LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Logarithmic derivative of the radial volume jacobian is the distance laplacian

Statement

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $(M, g)$ be a complete, connected, boundaryless Riemannian manifold of dimension $n \geq 2$ , let $p \in M$ and let $v \in S_{p} M$ be a unit tangent vector with cut time $c_{p} (v)$ . For every $t$ with $0 < t < c_{p} (v)$ , $\frac{d}{d t} \log J_{p} (t, v) = tr S_{v} (t) = Δ_{g} r_{p} (\exp_{p} (t v)),$ where $J_{p} (t, v)$ is the radial volume Jacobian of Radial volume jacobian, $S_{v} (t)$ is the radial Riccati operator of Radial riccati operator along $γ_{v}$ , and $r_{p} (x) = d_{g} (p, x)$ is the distance from $p$ .

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1], a complete connected boundaryless Riemannian manifold $(M, g)$ of dimension $n \geq 2$ , a point $p \in M$ , a unit vector $v \in S_{p} M$ with cut time $c_{p} (v)$ , the radial geodesic $γ_{v} (t) = \exp_{p} (t v)$ , the radial Jacobi tensor $A_{v}$ with parallel-frame matrix ${\overset{ˉ}{A}}_{v}$ , the radial volume Jacobian $J_{p} (t, v)$ and the Riccati operator $S_{v} (t)$ , and the distance function $r_{p}$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the cut-time, geodesic and exponential interfaces of Cut time in a unit tangent direction and Radial volume jacobian.

[F1]

Radial volume Jacobian: $J_{p} (t, v) = \det {\overset{ˉ}{A}}_{v} (t) > 0$ for $0 < t < c_{p} (v)$ , the determinant being taken in a compatible oriented orthonormal frame; ${\overset{ˉ}{A}}_{v}$ is smooth there, and ${\overset{ˉ}{A}}_{v} (t)$ is invertible because its determinant is nonzero (Radial volume jacobian, Radial jacobi tensor is invertible before the first conjugate point).

[F2]

Riccati operator: on the interval where it is defined, $S_{v} (t) = {\overset{ˉ}{A}}_{v}^{'} (t) {\overset{ˉ}{A}}_{v} (t)^{- 1}$ , equivalently the endomorphism $D_{t} A_{v} \circ A_{v}^{- 1}$ of the normal space at time $t$ , transported back to $N_{0}$ by parallel transport; its trace is therefore independent of which of these two representations is used (Radial riccati operator).

[F3]

Hessian of the distance: with $q = γ_{v} (t)$ , $0 < t < c_{p} (v)$ , the distance function $r_{p}$ is smooth near $q$ , and for every $X \in T_{q} M$ orthogonal to $T = {\dot{γ}}_{v} (t)$ the Levi-Civita Hessian satisfies $(\nabla^{2} r_{p})_{q} (X, Y) = g_{q} (D_{t} J_{X} (t), Y)$ , where $J_{X}$ is the normal Jacobi field along $γ_{v}$ with $J_{X} (0) = 0$ and $J_{X} (t) = X$ ; moreover $(\nabla^{2} r_{p})_{q} (T, \cdot) = 0$ and on $T^{⊥}$ the shape endomorphism $S = \nabla grad r_{p}$ is $D_{t} J (t) \circ J (t)^{- 1}$ (Hessian of distance in terms of radial jacobi fields).

[F4]

Laplace–Beltrami operator: for smooth $f$ , $Δ_{g} f (p) = {tr}_{g} (Hess f)_{p} = \sum_{i} Hess f (e_{i}, e_{i})$ in any orthonormal basis (Laplace–Beltrami operator as the trace of the Hessian).

[F5]

Jacobi's formula: for a differentiable family of invertible matrices $A (t)$ , $\frac{d}{d t} \det A (t) = \det A (t) tr (A (t)^{- 1} A^{'} (t))$ , the differential of the determinant being $D \det (A) [H] = \det (A) tr (A^{- 1} H)$ (The determinant differential is $D \det (A) [H] = tr (adj (A) H)$ at every matrix, and Jacobi's formula holds on the invertible locus).

Proof

Proof technique: direct: differentiate the determinant of the radial Jacobi matrix with Jacobi's formula, identify the trace of ${\overset{ˉ}{A}}^{'} {\overset{ˉ}{A}}^{- 1}$ with the trace of the shape operator of the distance sphere, and use that the Laplace–Beltrami operator is the trace of the Hessian.

