On Some Geometric Structures Associated to a k-Symplectic Manifold

TL;DR

This paper investigates the properties of a canonical connection on k-symplectic manifolds, exploring its geometric applications, including the existence of Ehresmann connections and vanishing theorems for characteristic classes.

Contribution

It introduces and analyzes a canonical connection on k-symplectic manifolds and proves the existence of Ehresmann connections under certain conditions.

Findings

Existence of a canonical connection on k-symplectic manifolds

Conditions under which an Ehresmann connection exists

Vanishing theorems for characteristic classes

Abstract

A canonical connection is attached to any k-symplectic manifold. We study the properties of this connection and its geometric applications to k-symplectic manifolds. In particular we prove that, under some natural assumption, any ksymplectic manifold admits an Ehresmann connection, discussing some corollaries of this result, and we find vanishing theorems for characteristic classes on a k-symplectic manifold.