Path spaces and induced maps on tangent spaces

Question

Let $M$ be a smooth manifold and $\Omega(M)$ the set of all piecewise smooth path in $M$. Let be $$ f: \Omega(M) \rightarrow \mathbb{R} $$ How can I define $$ f^*: T_{\omega}\Omega \rightarrow T_{f(\omega)}\mathbb{R} $$ on the tangent spaces?

Answer 1

You define such a map the way you always define a derivative $f_* : TM \to TN$; that is to say, given a tangent vector $X \in TM$, choose a path $\gamma$ in $M$ with $\gamma'(0) = X$ and define $$ f_* X = \frac{d}{dt}\Big|_{t = 0} f(\gamma(t)). $$

To be more specific to this example: If $\omega$ is a piecewise smooth path in $M$, then an element of $T_\omega \Omega$ is identified with a (piecewise smooth) vector field $V(s)$ along $\omega$. One chooses some (again piecewise smooth) variation $(t, s) \mapsto \omega_t(s)$ of paths such that $\omega_0 = \omega$, and $V(s) = \partial_t \omega_t(s)$. Then by definition, $$ f_* V = \frac{d}{dt}\Big|_{t = 0} f(\omega_t). $$ The statement that the map $f$ is differentiable (as a map $\Omega(M) \to \mathbb{R}$) is exactly the statement that $f_* V$ as defined above doesn't depend on the choice of variation $\omega_t$.