Let $(M,g)$ be a complete Riemannian manifold and $N$ a closed (orientable) hypersuface of $M$. Let $d$ be the signed distance from $N$ and $N_r=\{x\in M: 0<d(x,N)<r\}$. For $r$ small enough $$\Phi:N\times[0,r)\to N_r$$ $$(x,t)\mapsto (x,\exp_x(t\,\textbf{n}(x)))$$ is a diffeomorphism. Here $\textbf{n}$ is the inward unit normal. The following formula always holds: $$|\det D\Phi(x,t)|= 1+O(t),$$ by the tylor expansion of $\det D\Phi(x,t)$ and the fact that $\det D\Phi(x,0)$=1. I wonder if there is an explicite formula for the Jacobian of $\Phi$, $|\det D\Phi(x,t)|$ in general. I found some formulas under some curvature conditions in Gray's Book: Tubes. What about a general case?

Let $(M,g)$ be a complete Riemannian manifold and $N$ a closed (orientable) hypersuface of $M$. Let $d$ be the signed distance from $N$ and $N_r=\{x\in M: 0<d(x,N)<r\}$. For $r$ small enough $$\Phi:N\times[0,r)\to N_r$$ $$(x,t)\mapsto (x,\exp_x(t\,\textbf{n}(x)))$$ is a diffeomorphism. Here $\textbf{n}$ is the inward unit normal. The following formula always holds: $$|\det D\Phi(x,t)|= 1+O(t),$$ by the Taylor expansion of $\det D\Phi(x,t)$ and the fact that $\det D\Phi(x,0)$=1. I wonder if there is an explicit formula for the Jacobian of $\Phi$, $|\det D\Phi(x,t)|$ in general. I found some formulas under some curvature conditions in Gray's book Tubes. What about the general case?