(x,\xi) $.

1. The operators
$ b_s(x,\xi) $
and
$ b_u(x,\xi) $
continuously depend on
$ (x,\xi)\in \Omega M $.
If the stable and unstable distributions belong to the class
$ W^1_p $,
then the functions
$ (x,\xi)\rightarrow b_s(x,\xi) $
and
$ (x,\xi)\rightarrow b_u(x,\xi) $
belong also to
$ W^1_p $.

2. The operators
$ b_s $
and
$ b_u $
are selfdual. Indeed, let
$ \eta_i\in N_{(x,\xi)} $
and
$ \zeta_i=b_s(x,\xi)\eta_i\ (i=1,2) $.
Then there are
$ v_i\in X_s(x,\xi) $
such that
$ d\pi(v_i)=\eta_i $
and
$ Kv_i=\zeta_i $.
Therefore
$$
\langle b_s\eta_1,\eta_2\rangle -\langle \eta_1,b_s\eta_2\rangle
= \langle \zeta_1,\eta_2\rangle -\langle \eta_1,\zeta_2\rangle
$$
$$
=\langle Kv_1,d\pi(v_2)\rangle -\langle d\pi(v_1),Kv_2\rangle
=\omega (v_1,v_2)=0
$$
since
$ X_s(x,\xi) $
is a Lagrangian space.

3. The operator
$ b_s-b_u $
is nondegenerate. Indeed, let
$ b_s\eta =b_u\eta  $
for some vector
$ \eta \in N_{(x,\xi )} $.
There exist vectors
$ v\in X_s $
and
$ w\in X_u $
such that
$$
d\pi(v)=\eta =d\pi (w),\quad Kv=b_s\eta =b_u\eta =Kw.
$$
These relations imply that
$ v=w\in X_s\cap X_u=0 $.
Consequently,
$ v=w=0 $
and
$ \eta =0 $.

4. Along every unit speed geodesic
$ \gamma :{\bold R}\rightarrow M $,
each of the operator functions
$$
b_s(t)=b_s(\gamma(t),\dot\gamma(t)):
N_{(\gamma(t),\dot\gamma(t))}\rightarrow N_{(\gamma(t),\dot\gamma(t))},
$$
$$
b_u(t)=b_u(\gamma(t),\dot\gamma(t)):
N_{(\gamma(t),\dot\gamma(t))}\rightarrow N_{(\gamma(t),\dot\gamma(t))}
$$
satisfies the Riccati equation
$$
b'+b^2+R=0.
      \eqno{(3.3)} $$
Indeed, fix a geodesic
$ \gamma  $
and denote
$ N_t=N_{(\gamma(t),\dot\gamma(t))} $.
Define the operator function
$ D(t):N_t\rightarrow N_t $
as the solution to the Jacobi equation
$$
D''+RD=0
      \eqno{(3.4)} $$
satisfying the initial conditions
$$
D(0)=E\ (\,=\ \text{identity}),\quad D'(0)=b_s(0).
$$
Establish validity of the equality
$$
D'(t)=b_s(t)D(t).
      \eqno{(3.5)} $$
To this end fix a vector
$ \eta \in N_0 $
and denote by
$ \eta (t)\in N_t $
the result of parallel translating the vector
$ \eta  $
along
$ \gamma  $.
Then the vector function
$ Y(t)=D(t)\eta (t) $
is the Jacobi vector field along
$ \gamma  $
satisfying the initial conditions
$$
Y(0)=\eta ,\quad Y'(0)=b_s(0)\eta .
$$
On the other hand, if a vector
$ v\in X_s(\gamma(0),\dot\gamma(0)) $
is such that
$ d\pi (v)=\eta  $,
then
$$
D(t)\eta(t)=Y(t)=d\pi \circ dG^t(v),\quad
D'(t)\eta (t)=Y'(t)=K\circ dG^t(v).
$$
The vector
$ v_t=dG^t(v) $
belongs to
$ X_s(\gamma(t),\dot\gamma(t)) $
because the distribution
$ X_s $
is invariant with respect to the geodesic flow. The proceeding
equalities can be rewritten as follows:
$$
D(t)\eta (t)=d\pi (v_t),\quad
D'(t)\eta (t)=Kv_t,\quad
v_t\in X_s(\gamma (t),\dot\gamma (t)).
$$
These relations imply that
$$
D'(t)\eta (t)=b_s(t)D(t)\eta (t).
$$
The latter equality is equivalent to (3.5) because
$ \eta  $
is an arbitrary vector.

The operator
$ D(t) $
is nondegenerate for sufficiently small
$ |t| $,
and equality (3.5) can be rewritten as follows:
$ b_s(t)=D'(t)D^{-1}(t) $.
>From this and Jacobi equation (3.4), we see that
$ b_s(t) $
satisfies Riccati equation (3.3) at least for sufficiently small
$ |t| $.
Since the Riccati equation is invariant with respect to the shift
$ t\mapsto t+t_0 $,
it is satisfied for all
$ t $.

Given
$ (x,\xi )\in \Omega M $,
we define the operators
$$
\overset s\to a(x,\xi):T_xM\rightarrow T_xM,\quad
\overset u\to a(x,\xi):T_xM\rightarrow T_xM
$$
by the equalities
$$
\overset s\to a(x,\xi)|_{N_{(x,\xi)}}=b_s(x,\xi),\quad
\overset s\to a(x,\xi)\xi =0;
$$
$$
\overset u\to a(x,\xi)|_{N_{(x,\xi)}}=b_u(x,\xi),\quad
\overset u\to a(x,\xi)\xi =0.
$$
We then extend the functions
$ \overset s\to a $
and
$ \overset u\to a $
to
$ T^0M $
in such the way that they are positively homogeneous in
$ \xi  $
$$
\overset s\to a(x,t\xi)=t\overset s\to a(x,\xi),\quad
\overset u\to a(x,t\xi)=t\overset u\to a(x,\xi)\quad
\text{for}\quad t>0.
$$
We thus have constructed the semibasic tensor fields
$ \overset s\to a=(\overset s\to a{}^i_j(x,\xi))
\in C(\beta^1_1M;T^0M) $
and
$ \overset u\to a=(\overset u\to a{}^i_j(x,\xi))
\in C(\beta^1_1M;T^0M) $.
The above proved properties of
$ b_s $
and
$ b_u $
imply that the semibasic tensor fields
$ \overset s\to a{}_{ij}=g_{ik}\overset s\to a{}^k_j $
and
$ \overset u\to a{}_{ij}=g_{ik}\overset u\to a{}^k_j $
satisfy all statements of Lemma 3.1.
\enddemo

{\bf Remark}.
Observe that, proving Lemma 3.1, we did not use the main
property of the distributions
$ X_s $
and
$ X_u $
participating in the definition of an Anosov flow:
$ X_s $
is contracted and
$ X_u $
is expanded by
$ dG^t $.
Nevertheless, this property is implicitly used in the proof of Theorem
1.5 via the above reference to the Liv\v cic theorem.

