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Register a new userGeodesic flows on spheres and the local Riemann-Roch numbers
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We calculate the local Riemann-Roch numbers of the zero sections of $T^*S^n$ and $T^*R P^n$, where the local Riemann-Roch numbers are defined by using the $S^1$-bundle structure on their complements associated to the geodesic flows.
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In the present paper we prove, that if the geodesic flow of a metric G on the torus T is quadratically integrable, then the torus T isometrically covers a torus with a Liouville metric on it, and describe the set of quadratically integrable geodesic flows on the Klein bottle.
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^infty$ Riemannian metric $g$, whose genus $mathfrak{g} geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of $M$ is ergodic with respect to the Liouville measure, under the assumption that the set of points on $M$ with negative curvature has at most finitely many connected components.
The classical Riemann-Roch theorem has been extended by N. Nadirashvili and then M. Gromov and M. Shubin to computing indices of elliptic operators on compact (as well as non-compact) manifolds, when a divisor mandates a finite number of zeros and allows a finite number of poles of solutions. On the other hand, Liouville type theorems count the number of solutions that are allowed to have a pole at infinity. Usually these theorems do not provide the exact dimensions of the spaces of such solutions (only finite-dimensionality, possibly with estimates or asymptotics of the dimension. An important case has been discovered by M. Avellaneda and F. H. Lin and advanced further by J. Moser and M. Struwe. It pertains periodic elliptic operators of divergent type, where, surprisingly, exact dimensions can be computed. This study has been extended by P. Li and Wang and brought to its natural limit for the case of periodic elliptic operators on co-compact abelian coverings by P. Kuchment and Pinchover. Comparison of the results and techniques of Nadirashvili and Gromov and Shubin with those of Kuchment and Pinchover shows significant similarities, as well as appearance of the same combinatorial expressions in the answers. Thus a natural idea was considered that possibly the results could be combined somehow in the case of co-compact abelian coverings, if the infinity is added to the divisor. This work shows that such results indeed can be obtained, although they come out more intricate than a simple-minded expectation would suggest. Namely, the interaction between the finite divisor and the point at infinity is non-trivial.
The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.
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