Nurlan S. Dairbekov

Integral geometry problem for nontrapping manifolds

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a certain upper curvature bound, we prove that a tensor field, integrating to zero over geodesics between boundary points, is potential. For functions and 1-forms, this is shown to be true for arbitrary nontrapping manifolds with no conjugate points. As a consequence, we also establish deformation boundary rigidity for strongly geodesic minimizing manifolds with a certain upper curvature bound.

Some problems of integral geometry on Anosov manifolds

In this paper we prove that on an Anosov manifold the space of symmetric m -tensor fields of vanishing energy is finite dimensional modulo the space of potential tensor fields for an arbitrary m and coincides with the latter for m=0 and m=1 . For m=2 this question relates to the spectral rigidity problem.

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.

Entropy Production in Gaussian Thermostats

We show that an arbitrary Anosov Gaussian thermostat on a surface is dissipative unless the external field has a global potential. This result is obtained by studying the cohomological equation of more general thermostats using the methods in [3].

Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface Σ in the cotangent bundle τ : T ∗M → M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only τ∗θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our work in [7]. Other rigidity issues are also discussed.

Lengths and volumes in Riemannian manifolds

We consider the question of when an inequality between lengths of “corresponding” geodesics implies a corresponding inequality between volumes. We prove this in a number of cases for compact manifolds with and without boundary. In particular, we show that for two Riemannian metrics of negative curvature on a compact surface without boundary, an inequality between the marked length spectra implies the same inequality between the areas with equality implying isometry.

Entropy Production in Thermostats II

We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies

Local boundary rigidity of a compact Riemannian manifold with curvature bounded above

K_w(\sigma)+\frac14|E_\sigma|^2<0