\head
4. Smoothing the tensor fields
$ \overset s\to a $
and
$ \overset u\to a $
\endhead

We would like to use two modified horizontal derivatives defined by the
scheme of Section 8.2 of \cite{Sh1} (see also \cite{Sh3}) with the
modifying tensors
$ \overset s\to a $
and
$ \overset u\to a $
constructed in the previous section. Unfortunately, the fields
$ \overset s\to a $
and
$ \overset u\to a $
are only continuous but are not smooth. For constructing a modified
horizontal derivative, we need at least
$ C^2 $-smoothness of the modifying tensor because the definition of
the curvature tensor assumes existence of second order derivatives.
Therefore we have to smooth the tensor fields
$ \overset s\to a $
and
$ \overset u\to a $.
We will choose the smoothing tensors in such a way that they would
satisfy all statements of Lemma 3.1 with the following exception:
Riccati equation (3.1) will be satisfied approximately.

First of all we will discuss some questions concerning smoothing
sections of a vector bundle.

Let
$ \pi :E\rightarrow N $
be a smooth
$ m $-dimensional vector bundle over a compact manifold
$ N $.
Choose a finite atlas
$ \{U_a,\varphi_a\}^A_{a=1} $
of the manifold
$ N $,
a partition of unity
$ \{\mu_a\}^A_{a=1} $
subordinated to the atlas, and local trivializations
$ (e^a_1,\dots,e^a_m) $
of the bundle over
$ U_a $
(this means that
$ e^a_\alpha \in C^\infty(E;U_a) $,
and the vectors
$ e^a_1(x),\dots,e^a_m(x) $
constitute a basis of the fiber
$ E_x $
for every point
$ x\in U_a $).
Every section
$ f\in C^\infty(E) $
can be uniquely represented in the form
$$
f(x)=\sum\limits_{\alpha =1}^{m}f^\alpha_a (x)e^a_\alpha (x),
\quad x\in U_a.
      \eqno{(4.1)} $$
For
$ 0\leq k<\infty  $,
let
$ C^k(E) $
be the space of sections
$ f $
such that
$ (\mu_af^\alpha_a)\circ\varphi^{-1}_a\in C^k({\bold R}^n) $.
The norm on the space is defined by the equality
$$
\|f\|_{C^k(E)}=\sum\limits_{a=1}^{A}\sum\limits_{\alpha=1}^{m}
\|(\mu_af^\alpha_a)\circ\varphi^{-1}_a\|_{C^k({\bold R}^n)}.
$$
The
$ W^1_p $-norm on
$ C^1(E) $
is defined in the similar way
$$
\|f\|_{W^1_p(E)}=\sum\limits_{a=1}^{A}\sum\limits_{\alpha=1}^{m}
\|(\mu_af^\alpha_a)\circ\varphi^{-1}_a\|_{W^1_p({\bold R}^n)}.
$$
Up to equivalence, the norms are independent of the choice of the atlas,
partition of unity, and trivializations.

Let
$ H\in C^\infty(\tau _N) $
be a smooth vector field on
$ N $.
Choosing a connection on
$ E $,
we can define the derivative
$ Hf $
of a section
$ f $
with respect to
$ H $.
A section
$ f\in C(E) $
is said {\it differentiable along}
$ H $
if the derivative
$ Hf $
exists and belongs to
$ C(E) $.
In such the case the norm
$ \|Hf\|_{C(E)} $
is defined and independent, up to equivalence, of the choice of the
connection.

\proclaim{Lemma 4.1}
Let
$ H\in C^\infty(\tau _N) $
be a smooth vector field on a compact manifold
$ N $
which does not vanish at any point, and
$ E $
be a smooth vector bundle over
$ N $.
Fix a
$ C $-norm on
$ C(E) $
and a connection on
$ E $.
For a differentiable along
$ H $
section
$ f\in C(E) $
and for
$ \varepsilon >0 $,
there exists a smooth section
$ \widetilde{f}\in C^\infty(E) $
such that
$$
\|f-\widetilde{f}\|_{C(E)}<\varepsilon ,\quad
\|H(f-\widetilde{f})\|_{C(E)}<\varepsilon.
$$
Moreover, if the section
$ f $
belongs to the class
$ W^1_p $, then for a fixed
$ W^1_p $-norm the estimate
$$
\|\widetilde{f}\|_{W^1_p(E)}\leq C
$$
holds with some constant
$ C $
independent of
$ \varepsilon  $.
\endproclaim

\demo{Proof}
A chart
$ (U,\varphi) $
of the manifold
$ N $
is called {\it straightening the vector field}
$ H $
if
$ \varphi _*H $
coincides with the coordinate vector field
$ \partial /\partial x^1 $
on the range
$ \varphi (U)\subset{\bold R}^n $.
There exists a finite atlas
$ \{U_a,\varphi_a\}_{a=1}^A $
consisting of straightening charts \cite{Ar}. Choose a partition of
unity subordinated to the atlas and trivializations of
$ E $
over
$ U_a $.
Given a differentiable along
$ H $
section
$ f\in C(E) $,
representation (4.1) holds where, for every
$ a $
and
$ \alpha  $,
$ \widetilde{f}{}^\alpha _a=(\mu_a f^\alpha)\circ\varphi^{-1}_a $
is a continuous compactly supported function on
$ {\bold R}^n $
with the continuous derivative
$ \partial \widetilde{f}{}^\alpha_a/\partial x^1 $.
If
$ f $
belongs to the class
$ W^1_p $, then every
$ \widetilde{f}{}^\alpha_a $
belongs to
$ W^1_p({\bold R}^n) $.

