Metric degeneracies and gradient flows on symplectic leaves

TL;DR

This paper explores how degeneracies in pseudo-Riemannian metrics on Poisson manifolds affect gradient flows on symplectic leaves, introducing the generalized double bracket vector field and identifying regions where the metric remains non-degenerate.

Contribution

It introduces the generalized double bracket vector field for indefinite metrics and characterizes admissible regions with non-degenerate double bracket metrics on symplectic leaves.

Findings

Identification of admissible regions with non-degenerate metrics

Extension of gradient flow concepts to indefinite metric settings

Examples illustrating metric degeneracies and their implications

Abstract

For a Poisson manifold endowed with a pseudo-Riemannian metric, we investigate degeneracies arising when the metric is restricted to symplectic leaves. Central to this work is the generalized double bracket (GDB) vector field-a geometric construct introduced in our earlier work-which generalizes gradient dynamics to indefinite metric settings. We identify admissible regions where the so-called double bracket metric remains non-degenerate on symplectic leaves, enabling the GDB vector field to function as a gradient flow on the admissible regions with respect to this metric. We illustrate these concepts with a variety of examples and carefully discuss the complications that arise when the pseudo-Riemannian metric fails to induce a non-degenerate metric on certain regions of the symplectic leaves.