1.1F1F2F5

The logarithmic derivative of the radial volume Jacobian is the trace of $S_{v}$ . [F1, F2, F5] On $0 < t < c_{p} (v)$ the matrix family ${\overset{ˉ}{A}}_{v}$ is smooth and invertible by [F1]. Jacobi's formula of [F5], applied to $A (t) = {\overset{ˉ}{A}}_{v} (t)$ and divided by the nonvanishing determinant, gives $\frac{d}{d t} \log J_{p} (t, v) = \frac{1}{\det {\overset{ˉ}{A}}_{v} (t)} \frac{d}{d t} \det {\overset{ˉ}{A}}_{v} (t) = tr ({\overset{ˉ}{A}}_{v} (t)^{- 1} {\overset{ˉ}{A}}_{v}^{'} (t)) .$ The trace is invariant under cyclic permutation of a product, so $tr ({\overset{ˉ}{A}}_{v}^{- 1} {\overset{ˉ}{A}}_{v}^{'}) = tr ({\overset{ˉ}{A}}_{v}^{'} {\overset{ˉ}{A}}_{v}^{- 1}) = tr S_{v} (t)$ by the definition of the Riccati operator in [F2].

2.1F2F3step 1.1

The trace of the Hessian of the distance equals the trace of $S_{v}$ . [F2, F3, step 1.1] Put $q = γ_{v} (t)$ with $0 < t < c_{p} (v)$ and $T = {\dot{γ}}_{v} (t)$ . Choose an orthonormal basis of $T_{q} M$ consisting of $T$ together with an orthonormal basis $e_{1}, \dots, e_{n - 1}$ of $T^{⊥}$ . By [F3] the Hessian entries in the radial direction vanish, $(\nabla^{2} r_{p})_{q} (T, \cdot) = 0$ , and on $T^{⊥}$ the endomorphism representing the Hessian is $D_{t} J (t) \circ J (t)^{- 1}$ . In the parallel frame of [F2] this endomorphism is exactly $S_{v} (t)$ , and conjugation by the isometry $P_{t}$ preserves the trace, so ${tr}_{g} (Hess r_{p})_{q} = \sum_{i = 1}^{n - 1} g_{q} (D_{t} J_{e_{i}} (t), e_{i}) = tr S_{v} (t) .$

3.1F4step 1.1step 2.1

Conclusion. [F4, step 1.1, step 2.1] By [F4] the Laplace–Beltrami operator of $r_{p}$ at $q$ is the trace of the Hessian, and by step 2.1 this trace equals $tr S_{v} (t)$ ; by step 1.1 the logarithmic derivative of $J_{p} (t, v)$ equals the same trace. Therefore $\frac{d}{d t} \log J_{p} (t, v) = tr S_{v} (t) = Δ_{g} r_{p} (\exp_{p} (t v))$ for every $0 < t < c_{p} (v)$ .

4.1F1F2F3step 1.1step 2.1step 3.1∎

Boundary cases. [F1, F2, F3, step 1.1, step 2.1, step 3.1] The identity is asserted on the open interval $0 < t < c_{p} (v)$ : at $t = 0$ the Jacobian vanishes, $\log J_{p}$ is not defined at $0$ and $r_{p}$ is not differentiable at $p$ ; at $t = c_{p} (v)$ (when the cut time is finite and attained) the distance function need not be smooth, and [F3] does not apply. If $c_{p} (v) = + \infty$ the identity holds on all of $(0, \infty)$ . In dimension $n = 2$ the normal space is one-dimensional, $S_{v} (t)$ is a scalar and $tr S_{v} (t)$ is that scalar; the argument above already covers this case because the basis $e_{1}, \dots, e_{n - 1}$ then has one element. The statement is for unit $v$ ; for a general $w \neq 0$ one applies it to $v = w / ∣ w ∣$ and the affine reparametrization $t \mapsto ∣ w ∣ t$ , while $w = 0$ has no radial direction. No choice beyond the inherited [A1] is used.

Source locator

Datar §27.2 and §28.1, pp.200–209, and Eschenburg §4–5, pp.15–20, derive the identity $\frac{d}{d t} \log J = tr S = Δ r$ for the radial jacobian before the cut locus. The proof above is assembled from the in-run radial Jacobian, Riccati and Hessian items and Jacobi's formula.

Depends on

Used by

Rigidity in bishop gromov on an interval Proposition
Relative volume density comparison Theorem

Dependency tree · two levels

69 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)
J.-H. Eschenburg, Comparison Theorems in Riemannian Geometry (standard reference, not scraped)