Fix a nonnegative function
$ \lambda \in C^\infty_0({\bold R}^n) $
such that
$ \int_{{\bold R}^n}\lambda \,dx=1 $,
and put
$ \lambda_\delta(x)=\lambda(x/\delta)/\delta^n $
for
$ \delta >0 $.
For every indices
$ a $
and
$ \alpha  $,
the function
$ \overset \delta \to f{}^\alpha_a = f^\alpha_a*\lambda_\delta  $
is
$ C^\infty  $-smooth on
$ {\bold R}^n $,
and
$ \text{supp}\,\overset \delta \to f{}^\alpha_a\subset\varphi_a(U_a) $
for sufficiently small
$ \delta  $.
The differences
$ \overset \delta \to f{}^\alpha_a-\widetilde{f}{}^\alpha_a $
and
$ \partial(\overset \delta \to f{}^\alpha_a-
\widetilde{f}{}^\alpha_a)/\partial x^1 $
tend to zero uniformly on
$ {\bold R}^n $
as
$ \delta \rightarrow 0 $.
If
$ f $
belongs to the class
$ W^1_p $,
then the difference
$ \overset \delta \to f{}^\alpha_a-\widetilde{f}{}^\alpha_a $
tends to zero in the space
$ W^1_p({\bold R}^n) $.
Therefore the section
$$
\widetilde{f}=\sum\limits_{a=1}^{A}\sum\limits_{\alpha =1}^{m}
\mu_a(\overset \delta \to f{}^\alpha_a\circ\varphi_a)e^a_\alpha
$$
possesses all the desired properties for sufficiently small
$ \delta >0 $.
The lemma is proved.
\enddemo

Given an Anosov manifold
$ (M,g) $,
let
$ \beta^0_2M|_{\Omega M} $
be the restriction of the bundle
$ \beta^0_2M $
to the compact manifold
$ N=\Omega M $,
and
$ E $
be its subbundle consisting of semibasic tensors
$ f=(f_{ij}) $
satisfying the conditions
$ f_{ij}=f_{ji} $
and
$ \xi^if_{ij}=0 $.
Let
$ \overset s\to a, \overset u\to a\in C(E) $
be the sections constructed in Lemma 3.1, and
$ H $
be the geodesic vector field on
$ \Omega M $.
Applying Lemma 4.1 to
$ \overset s\to a $
and
$ \overset u\to a $
and extending the so-obtained smooth fields to
$ T^0M $
by homogeneity, we obtain the following

\proclaim{Lemma 4.2}
Let
$ (M,g) $
be an
$ n $-dimensional Anosov manifold. For every
$ \varepsilon >0 $,
there exist smooth semibasic tensor fields
$ \overset s\to a=(\overset s\to a{}_{ij}(x,\xi))\in
C^\infty(\beta^0_2M;T^0M) $
and
$ \overset u\to a=(\overset u\to a{}_{ij}(x,\xi))\in
C^\infty(\beta^0_2M;T^0M) $
satisfying statements (1)--(3) of Lemma 3.1 and the following ones:

($4'$) along every geodesic
$ \gamma :{\bold R}\rightarrow M $,
the fields
$ \overset s\to a{}^i_j(t)=(g^{ik}\overset s\to a{}_{kj})
(\gamma(t),\dot\gamma(t)) $
and
$ \overset u\to a{}^i_j(t)=(g^{ik}\overset u\to a{}_{kj})
(\gamma(t),\dot\gamma(t)) $
satisfy the inequality
$$
|a'+a^2+R|<\varepsilon
$$
for all
$ t\in{\bold R} $;

($4''$) the estimates
$$
\|\overset s\to a\|_{C(\beta^0_2M|_{\Omega M})}\leq C,\quad
\|\overset u\to a\|_{C(\beta^0_2M|_{\Omega M})}\leq C
$$
hold with some constant
$ C $
independent of
$ \varepsilon  $;

($5'$) if the stable and unstable distributions belong to the class
$ W^1_p $,
then the estimates
$$ \|\overset s\to a\|_{W^1_p(\beta^0_2M|_{\Omega M})}
\leq C,\quad
\|\overset u\to a\|_{W^1_p(\beta^0_2M|_{\Omega M})}\leq C
$$
hold with some constant
$ C $
independent of
$ \varepsilon  $.
\endproclaim

\head
5. The modified horizontal derivatives
$ \overset s\to \nabla  $
and
$ \overset u\to \nabla  $
\endhead

Here we will use the vertical and horizontal derivatives
$ \overset v\to \nabla,\overset h\to \nabla:
C^\infty(\beta^r_sM)\rightarrow C^\infty(\beta^r_{s+1}M)  $
defined for a Riemannian manifold
$ (M,g) $.
The definition of the derivatives is presented in Chapter 3 of
\cite{Sh1} (see also \cite{PSh}). There are canonical isomorphisms
$ \beta^r_{s}M\cong\beta^{r+s}_{0}M\cong\beta^{0}_{r+s}M $
defined by the metric
$ g $,
and we will sometimes identify these bundles.

Let us now recall the construction of a modified horizontal derivative
exposed in Chapter 8 of \cite{Sh1} and in \cite{Sh3}. A {\it modifying
tensor} is a semibasic tensor field
$ a=(a^{ij}(x,\xi))\in C^\infty(\beta^2_0M) $
which is symmetric,
$ a^{ij}=a^{ji} $;
orthogonal to
$\xi$,
$ \xi_ia^{ij}(x,\xi)=0 $;
and positively homogeneous of degree one in
$ \xi  $.
Given such a field, the {\it modified horizontal derivative} is the
first order differential operator
$$
{\overset a\to{\nabla}}:
C^\infty(\beta^r_sM)\rightarrow C^\infty(\beta^{r+1}_sM)
$$
which is defined in local coordinates as follows. For
$ u=(u^{i_1\dots i_r}_{j_1\dots j_s})\in C^\infty(\beta^r_sM) $,
$$
{\overset a\to{\nabla}}{}^k
u^{i_1\dots i_r}_{j_1\dots j_s}
=
{\overset h\to{\nabla}}{}^k
u^{i_1\dots i_r}_{j_1\dots j_s}
+
a^{kp}{\overset v\to{\nabla}}{}_{\!p}
u^{i_1\dots i_r}_{j_1\dots j_s}
$$
$$
-
\sum\limits_{\alpha =1}^{r}
{\overset v\to{\nabla}}{}_{\!p}a^{ki_\alpha}\cdot
u^{i_1\dots i_{\alpha-1}pi_{\alpha+1}\dots i_r}_{j_1\dots j_s}
+
\sum\limits_{\alpha =1}^{s}
{\overset v\to{\nabla}}{}_{\!j_\alpha}a^{kp}\cdot
u^{i_1\dots i_r}_{j_1\dots j_{\alpha-1}pj_{\alpha+1}\dots j_s}.
      \eqno{(5.1)} $$
In contrast to the ordinary horizontal derivative, the metric tensor
$ g $
is not parallel with respect to
$ {\overset a\to{\nabla}} $.
This is just the reason why
$ k $
is an upper index in definition (5.1). Nevertheless, we will also use
the operator
$ {\overset a\to{\nabla}}_{\!k}=
g_{kl}{\overset a\to{\nabla}}{}^l $.

