nLab
tangent spaces of mapping spaces

Contents

Idea

As shown in manifold structure of mapping spaces, the space of smooth maps from a sequentially compact Frölicher space (or diffeological or Chen space) in to a smooth manifold is again a smooth manifold. Its tangent space is straightforward to identify. A tangent vector is an infinitesimal deviation of a smooth map; that is, it defines a direction in which to deform that map. As a smooth map is determined by its values at points, when deforming a smooth map it is enough to explain how to deform each point. Thus a tangent vector at α:SM\alpha \colon S \to M defines, for each pSp \in S, a tangent vector at α(p)M\alpha(p) \in M. Thus we obtain a map STMS \to T M. It is not unbelievable that this map is again smooth, whence we have:

(1)TC (S,M)C (S,TM). T C^\infty(S,M) \to C^\infty(S, T M).

The claim of this page is that this is a diffeomorphism, and further a vector bundle isomorphism, where each is considered a vector bundle over C (S,M)C^\infty(S, M).

Revised on June 3, 2011 08:36:48 by Urs Schreiber (89.204.137.115)