Metric degeneracies and gradient flows on symplectic leaves

TL;DR

This work analyzes how a pseudo-Riemannian metric $g$ degenerates when restricted to symplectic leaves of a Poisson manifold and develops a gradient-flow framework via the generalized double bracket (GDB) vector field. It introduces good leaves, green zones, and red zones to partition leaves by the nondegeneracy of the induced metric and the metriplectic tensor $\mathcal{M}$, proving gradient-like behavior of the GDB flow on admissible regions and detailing how degeneracies arise in indefinite settings. The authors construct and study a broad class of Poisson structures on $\\mathbb{R}^3$ (including linear, quadratic, and Poisson-Lie deformations of $\\sl_2^*$), analyze the topology of their leaves, and characterize the degeneracy locus $\\mathcal{R}$ (red zones) where the DB metric fails. A comprehensive geometric interpretation is provided via Minkowski-like coordinates, identifying red lines that separate Euclidean and Lorentzian green zones and showing that, on green zones, the GDB vector field is the gradient of $G$ with respect to the DB metric, with applications to gradient-like dynamics on Poisson manifolds. These results offer a structured view of gradient dynamics in nondegenerate regions and illuminate the complex interplay between Poisson geometry, indefinite metrics, and leaf topology.

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.

Figures (12)

• Figure 1: Symplectic leaves on $\sl(2)^*$: (A) for $c=-1$, (B) for $c=0$, and (C) for $c=1$
• Figure 2: Quadratic bracket: symplectic leaf for $c=1$
• Figure 3: Quadratic bracket: symplectic leaf $S$ for $c=0$: (A) seen from the side (B) seen from above. Topologically $S$ is a punctured torus.
• Figure 4: Symplectic leaves on Poisson-Lie group for the deformation parameter $\eta =1$: (A) for $c=-1$, (B) for $c=0$, and (C) for $c=1$. For $\eta \to 0$, they more and more approach the leaves shown in Fig. \ref{['fig1']}.
• Figure 5: (A) Function $h_c$ for the linear Poisson structure on $\sl^*_2$, orange color for $h_{-1}$, black color for $h_{0}$ and blue color for $h_{1}$. (B) Function $h_c$ for the quadratic Poisson structure, orange color for $h_{0}$ and blue color for $h_{1}$.

Theorems & Definitions (7)

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