Since the modifying tensor is orthogonal to
$ \xi  $,
the geodesic vector field
considered as the differential operator
$ H:C^\infty(TM)\rightarrow C^\infty(TM) $
can be expressed in terms of
$ {\overset a\to{\nabla}} $
$$
Hu=\xi_k{\overset a\to{\nabla}}{}^ku\quad \quad
(u\in C^\infty(TM)).
      \eqno{(5.2)} $$

The curvature tensor
$ \overset a\to R=(\overset a\to R_{ijkl}(x,\xi))\in
C^\infty(\beta^0_4M) $
corresponding to
$ {\overset a\to{\nabla}} $
is defined by the equality
$$
\overset a\to R_{ijkl}
=
R_{ijkl}
+
{\overset h\to{\nabla}}_{\!l}{\overset v\to{\nabla}}_{\!j}a_{ik}
-
{\overset h\to{\nabla}}_{\!k}{\overset v\to{\nabla}}_{\!j}a_{il}
+
$$
$$
+
a_{lp}{\overset v\to{\nabla}}{}^p{\overset v\to{\nabla}}_{\!j}a_{ik}
-
a_{kp}{\overset v\to{\nabla}}{}^p{\overset v\to{\nabla}}_{\!j}a_{il}
+
{\overset v\to{\nabla}}{}^pa_{ik}\cdot
{\overset v\to{\nabla}}_{\!j}a_{lp}
-
{\overset v\to{\nabla}}{}^pa_{il}\cdot
{\overset v\to{\nabla}}_{\!j}a_{kp},
$$
where
$ (R_{ijkl}) $
is the Riemannian curvature tensor. This formula implies that
$$
\overset a\to R_{ijkl}\xi^j\xi^l
=
(Ha)_{ik}
+
a^l_ia_{kl}
+
R_{ijkl}\xi^j\xi^l.
      \eqno{(5.3)} $$

Let now
$ (M,g) $
be an Anosov manifold, and
$ \overset s\to a\ (\overset u\to a) $
be the semibasic tensor field constructed in Lemma 4.2. Setting
$ a=\overset s\to a\ (\overset u\to a) $
in (5.1),
we define the modified horizontal derivative on
$ C^\infty(\beta^r_sM;T^0M) $
which will be denoted by
$ {\overset s\to{\nabla}}\ ({\overset u\to{\nabla}}) $.
The corresponding curvature tensor will be denoted by
$ \overset s\to R\ (\overset u\to R) $.

Comparing (5.3) with statement ($4'$) of Lemma 4.2, we arrive at the
following important conclusion: for every semibasic vector fields
$ v,w\in C^\infty(\beta^1_0M) $,
the estimates
$$
|\overset s\to R_{ijkl}\xi^i\xi^kv^jw^l|<\varepsilon |v||w|,\quad
|\overset u\to R_{ijkl}\xi^i\xi^kv^jw^l|<\varepsilon |v||w|
      \eqno{(5.4)} $$
hold uniformly on
$ \Omega M $.

By statement (3) of Lemma 4.2, the set
$$
{\overset s\to{\nabla}}{}^iu,\quad
{\overset u\to{\nabla}}{}^iu \ (i=1,\dots,n),\quad
\xi^k{\overset v\to{\nabla}}_{\!k}u
$$
is a full family of derivatives of a function
$ u\in C^\infty(T^0M) $,
i.e., every first order derivative of
$ u $
is a linear combination of the set. This observation is specified by
the following

\proclaim{Lemma 5.1}
For every function
$ u\in C^\infty(T^0M) $,
the estimates
$$
|{\overset v\to{\nabla}}u-
\langle \xi,{\overset v\to{\nabla}}u\rangle \xi | \leq
C(|{\overset s\to{\nabla}}u|+
|{\overset u\to{\nabla}}u|),
                                                  \eqno{(5.5)} $$
$$
|{\overset h\to{\nabla}}u| \leq
C(|{\overset s\to{\nabla}}u|+|{\overset u\to{\nabla}}u|)
                                                  \eqno{(5.6)} $$
hold on
$ \Omega M $ with some constant
$ C $
independent of
$ u $.
\endproclaim

\demo{Proof}
It suffices to consider a function
$ u $
whose support is contained in the domain
$ U\subset T^0M $
of a local coordinate system. The semibasic vector field
$ y={\overset v\to{\nabla}}u-
\langle \xi ,{\overset v\to{\nabla}}u\rangle \xi  $
is orthogonal to
$ \xi  $
on
$ \Omega M $.
By the definition of the modified derivatives
$$
{\overset s\to{\nabla}}{}^iu=
{\overset h\to{\nabla}}{}^iu+
\overset s\to a{}^{ij}{\overset v\to{\nabla}}_{\!j}u,\quad
{\overset u\to{\nabla}}{}^iu=
{\overset h\to{\nabla}}{}^iu+
\overset u\to a{}^{ij}{\overset v\to{\nabla}}_{\!j}u.
      \eqno{(5.7)} $$
Substituting
$ {\overset v\to{\nabla}}_{\!j}u
=y_j+\langle \xi ,{\overset v\to{\nabla}}u\rangle \xi_j $
into these equalities and using orthogonality of
$ \overset s\to a $
and
$ \overset u\to a $
to
$ \xi  $,
we obtain
$$
{\overset s\to{\nabla}}{}^iu={\overset h\to{\nabla}}{}^iu+
\overset s\to a{}^{ij}y_j,\quad
{\overset u\to{\nabla}}{}^iu={\overset h\to{\nabla}}{}^iu+
\overset u\to a{}^{ij}y_j.
$$
This implies that
$$
(\overset s\to a{}^{ij}-\overset u\to a{}^{ij})y_j
= {\overset s\to{\nabla}}{}^iu-{\overset u\to{\nabla}}{}^iu.
      \eqno{(5.8)} $$
By statements (1) and (3) of Lemma 4.2, the operator
$ \overset s\to a-\overset u\to a $
is an automorphism of the space
$ N_{(x,\xi)}=\{\eta \in T_xM\mid\langle \xi,\eta\rangle =0\} $.
The right-hand side of (5.8) belongs to this space because
$ \langle \xi, {\overset s\to{\nabla}}u-
{\overset u\to{\nabla}}u\rangle
=\xi_i{\overset s\to{\nabla}}{}^iu-
\xi _i{\overset u\to{\nabla}}{}^iu=Hu-Hu=0. $
Therefore equation (5.8) has a unique solution
$$
y_i=\alpha_{ij}({\overset s\to{\nabla}}{}^ju-
{\overset u\to{\nabla}}{}^ju) $$
with some coefficients
$ \alpha_{ij} $
that are smooth functions in
$ U $
and are independent of
$ u $.
>From this we obtain (5.5). The estimate (5.6) follows from (5.5) and
(5.7).
The lemma is proved.
\enddemo

