Let $M$ be a complete Riemannian manifold, endowed with its exponential map $\exp: TM \longrightarrow M$. For any $C^k$- function $u$, we get the Pullback $$ \exp^* u = u \circ \exp$$ which is in $C^k(TM)$, as $\exp$ is smooth.
If $u$ is only in $L^p_{\mathrm{loc}}$, however, the pullback is harder to define: For example, one could do it by continuous approximation: If $u_k \in C^0$ converge to $u$ with respect to the seminorm $L^p(K)$ for $K \subseteq M$ compact, then $\exp^* u$ is defined (if it exists) as the limit $\exp^* u_k$ with respect to the $L^p$ norm on compact subsets of $\exp^{-1}(K)$.
So, when does the pullback of an $L^p_{\mathrm{loc}}$ function under $\exp$ exist (and what space is it in)?
Clearly, the only points that cause problems are cutpoints. For $L^1$ on, say $S^n$, the set of nonregular points of $\exp$ is exactly the sphere bundle of radius $\pi$ over $S^n$. Now a singularity of order $r^{1-n}$ at a point gets blown up to a singularity along the corresponding cutpoints, which is an $n$-dimensional sphere. The codimension, however, is $n$, so this is still integrable. I count this as evidence that $L^p_{\mathrm{loc}}$-functions on $M$ indeed pullback to $L^p_{\mathrm{loc}}$-functions on $TM$.
If put this under "reference request" as I guess people should have thought about this...
