Canonical Torsion-Free Connections on the Total Space of the Tangent and the Cotangent Bundle

TL;DR

This paper introduces a class of torsion-free connections on tangent and cotangent bundles that preserve fiber structures, are flat on fibers, and have applications in metric and symplectic geometry.

Contribution

It defines a new class of torsion-free connections on (co-)tangent bundles with specific geometric properties and analyzes their structure and special cases.

Findings

Fiber tangent spaces are flat under these connections.

The projection to the base manifold is totally geodesic.

Connections can be metric with signature (n,n) or symplectic.

Abstract

In this paper we define a class of torsion-free connections on the total space of the (co-)tangent bundle over a base-manifold with a connection and for which tangent spaces to the fibers are parallel. Each tangent space to a fiber is flat for these connections and the canonical projection from the (co-)tangent bundle to the base manifold is totally geodesic. In particular cases the connection is metric with signature (n,n) or symplectic and admits a single parallel totally isotropic tangent n-plane.