Structure of some mapping spaces

TL;DR

The paper develops a comprehensive framework for understanding spaces of paths and maps on differentiable manifolds as infinite-dimensional manifolds. It constructs explicit diffeomorphisms, notably the map $P$ that identifies path spaces with tangent-path spaces, and extends these ideas to higher-dimensional mapping spaces, yielding direct-sum models in the Fréchet setting. It further establishes fiber-bundle structures for mapping spaces and path fibrations, with smooth local trivializations built from parallel transport and exponential maps. Finally, it shows that mapping spaces from compact spaces into manifolds are Banach (or complex Banach) manifolds, providing a unifying treatment that connects ODE theory, exponential laws, and infinite-dimensional geometry. These results advance the understanding of the geometric and topological structure of mapping spaces and have potential implications for global analysis and shape spaces.

Abstract

We prove that the path space of a differentiable manifold is diffeomorphic to a Fréchet space, endowing the path space with a linear structure. Furthermore, the base point preserving mapping space consisting of maps from a cube to a differentiable manifold is also diffeomorphic to a Fréchet space. As a corollary of a more general theorem, we prove that the path fibration becomes a fibre bundle for manifolds M. Additionally, we discuss the mapping space from a compact topological space to a differentiable manifold, demonstrating that this space admits the structure of a smooth Banach manifold.

Theorems & Definitions (87)

• Proposition 1.1
• proof
• Definition 2.1
• Lemma 2.1: smoothness of solution to ordinary differential equations.
• proof
• Lemma 2.2: Parallel transport of smooth vector fields
• proof
• Lemma 2.3
• proof
• Lemma 2.4