For a semibasic tensor field
$ u\in C^\infty(\beta^r_sM;T^0M) $,
we will use the notations
$$
\|u\|^2_{L_2}=\int\limits_{\Omega M}|u(x,\xi)|^2\,d\Sigma(x,\xi),
\quad
\|u\|^2_{H^1}=\|u\|^2_{L_2}+
\|{\overset h\to{\nabla}}u\|^2_{L_2}+
\|{\overset v\to{\nabla}}u\|^2_{L_2},
$$
where
$ d\Sigma(x,\xi)=d\omega_x(\xi)\wedge dV^n(x) $
is the symplectic volume form on
$ \Omega M $.
Lemma 5.1 has the following

\proclaim{Corollary 5.2}
The two norms
$ \|u\|_{H^1} $
and
$ (\|{\overset s\to{\nabla}}u\|^2_{L_2}+
\|{\overset u\to{\nabla}}u\|^2_{L_2}+
\|u\|^2_{L_2})^{1/2} $
are equivalent on the space of positively homogeneous in
$ \xi  $
functions
$ u(x,\xi)\in C^\infty(T^0M) $
with the same degree of homogeneity.
\endproclaim


Indeed,
$ \langle \xi ,{\overset v\to{\nabla}}u\rangle =\lambda u $
if
$ u $
is homogeneous of degree
$ \lambda  $.

\smallpagebreak

{\bf Remark.}
The numbers
$ \varepsilon  $
and
$ C $
participating in (5.4)--(5.6) are independent in the following
sense:
$ \varepsilon  $
can be chosen arbitrary small with a fixed value of
$ C $.
Indeed,
$ C $
is determined by the continuous fields
$ \overset s\to a $
and
$ \overset u\to a $
constructed in Lemma 3.1, while
$ \varepsilon  $
depends on the degree of approximating these fields by smooth ones.

\head
6. Proof of Lemma 2.1
\endhead

Let a tensor field
$ f\in C^\infty(S^m\tau'_M) $
and a function
$ u\in C^\infty(\Omega M) $
satisfy equation (2.1). We extend the function
$ u $
to
$ T^0M $ in such the way that it is positively homogeneous of degree
$ m-1 $
in
$ \xi  $.
Then equation (2.1) holds on
$ T^0M $.
By (5.2), this equation can be rewritten in the form
$$
Hu=\xi_k{\overset s\to{\nabla}}{}^ku=
\xi_k{\overset u\to{\nabla}}{}^ku=
\langle f(x),\xi ^m\rangle .
      \eqno{(6.1)} $$

For the modified horizontal derivative
$ {\overset s\to{\nabla}} $,
the following
{\it Pestov identity} is valid (see Section 8.2 of \cite{Sh1} or
\cite{Sh3}):
$$
2\langle {\overset s\to{\nabla}}u,
H{\overset v\to{\nabla}}u\rangle
=
|{\overset s\to{\nabla}}u|^2+
{\overset s\to{\nabla}}{}^iv_i+
{\overset v\to{\nabla}}_{\!i}w^i-
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu,
      \eqno{(6.2)} $$
where
$$
v_i=\xi_i{\overset s\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}_{\!j}u-
\xi_j{\overset v\to{\nabla}}_{\!i}u\cdot
{\overset s\to{\nabla}}{}^ju,
      \eqno{(6.3)} $$
$$
w^i=\xi_j{\overset s\to{\nabla}}{}^iu\cdot
{\overset s\to{\nabla}}{}^ju.
      \eqno{(6.4)} $$

We transform the left-hand side of identity (6.2). From (6.1) we obtain
$$
{\overset v\to{\nabla}}_{\!k}(Hu)=
{\overset v\to{\nabla}}_{\!k}
(\langle f(x),\xi^m\rangle) =
mf_{ki_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}.
$$
With the help of the relation
$ {\overset s\to{\nabla}}{}^i\xi_j=0 $,
the latter formula implies
$$
2\langle {\overset s\to{\nabla}}u,
{\overset v\to{\nabla}}Hu\rangle =
2m{\overset s\to{\nabla}}{}^iu\cdot
f_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}
=
$$
$$
=
{\overset s\to{\nabla}}{}^i
(2muf_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m})
-2mu{\overset s\to{\nabla}}{}^i
(f_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}).
$$
Introducing the notation
$$
\widetilde{v}_i=2muf_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m},
$$
we have
$$
2\langle {\overset s\to{\nabla}}u,
{\overset v\to{\nabla}}Hu\rangle =
{\overset s\to{\nabla}}{}^i\widetilde{v}_i
-2mu{\overset s\to{\nabla}}{}^i
(f_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}).
      \eqno{(6.5)} $$
By the definition of the modified derivative
$$ {\overset s\to{\nabla}}{}^i
(f_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m})
=
{\overset h\to{\nabla}}{}^i
(f_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m})
+
$$
$$
+
\overset s\to a{}^{ip}
{\overset v\to{\nabla}}_{\!p}
(f_{ii_2\dots i_m}\xi^{i_2}\dots\xi^{i_m})
+
{\overset v\to{\nabla}}_{\!i}
\overset s\to a{}^{ip}\cdot
(f_{pi_2\dots i_m}\xi^{i_2}\dots\xi^{i_m})
=
$$
$$
=
(\delta f)_{i_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}
+
(m-1)\overset s\to a{}^{ip}
f_{ipi_3\dots i_m}\xi^{i_3}\dots\xi^{i_m}
+
{\overset v\to{\nabla}}_{\!i}
\overset s\to a{}^{ip}\cdot
f_{pi_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}.
$$
Substituting the latter expression into (6.5), we obtain
$$
2\langle {\overset s\to{\nabla}}u,
{\overset v\to{\nabla}}Hu\rangle =
{\overset s\to{\nabla}}{}^i\widetilde{v}_i
-2mu
(\delta f)_{i_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}
-
$$
$$
-
2m(m-1)\overset s\to a{}^{ip}u
f_{ipi_3\dots i_m}\xi^{i_3}\dots\xi^{i_m}
-
2m{\overset v\to{\nabla}}_{\!i}
\overset s\to a{}^{ip}\cdot u
f_{pi_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}.
$$

