Radial volume jacobian

Assume the inherited Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ once and for all through the cut-time and polar-coordinate interfaces named below. Let $(M,g)$ be a complete connected boundaryless Riemannian manifold of dimension $n\ge 2$, let $p\in M$, let $v\in S_{p}M$ be a unit tangent vector and let ${\gamma }_v(t):={\exp }_p(tv)$ be the radial geodesic, with cut time $c_p(v)$ (Cut time in a unit tangent direction). Let $A_v(t):N_0\to N_t$ be the radial Jacobi tensor of ${\gamma }_v$ (Radial Jacobi tensor) and let ${\bar{A}}_v(t)=P_t^{-1}\circ A_v(t)$ be its parallel-frame matrix.

The radial volume Jacobian is $J_p(t,v):=\det {\bar{A}}_v(t)>0,0<t<c_p(v),$ the determinant of $A_v$ computed in any oriented orthonormal basis of $N_0$ and its parallel transport, whose orientations are declared compatible at $t=0$. The value does not depend on the choice of oriented orthonormal basis: a change of basis with orthogonal matrix $C$ replaces ${\bar{A}}_v$ by $C^{-1}{\bar{A}}_{v}C$, whose determinant is unchanged (Levi civita parallel transport preserves lengths angles and volume).

The defining properties recorded here are:

1. Positivity. $J_p(t,v)>0$ for $0<t<c_p(v)$. Indeed, as $t\downarrow 0$, ${\bar{A}}_v(t)=t\mathrm{i}\mathrm{d}+O(t^2)$, so $\det {\bar{A}}_v(t)=t^{n-1}(1+O(t))>0$ for small $t$; the determinant is continuous and cannot vanish on $(0,c_p(v))$, because the cut time does not exceed the first conjugate time and the radial Jacobi tensor is invertible before the first conjugate instant (Radial jacobi tensor is invertible before the first conjugate point).
2. Normalisation. $J_p(t,v)\mathrm{/}{t}^{n-1}\to 1$ as $t\downarrow 0$, by the same expansion.
3. Geometric role. On the polar chart before the cut time, Riemannian volume is the pushforward of $J_p(t,v)dtd{\sigma }_p(v)$, where ${\sigma }_p$ is the surface measure on $S_{p}M$; this is the polar integration formula (Polar integration may discard the cut locus). Relative to Euclidean polar measure $t^{n-1}dtd{\sigma }_p(v)$, the density ratio is $J_p(t,v)\mathrm{/}{t}^{n-1}$.

No sign or estimate beyond positivity is asserted here, and the value at $t\ge c_p(v)$ is not defined. The integer $n\ge 2$ keeps the normal space nonempty; for $n=1$ the normal space is zero-dimensional and $J_p(t,v)=1$ by the empty-determinant convention, which is the convention used by the polar integration formula.