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Pestov identity for geodesic data on a simple surface

Statement

Let $(M, g)$ be a compact, connected, nonempty, oriented smooth Riemannian surface with smooth strictly convex boundary, no conjugate points, and nontrapping geodesic flow. Set $S M = {(x, v) : x \in M, g_{x} (v, v) = 1}$ and let $φ_{t}$ be the unit-speed geodesic flow until boundary exit. For $w \in C^{\infty} (S M, R)$ define $X w = {\frac{d}{d t} ∣}_{0} w (φ_{t} (x, v))$ . Define $V w (x, v) = {\frac{d}{d θ} ∣}_{0} w (x, \cos θ v + \sin θ v^{⊥})$ , where $v^{⊥}$ is the positive $90$ -degree rotation of $v$ , and set $X^{⊥} = [X, V]$ . For a positive orthonormal frame $e_{1}, e_{2}$ , let $K (x) = g (R (e_{1}, e_{2}) e_{2}, e_{1})$ , where $R (A, B) C = \nabla_{A} \nabla_{B} C - \nabla_{B} \nabla_{A} C - \nabla_{[A, B]} C$ ; this value is independent of the frame. Let $d Σ = d V_{g} (x) d S_{x} (v)$ be the Liouville measure and define $(a, b) = \int_{S M} a b d Σ$ and $∥ a ∥^{2} = (a, a)$ . For every real-valued $u \in C^{\infty} (S M)$ with $u = 0$ on $\partial S M$ ,

∥ V X u ∥^{2} = ∥ X V u ∥^{2} - (K V u, V u) + ∥ X u ∥^{2} .

Statement references

Definition of a simple surface and its boundary conventions

Surface geodesic-flow commutators and integration by parts

Facts & Assumptions

Given: A compact, connected, nonempty, oriented simple Riemannian surface with smooth boundary and a real-valued smooth function $u$ on $S M$ that vanishes on $\partial S M$ .

[F1]

Simple surface and geodesic X-ray transform: a simple surface is compact, connected, and nonempty, has smooth strictly convex boundary, has no conjugate points, and is nontrapping; $S M$ consists of unit tangent vectors based in $M$ .

[F2]

Surface geodesic flow, index form, and transport primitive: $[X, V] = X^{⊥}$ , $[V, X^{⊥}] = X$ , and $[X, X^{⊥}] = - K V$ .

[F3]

Surface geodesic flow, index form, and transport primitive: $V$ is tangent to $\partial S M$ ; integration by parts holds for $X$ and $X^{⊥}$ when the product of the factors vanishes on the boundary, and it holds for $V$ for all smooth factors.

Proof

1.1F1F3given

Since $V$ rotates the vector within its fixed fibre, it is tangent to $\partial S M$ . The restriction of $u$ to $\partial S M$ is zero, so its derivative along every tangent vector there is zero; hence $V u = 0$ on $\partial S M$ . In particular, the boundary products $(V u) (X^{⊥} u)$ and $(V u) (X u)$ vanish, as needed for the $X$ and $X^{⊥}$ integration-by-parts formulas in [F3].

2.1F2step 1.1algebra

Put $P = X V u$ and $Q = X^{⊥} u$ . From $[X, V] = X V - V X = X^{⊥}$ we have $V X u = P - Q$ , and therefore $∥ V X u ∥^{2} = ∥ P ∥^{2} - 2 (P, Q) + ∥ Q ∥^{2}$ .

3.1F2F3step 1.1step 2.1algebra

Integrating $X$ by parts with boundary factor $(V u) (X^{⊥} u) = 0$ , then using $X X^{⊥} = X^{⊥} X - K V$ , gives $(P, Q) = - (V u, X X^{⊥} u) = - (V u, X^{⊥} X u) + (K V u, V u)$ . Integrating $X^{⊥}$ by parts with boundary factor $(V u) (X u) = 0$ turns the first term into $(X^{⊥} V u, X u)$ .

4.1F2F3step 3.1algebra

The commutator $[V, X^{⊥}] = X$ implies $X^{⊥} V u = V X^{⊥} u - X u$ . Thus $(X^{⊥} V u, X u) = (V X^{⊥} u, X u) - ∥ X u ∥^{2} = - (X^{⊥} u, V X u) - ∥ X u ∥^{2}$ , where the last equality is integration by parts for $V$ , whose vector field is tangent to $\partial S M$ . As $V X u = P - Q$ , this gives $(P, Q) = - (Q, P - Q) - ∥ X u ∥^{2} + (K V u, V u) = - (P, Q) + ∥ Q ∥^{2} - ∥ X u ∥^{2} + (K V u, V u)$ . Solving gives $2 (P, Q) = ∥ Q ∥^{2} - ∥ X u ∥^{2} + (K V u, V u)$ .

5.1step 2.1step 4.1algebra∎

Substitute this value in step 2.1 and cancel $∥ Q ∥^{2}$ . The result is $∥ V X u ∥^{2} = ∥ P ∥^{2} + ∥ X u ∥^{2} - (K V u, V u) = ∥ X V u ∥^{2} - (K V u, V u) + ∥ X u ∥^{2}$ , which is the asserted identity.

Source locator

The source comparisons and item-specific dispositions are recorded in research/inverse-problems-knot-theory-all-109-2026-10-19-batch-38.coverage.json and the batch notes.

Depends on

Used by

Unweighted scalar geodesic X-ray injectivity on simple surfaces Theorem

Dependency tree · two levels

16 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

Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann, Geometric inverse problems, with emphasis on two dimensions (standard reference, not scraped)
Joonas Ilmavirta, Geometry of geodesics, full lecture notes (standard reference, not scraped)
Mikko Salo, On geometric inverse problems and microlocal analysis, author-hosted lecture notes (standard reference, not scraped)