This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group Ham(M,ω) to the identity component Symp0(M,ω) of the symplectomorphism group. In particular, we prove that the Hofer metric on Ham(M,ω) does not extend to a bi-invariant metric on Symp0(M,ω) for many symplectic manifolds. We also show that for the torus T2n with the standa...

If D is a bounded convex domain in C , then the work of Lempert [L] and Royden-Wong [RW] (see also [A]) show that given any point p ∈ D and any non-zero tangent vector v ∈ C at p, there exists a holomorphic map φ:U → D from the unit disk U ⊂ C into D passing through p and tangent to v in p which is an isometry with respect to the hyperbolic distance of U and the Kobayashi distance of D. Further...

In this paper, we study a class of Finsler metrics with orthogonal invariance. We find an equation that characterizes these Finsler metrics of almost vanishing H-curvature. As a consequence, we show that all orthogonally invariant Finsler metrics of almost vanishing H-curvature are of almost vanishing Ξ-curvature and corresponding one forms are exact, generalizing a result previously only known...

In this paper we consider a dominating Finsler metric on a complete Riemannian manifold. First we prove that the energy integral of the Finsler metric satisfies the Palais-Smale condition, and ask for the number of geodesics with endpoints in two given submanifolds. Using Lusternik-Schnirelman theory of critical points we obtain some multiplicity results for the number of Finsler-geodesics betw...

Following the study on volume of indicatrices in a real Finsler space, in this paper we are investigating some volume properties of the indicatrix considered in an arbitrary fixed point of a complex Finsler manifold. Since for each point of a complex Finsler space the indicatrix is an embedded CR-hypersurface of the punctured holomorphic tangent bundle, by means of its normal vector, the volume...

Berwald and Wagner manifolds are two important classes of spaces in Finsler geometry. They are closely related to each other via the conformal change of the metric. After discussing the basic definitions and the elements of the theory we present general methods to construct examples of them.

Anisotropic cosmological models with a generalized Chaplygin gas in a Finsler space-time geometry are considered and a class of exact solutions as well as cosmological parameters behaviors are studied. Moreover, the viability and stability criteria for a general solution are also discussed.

The notion of isometric submersion is extended to Finsler spaces and it is used to construct examples of Finsler metrics on complex and quaternionic projective spaces all of whose geodesics are (geometrical) circles.

We describe the action of fundamental group a closed Finsler surface negative curvature on geodesics in universal covering terms flat symplectic connection and consider first order deformation theory about hyperbolic metric. A construction O.Biquard yields family metrics which give nontrivial deformations holonomy, extending representation from SL(2,R) into Hamiltonian diffeomorphisms S^1 x R, ...

This thesis studies the role of Finsler geometry in quantum time optimal control of systems with constrained control field power and other constraints. The systems considered are all finite dimensional systems with pure states. A Finsler metric is constructed such that its geodesics are the time optimal trajectories for the quantum time evolution operator on the special unitary group. This metr...