With the help of the latter formula, we transform Pestov identity (6.2)
to the form
$$
|{\overset s\to{\nabla}}u|^2
=
{\overset s\to{\nabla}}{}^i(\widetilde{v}_i-v_i)
-
{\overset v\to{\nabla}}_{\!i}w^i
-2mu
(\delta f)_{i_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}
-
2m(m-1)\overset s\to a{}^{ip}u
f_{ipi_3\dots i_m}\xi^{i_3}\dots\xi^{i_m}
$$
$$
-
2m{\overset v\to{\nabla}}_{\!i}
\overset s\to a{}^{ip}\cdot u
f_{pi_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}
+
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu.
      \eqno{(6.6)} $$

With the help of statement ($4''$) of Lemma 4.2 and inequality
(5.4), the last tree terms on the right-hand side of (6.6) can be
estimated at a point
$ (x,\xi)\in \Omega M $
as follows:
$$
|\overset s\to a{}^{ip}u
f_{ipi_3\dots i_m}\xi^{i_3}\dots\xi^{i_m}|
\leq C|u||f|,
$$
$$
|{\overset v\to{\nabla}}_{\!i}
\overset s\to a{}^{ip}\cdot u
f_{pi_2\dots i_m}\xi^{i_2}\dots\xi^{i_m}|
\leq |{\overset v\to{\nabla}}a||u||f|,
$$
$$
|\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu|
<\varepsilon |{\overset v\to{\nabla}}u|^2.
$$
With the help of these estimates, (6.6) gives the inequality
$$
|{\overset s\to{\nabla}}u|^2
\leq
{\overset s\to{\nabla}}{}^i(\widetilde{v}_i-v_i)
-
{\overset v\to{\nabla}}_{\!i}w^i
+
C|u||f|+2m|u||\delta f|+
\varepsilon |{\overset v\to{\nabla}}u|^2
+
2m|{\overset v\to{\nabla}}a||u||f|
      \eqno{(6.7)} $$
that is valid on
$ \Omega M $.

We integrate inequality (6.7) over
$ \Omega M $
and transform the integrals of divergent terms by the
Gauss---Ostrogradski\u{\i} formulas (Theorem 3.6.3 and formula (8.2.30)
of \cite{Sh1}). The integral of
$ {\overset s\to{\nabla}}{}^i(\widetilde{v}_i-v_i) $
equals to zero because
$ \Omega M $
is a closed manifold. The integral of the second term is nonpositive;
indeed
$$
-\int\limits_{\Omega M}{\overset v\to{\nabla}}_{\!i}w^i\,d\Sigma
=
-(n+2m-2)\int\limits_{\Omega M}\langle w,\xi \rangle \,d\Sigma =
-(n+2m-2)\int\limits_{\Omega M}|Hu|^2\,d\Sigma \leq 0.
$$
The latter equality is written because
$ \langle w,\xi \rangle =|Hu|^2 $
as is seen from (6.4). Thus, after integration (6.7) gives us the
inequality
$$
\|{\overset s\to{\nabla}}u\|^2_{L_2}
\leq
C(\|u\|_{L_2}\cdot\|f\|_{L_2}+\|u\|_{L_2}\cdot\|\delta f\|_{L_2})
+\varepsilon \|u\|^2_{H^1}
+
\int\limits_{\Omega M}|{\overset v\to{\nabla}}a||u||f|\,d\Sigma .
      \eqno{(6.8)} $$

Let
$ p>2n-1 $
be such that the stable and unstable distributions belong to the class
$ W^1_p $,
and let
$ q=2p/(p-2) $.
Applying the H\"older inequality, we estimate the integral on (6.8)
$$
\int\limits_{\Omega M}|{\overset v\to{\nabla}}a||u||f|\,d\Sigma
\leq
\left(
\int\limits_{\Omega M}|{\overset v\to{\nabla}}a|^p\,d\Sigma
\right)^{1/p}
\|u\|_{L_q}\cdot\|f\|_{L_2}.
$$
By statement ($5'$) of Lemma 4.2, the integral on the right-hand side
can be estimated by some constant
$ C $
independent on
$ u $,
and we obtain
$$
\int\limits_{\Omega M}|{\overset v\to{\nabla}}a||u||f|\,d\Sigma
\leq C\|u\|_{L_q}\cdot\|f\|_{L_2}.
                                                \eqno{(6.9)} $$

With the help of the latter estimate, (6.8) implies the inequality
$$
\|{\overset s\to{\nabla}}u\|^2_{L_2}
\leq
C(\|u\|_{L_q}\cdot\|f\|_{L_2}+\|u\|_{L_2}\cdot\|\delta f\|_{L_2})
+\varepsilon \|u\|^2_{H^1}.
      \eqno{(6.10)} $$

We can repeat our arguments with
$ {\overset s\to{\nabla}} $
replaced with
$ {\overset u\to{\nabla}} $.
In such the way we obtain the following analog of (6.10):
$$
\|{\overset u\to{\nabla}}u\|^2_{L_2}
\leq
C(\|u\|_{L_q}\cdot\|f\|_{L_2}+\|u\|_{L_2}\cdot\|\delta f\|_{L_2})
+\varepsilon \|u\|^2_{H^1}.
      \eqno{(6.11)} $$

With the help of Corollary 5.2, inequalities (6.10) and (6.11) give
$$
\|u\|^2_{H^1}
\leq
CC'(\|u\|_{L_q}\cdot\|f\|_{L_2}+\|u\|_{L_2}\cdot\|\delta f\|_{L_2})
+C'\|u\|^2_{L_2}+C'\varepsilon  \|u\|^2_{H^1}.
      \eqno{(6.12)} $$

As we have emphasized, the number
$ \varepsilon  $
can be chosen arbitrary small with some fixed values of
$ C $
and
$ C' $.
In particular, we can assume that
$ C'\varepsilon <1 $,
and inequality (6.12) can be rewritten in the form
$$
\|u\|^2_{H^1}
\leq
C(\|u\|_{L_q}\cdot\|f\|_{L_2}+\|u\|_{L_2}\cdot\|\delta f\|_{L_2}
+\|u\|^2_{L_2})
      \eqno{(6.13)} $$
with some new constant
$ C $
independent of
$ u $.

