Vector Fields on Product Manifolds

TL;DR

This paper explores fundamental properties of smooth vector fields on product manifolds, demonstrating a natural decomposition into horizontal and vertical components and their relation to vector fields on individual factors.

Contribution

It establishes the existence of a direct sum decomposition of vector fields on product manifolds and links these to smooth families of vector fields on the factors, clarifying their structure.

Findings

Existence of a direct sum decomposition into horizontal and vertical vector fields.

Horizontal and vertical vector fields are isomorphic to smooth families on the factors.

Vector fields are derivations of the algebra of smooth functions.

Abstract

This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth families of vector fields defined on the factors. Vector fields are regarded as derivations of the algebra of smooth functions.