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...



Pullback of $L^p$ functions via exponential map
Let us denote the geodesic flow for time $t$ by $\Theta^t$,
this is the map $TM\to TM$
and let $\sigma$ be the projection $TM\to M$.
Then $\exp=\sigma\circ \Theta^1$.


Note that $\Theta^1$ preserves volume on $TM$. 
Threfore the $\Theta^1$-pullback of $L^p\_{loc}$-function on $TM$ is $L^p\_{loc}$.
Hence your statement follows.