The kinetic equation
$ Hu=\xi^i{\overset h\to{\nabla}}_{\!i}u=\langle f,\xi^m\rangle  $
implies the estimate
$ \|f\|_{L_2}\leq C\|{\overset h\to{\nabla}}u\|_{L_2}
\leq C\|u\|_{H^1} $
that allows us to rewrite (6.13) in the form
$$
\|u\|^2_{H^1}
\leq
C(\|u\|_{L_q}\cdot\|u\|_{H^1}+\|u\|_{L_2}\cdot\|\delta f\|_{L_2}
+\|u\|^2_{L_2}).
      \eqno{(6.14)} $$

Considering (6.14) as a quadratic inequality in
$ x=\|u\|_{H^1} $,
one can easily see that it implies (2.2) with another constant
$ C $.
The lemma is proved.

\smallpagebreak

Observe that the hypothesis on membership of the stable and
unstable distributions in the class
$ W^1_p $
was used in the proof only once, in estimate
(6.9). As we will see, there is no need of this estimate in the proofs
of Theorems 1.1 and 1.3.

\smallpagebreak

\demo{Proof of Theorem 1.1}
Let a function
$ f\in C^\infty(M) $
integrates to zero over every closed geodesic. Applying the Liv\v cic
theorem, we obtain the function
$ u\in C^\infty(\Omega M) $
satisfying the kinetic equation
$$
Hu(x,\xi)=f(x)
      \eqno{(6.15)} $$
on $ \Omega M $.
Extending
$ u $
to
$ T^0M $
in such the way as
$ u(x,t\xi)=t^{-1}u(x,\xi) $
for
$ t>0 $,
equation (6.15) is satisfied on
$ T^0M $.
The left-hand side of the Pestov identity (6.2) is identical zero in
our case. After integration over
$ \Omega M $,
the identity gives
$$
\|{\overset s\to{\nabla}}u\|^2_{L_2}
+(n-2)\|Hu\|^2_{L_2} =
\int\limits_{\Omega M}
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu\,d\Sigma .
      \eqno{(6.16)} $$

Let
$ y^i = {\overset v\to{\nabla}}{}^iu
-\langle \xi ,{\overset v\to{\nabla}}u\rangle \xi^i $.
Then, using the symmetries of the curvature tensor, we obtain
$$
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu
=
\overset s\to R_{ijkl}\xi^i\xi^ky^jy^l.
$$
With the help of (5.4) and (5.5), the latter equality implies the
estimate
$$
|\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu|
\leq \varepsilon |y|^2\leq C\varepsilon (
|{\overset s\to{\nabla}}u|^2+|{\overset u\to{\nabla}}u|^2).
$$
Combining this estimate with (6.16), we obtain
$$
\|{\overset s\to{\nabla}}u\|^2_{L_2}
\leq
C\varepsilon (\|{\overset s\to{\nabla}}u\|^2_{L_2}
+\|{\overset u\to{\nabla}}u\|^2_{L_2}).
      \eqno{(6.17)} $$

In the same way we obtain the similar estimate
$$
\|{\overset u\to{\nabla}}u\|^2_{L_2}
\leq
C\varepsilon (\|{\overset s\to{\nabla}}u\|^2_{L_2}
+\|{\overset u\to{\nabla}}u\|^2_{L_2}).
      \eqno{(6.18)} $$

If
$ C\varepsilon <1/2 $,
inequalities (6.17) and (6.18) imply that
$ {\overset s\to{\nabla}}u=
{\overset u\to{\nabla}}u=0 $.
Consequently,
$ f=\xi_i{\overset s\to{\nabla}}{}^iu=0 $.
The theorem is proved.
\enddemo

\demo{Proof of Theorem 1.3}
In this case the kinetic equation looks as follows:
$$ Hu(x,\xi)=f_i(x)\xi ^i,
      \eqno{(6.19)} $$
and
$ {\overset v\to{\nabla}}Hu=f $.
Therefore the Pestov identity (6.2) has the form
$$
2\langle {\overset s\to{\nabla}}u,f\rangle
=
|{\overset s\to{\nabla}}u|^2+
{\overset s\to{\nabla}}{}^iv_i+
{\overset v\to{\nabla}}_{\!i}w^i-
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu.
$$
After integration over
$ \Omega M $
this gives
$$
\|{\overset s\to{\nabla}}u\|^2_{L_2}
-
2({\overset s\to{\nabla}}u,f)_{L_2}
+
n\|Hu\|^2_{L_2}
=
\int\limits_{\Omega M}
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu\,d\Sigma .
      \eqno{(6.20)} $$

>From (6.19), we obtain
$$
\|Hu\|^2_{L_2}=
\int\limits_{\Omega M}f_i(x)f_j(x)\xi^i\xi^j\,d\Sigma (x,\xi)
=
$$
$$
=
\int\limits_{M}f_i(x)f_j(x)\left[\int\limits_{\Omega_xM}
\xi^i\xi^j\,d\omega_x(\xi)\right]dV^n(x)=\frac {1} {n}
\|f\|^2_{L_2}.
$$
With the help of the latter equality, (6.20) takes the form
$$
\|{\overset s\to{\nabla}}u-f\|^2_{L_2}
=
\int\limits_{\Omega M}
\overset s\to R_{ijkl}\xi^i\xi^k
{\overset v\to{\nabla}}{}^ju\cdot
{\overset v\to{\nabla}}{}^lu\,d\Sigma .
$$
Estimating the right-hand side integral with the help of (5.4), we
obtain
$$
\|{\overset s\to{\nabla}}u-f\|^2_{L_2}
\leq\varepsilon C\|{\overset v\to{\nabla}}u\|^2_{L_2}.
      \eqno{(6.21)} $$

Repeating our arguments with interchanged
$ {\overset s\to{\nabla}} $
and
$ {\overset u\to{\nabla}} $,
we obtain the similar inequality
$$
\|{\overset u\to{\nabla}}u-f\|^2_{L_2}
\leq\varepsilon C\|{\overset v\to{\nabla}}u\|^2_{L_2}.
      \eqno{(6.22)} $$

