Holography of geodesic flows, harmonizing metrics, and billiards' dynamics

Integrable geodesic flows of non-holonomic metrics

In the present article we show how to produce new examples of integrable dynamical systems of differential geometry origin. This is based on a construction of a canonical Hamiltonian structure for the geodesic flows of Carnot–Carathéodory metrics ([7, 17]) via the Pontryagin maximum principle. This Hamiltonian structure is achieved by introducing Lagrange multipliers bundles being the phase spa...

The Dynamics of Geodesic Flows*

Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

It is shown that most, but not all, of the four dimensional metrics in the MultiCentre family with integrable geodesic flow may be recognized as belonging to spatially homogeneous Bianchi type A metrics. We show that any diagonal bi-axial Bianchi II metric has an integrable geodesic flow, and that the simplest hyperkähler metric in this family displays a finite dimensional W-algebra for its obs...

On the Integrability of Geodesic Flows of Submersion Metrics

Suppose we are given a compact Riemannian manifold (Q, g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on Q/G equipped with the submersion metric be integrable? Under one natural assumption, we prove that the answer is affirmative. New examples of manifolds with completely...

On geodesic propagators and black hole holography

One of the most challenging technical aspects of the dualities between string theory on anti-de Sitter spaces and conformal field theories is understanding how location in the interior of spacetime is represented in the field theory. It has recently been argued that the interior of the spacetime can be directly probed by using intrinsically non-local quantities in the field theory. In addition,...