Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIntegrable geodesic flows on the suspensions of toric automorphisms
91
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic flow, which has vanishing Liouville entropy and, moreover, is integrable but has positive topological entropy. The authors also discuss some open problems on integrability of geodesic flows and related subjects.
rate research
Read More
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The flows corresponding to left-invariant metrics and left-invariant distributions on Lie groups are reduced to Euler equations on Lie groups. Relation of these constructions to problems of analytical mechanics is discussed.
We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can construct a fundamental solution of the the Jacobi-Hill equation. This is done for quadratically integrable geodesic flows.
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.
An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^infty$ functions and has positive topological entropy is constructed.
We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some constant. This is an extension of our previous work. We use the toric condition in a more essential way to rule out bubbles.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
