Submanifolds in Koszul-Vinberg geometry

TL;DR

This paper explores the geometry of submanifolds within Koszul-Vinberg manifolds, which connect Poisson and pseudo-Riemannian geometries, extending previous work on contravariant pseudo-Hessian structures.

Contribution

It advances the understanding of submanifold theory in Koszul-Vinberg geometry, integrating concepts from Poisson submanifold theory and generalizing existing geometric frameworks.

Findings

Characterization of submanifolds in Koszul-Vinberg manifolds.

Extension of Poisson submanifold theory to Koszul-Vinberg setting.

New conditions for submanifold integrability and geometry.

Abstract

A Koszul-Vinberg manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our previous article [\textit{Contravariant Pseudo-Hessian manifolds and their associated Poisson structures}. \rm{Differential Geometry and its Applications} (2020)]. Our objective here will be to pursue our study by focusing in this setting on submanifolds by taking into account some developments in the theory of Poisson submanifolds.