I have some trouble on understanding the domain of the "volume density function" on Riemannian manifold. Putting the volume density function in quote means actually I am working on the function defined on the Riemannian manifold $M$, rather than its tangent space $T_pM$ at $p\in M$. The function is defined by (6.75), page 217 of T.J. Willmore's book "Riemannian Geometry". Anyway I will recall the definition.

Let $(M, g)$ be a connected compact $n$-dimensional Riemannian manifold. For simplicity assume $M$ is orientable and is oriented. Let $p\in M$, and $r_p0$ the injectivity radius at $p$, which is positive. Then $\exp_p:B(0; r_p)\to B(p; r_p)$ is a diffeomorphism.

Let $\omega_g\in\Omega^(M)$ be the Riemannian volume form on $M$, and still denote by $\omega_g$ the Riemannian volume form on $B(p; r_p)$ (obtained by pulling it back). Consider $T_pM$ with the inner product $g_p$ as a Riemannian manifold. The orientation on $M$ induces an orienation on $T_pM$. Denote by $\omega_p$ the Riemannian volume form on $(T_pM, \omega_p)$. Then there exists a unique smooth function $\delta_p:T_pM\to\mathbb{R}$ such that $$\exp_p^*\omega_g=\delta_p\omega_p.$$

The function $\delta_p$ is called the volume density function. Define a function $\theta_p:B(p; r_p)\to\mathbb{R}$ by $$\theta_p:=\delta_p\circ\exp_p^{-1}.$$ Thus we do obtain a function $\theta_p$ defined on an open geodesic ball in $M$.

My question is whether the function $\theta_p$ can be defined on all of $M$. I do not think so, while there exist some papers say yes by citing arguments in page 219-221 of T.J. Willmore's book "Riemannian Geometry". On page 154 of Authur Besse's book "Manifolds all of whose Geodesics are Closed" there seems to be a similar argument.

The related argument in T.J. Willmore's book is the function $\theta:TM\to\mathbb{R}$ defined by (6.91) on page 219, and Proposition (6.6.5) on page 221. The related argument in Authur Besse's book is the last paragraph on page 154. My understanding is these two arguments just say the function $\theta:TM\to\mathbb{R}$ is defined on all of $TM$ whenever $M$ is complete (which, under our assumption, is true). However, these two arguments do not say the function $\theta_p:B(p; r_p)\to\mathbb{R}$ can be extended to all of $M$ (otherwise it implies $\exp:T_pM\to M$ would be a diffeomorphism, which is definitely absurd). Is my understanding correct or did I miss something?

Thank you.