Let now
$ \varepsilon  $
tend to zero in (6.21) and (6.22). The vector fields
$ f $
and
$ {\overset v\to{\nabla}}u $
are independent of
$ \varepsilon  $
as well as the constant
$ C $.
The fields
$ {\overset s\to{\nabla}}u $
and
$ {\overset u\to{\nabla}}u $
tend respectively to
$$
{\overset s\to{\nabla}}_{\!i}u
={\overset h\to{\nabla}}_{\!i}u
+\overset s\to a{}^p_i{\overset v\to{\nabla}}_{\!p}u,\quad
{\overset u\to{\nabla}}_{\!i}u
={\overset h\to{\nabla}}_{\!i}u
+\overset u\to a{}^p_i{\overset v\to{\nabla}}_{\!p}u
      \eqno{(6.23)} $$
with the continuous tensors
$ \overset s\to a $
and
$ \overset u\to a $
constructed in Lemma 3.1. Thus, passing to limit in (6.21) and (6.22)
as
$ \varepsilon \rightarrow 0 $,
we obtain
$$
{\overset s\to{\nabla}}_{\!i}u=
f_i={\overset u\to{\nabla}}_{\!i}u.
      \eqno{(6.24)} $$
Now equalities (6.23) give us
$$
(\overset s\to a{}^p_i-\overset u\to a{}^p_i)
{\overset v\to{\nabla}}_{\!p}u =0.
                                                \eqno{(6.25)} $$

In our case the function
$ u(x,\xi) $
is positively homogeneous of zero degree, and therefore
$$
\xi^k{\overset v\to{\nabla}}_{\!k}u=0.
      \eqno{(6.26)} $$
With the help of statement (3) of Lemma 3.1, (6.25) and (6.26) imply
that
$ {\overset v\to{\nabla}}u=0 $,
i.e., the function
$ u $
is independent of
$ \xi ;\ u=u(x) $.
Equalities (6.23) and (6.24) take now the form
$ f_i={\overset h\to{\nabla}}_{\!i}u=\partial u/\partial x^i $.
Therefore
$ f $
is the exact form,
$ f=du $.
The theorem is proved.
\enddemo



\Refs
\widestnumber\key{LMM}

\ref \key{An}
\by D. Anosov
\book Geodesic Flows on Closed Riemannian Manifolds with Negative
Curvature, {\rm Proc. Steclov Inst. of Math.}
\yr 1967 \vol 90
\endref

\ref \key{AS}
\by D. V. Anosov, Ya. G. Sinay
\paper Some smooth ergodic systems
\jour Uspehi Mat. Nauk
\yr 1967
\vol 22
\issue 5
\lang Russian
\endref

\ref \key{Ar}
\by V. I. Arnold
\book Ordinary Differential Equations
\yr 1984
\publ Nauka
\publaddr Moscow
\lang Russian
\endref

\ref \key{CSh}
\by C. B. Croke, V. A. Sharafutdinov
\paper Spectral rigidity of a negatively curved manifold
\jour Topology
\yr to appear
\endref

\ref \key{Eb}
\by P. Eberlein
\paper When is a geodesic flow of Anosov type? I
\jour J. Differential Geometry
\yr 1973
\vol 8
\pages 437--463
\endref

\ref \key{Es}
\by J.-H. Eschenburg
\paper Horospheres and the stable part of the geodesic flow
\jour Math. Z.
\yr 1977
\vol 153
\pages 237--251
\endref

\ref \key{F}
\by P. Funk
\paper \"Uber eine geometrische Anwendung der Abelschen
Integralgleichung
\jour Math. Ann.
\vol 77
\yr 1916
\pages 129--135
\endref

\ref \key{G}
\by L. W. Green
\paper A theorem of E. Hopf
\jour Michigan Math. J.
\yr 1958
\vol 5
\pages 31--34
\endref

\ref \key{GK}
\by V. Guillemin, D. Kazhdan
\paper Some inverse spectral results for negatively curved 2-manifolds
\jour Topology
\yr 1980
\vol 19
\pages 301--312
\endref

\ref \key{H}
\by E. Hopf
\paper Closed surfaces without conjugate points
\jour Proc. Nat. Acad. Sci. U.S.A.
\yr 1948
\vol 34
\pages 47--51
\endref

\ref \key{Hl}
\by S. Helgason
\book Integral Geometry, Invariant Differential Operators, and
Spherical Functions
\yr 1984
\publ Academic Press, Inc.
\publaddr Orlando --- S\"ao Paulo
\endref



\ref \key{LMM}
\by R. de la Llave, J. M. Marco, R. Moriyon
\paper Canonical perturbation theory of Anosov Systems and regularity
results for the Livsic cohomology equation
\jour Annals of Math.
\yr 1986
\vol 123
\pages 537--611
\endref

\ref \key{M1}
\by R. Michel
\paper Les tenseurs sym\'etriques dont l'\'energie est nulle toutes les
g\'eod\'esiques des espaces sym\'etriques de rang 1
\jour J. Mat. Pures Appl.
\vol 45
\yr 1974
\pages 271--278
\endref

\ref \key{M2}
\by R. Michel
\paper  Tenseurs sym\'etriques et geodesiques
\jour C.~R.~Acad.~Sci.~S\'er~I~Math.
\vol 284
\yr 1977
\pages 1065--1068
\endref


\ref \key{PSh}
\by L. N. Pestov, V. A. Sharafutdinov
\paper Integral geometry of tensor fields on  manifold of negative
curvature
\jour Siberian Math. J.
\yr 1988
\vol 29
\issue 3
\pages 427--441
\endref

\ref \key{Sh1}
\by V. A. Sharafutdinov
\book  Integral Geometry of Tensor Fields
\yr 1994
\publ VSP
\publaddr Utrecht, the Netherlands
\endref

\ref \key{Sh2}
\by V. A. Sharafutdinov
\paper Finiteness theorem for the ray transform on a Riemannian
manifold
\jour Inverse Problems
\yr 1995
\vol 11
\pages 1039--1050
\endref

\ref \key{Sh3}
\by V. A. Sharafutdinov
\paper Modified horizontal derivative and some of its applications
\jour Siberian Math. J.
\yr 1995
\vol 36
\issue 3
\pages 664--700
\endref

\ref \key{Y}
\by K. Yosida
\book Functional Analysis
\yr 1965
\publ Springer-Verlag
\publaddr Berlin, G\"ottingen, Heidelberg
\endref



\endRefs

\enddocument