Geometry of Bounded Frechet Manifolds

TL;DR

This paper explores the geometric structure of bounded Fréchet manifolds, establishing conditions for tangent bundles to be vector bundles and proving the existence and uniqueness of integral curves.

Contribution

It introduces the geometry of bounded Fréchet manifolds, showing that their tangent bundles can be vector bundles and characterizing when the second order tangent bundle forms a vector bundle.

Findings

Bounded Fréchet tangent bundle admits a vector bundle structure.

The second order tangent bundle is a vector bundle iff the manifold has a linear connection.

Existence and uniqueness of integral curves of vector fields on bounded Fréchet manifolds.

Abstract

In this paper we develop the geometry of bounded Fr\'echet manifolds. We prove that a bounded Fr\'echet tangent bundle admits a vector bundle structure. But the second order tangent bundle $T^2M$ of a bounded Fr\'echet manifold $M$, becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As an application, we prove the existence and uniqueness of the integral curve of a vector field on $M$.