Geometry Via Sprays on Frechet Manifolds

TL;DR

This paper develops a geometric framework for Frechet manifolds by constructing connection maps and symmetric connections using sprays, establishing a correspondence between these structures and tangent bundle connections.

Contribution

It introduces a novel approach to characterize and relate sprays and symmetric connections on Frechet manifolds, expanding geometric tools for infinite-dimensional analysis.

Findings

Established a bijective correspondence between sprays and symmetric connections.

Characterized symmetric connections via bilinear symmetric mappings.

Provided an alternative tangent structure-based characterization.

Abstract

We construct connection maps and linear symmetric connections on tangent and second-order tangent bundles for \fr manifolds using the notion of a spray. For these manifolds, we characterize linear symmetric connections on tangent bundles in terms of bilinear symmetric mappings associated with sprays. We also provide an alternative characterization of these connections using tangent structures. Furthermore, we prove that a bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.