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Index form of a geodesic segment

Definition

Assume ${A C}_{ω}$ through the declared dependencies, including The Axiom of Countable Choice ( ${A C}_{ω}$ ), Algebraic symmetries of the Riemann tensor, and Second variation formula for energy. Let $a < b$ and let $γ : [a, b] \to M$ be an affinely parametrized geodesic in a Riemannian manifold. Let $X_{p w}^{1} (γ)$ be the real vector space of continuous vector fields along $γ$ that are $C^{1}$ on each piece of some finite subdivision of $[a, b]$ . Derivatives at included endpoints and the two traces at an interior breakpoint are taken one-sided.

For $V, W \in X_{p w}^{1} (γ)$ , choose a common finite subdivision on which both fields are $C^{1}$ and define $I_{γ} (V, W) : = \sum_{k = 1}^{m} \int_{t_{k - 1}}^{t_{k}} (g (D_{t} V, D_{t} W) - g (R (V, \dot{γ}) \dot{γ}, W)) d t .$ The derivatives are the covariant derivatives along $γ$ from Covariant derivative along a curve, and the curvature sign and slot order are those in Riemann curvature four-tensor. The finite sum is independent of the chosen common subdivision. The resulting index form is a symmetric bilinear form. Its fixed-endpoint subspace is $X_{0} (γ) : = {V \in X_{p w}^{1} (γ) : V (a) = V (b) = 0},$ and the index form on fixed-endpoint fields is its restriction to $X_{0} (γ) \times X_{0} (γ)$ .

For any fixed-endpoint two-parameter variation of $γ$ covered by Second variation formula for energy, its variation fields $V, W$ lie in $X_{0} (γ)$ and its mixed energy derivative is $I_{γ} (V, W)$ . This assertion is only about fields that arise from such a variation; no realization of every piecewise $C^{1}$ field is claimed here.

Facts & Assumptions

Given: The interval, affinely parametrized geodesic, metric, and continuous piecewise $C^{1}$ fields in the definition.

[A1]

Countable choice is the assumption ${A C}_{ω}$ defined by The Axiom of Countable Choice ( ${A C}_{ω}$ ). The algebraic-symmetry theorem carries it through its first-Bianchi supplier (Algebraic symmetries of the Riemann tensor), and the second-variation formula also assumes it (Second variation formula for energy). Here the local use is pair interchange and the pair skews that make the curvature bilinear term symmetric. No full Axiom of Choice or additional choice is used.

[F1]

For a fixed-endpoint two-parameter variation with central geodesic $γ$ , the mixed energy derivative is the derivative-product and curvature integral; its endpoint-acceleration term vanishes. This is Second variation formula for energy.

[F2]

Covariant differentiation along the curve is defined stripwise, with one-sided endpoint values, by Covariant derivative along a curve.

[F3]

$Rm (X, Y, Z, W) = g (R (X, Y) Z, W)$ by Riemann curvature four-tensor.

[F4]

The curvature four-tensor has first- and last-pair skewness and pair interchange by Algebraic symmetries of the Riemann tensor.

[F5]

A continuous real-valued function on a compact nondegenerate interval is Riemann integrable by A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion.

[F6]

The real Riemann integral is linear on integrable functions by Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ .

[F7]

The integral over an interval splits additively at every interior point by For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ .

Verification

Proof technique: verify the finite-piece definition, bilinearity, symmetry, and the fixed-endpoint Hessian interpretation.

1.1

The piecewise integrand is Riemann integrable on each common subdivision piece, and the finite sum is partition-independent. [F2, F3, F5, F7] $D_{t} V$ and $D_{t} W$ have continuous one-sided extensions by [F2], and the curvature expression is continuous there by [F3]. Thus each scalar integrand is Riemann integrable by [F5]. The sum is finite. If the subdivision is refined, [F7] leaves each piece's integral unchanged; the union of two finite breakpoint sets is a finite common refinement, so any two choices give the same sum. The expression is therefore well-defined.

1.2

The displayed form is bilinear in its two vector-field inputs. [F2, F3, F6] Covariant differentiation and curvature are linear in the field slots by [F2] and [F3], and $g$ is bilinear. Hence the displayed integrand is bilinear in $(V, W)$ on every piece. Applying [F6] to its finite integrals and summing proves that $I_{γ}$ is bilinear.

1.3

Curvature pair symmetry and symmetry of $g$ make the full form symmetric. [F3, F4] For every $t$ on each piece, [F3] and the pair interchange and pair skews in [F4] give $g (R (V, \dot{γ}) \dot{γ}, W) = Rm (V, \dot{γ}, \dot{γ}, W) = Rm (W, \dot{γ}, \dot{γ}, V) = g (R (W, \dot{γ}) \dot{γ}, V) .$ The derivative-product term is symmetric because $g$ is symmetric. Integrating these pointwise equalities proves $I_{γ} (V, W) = I_{γ} (W, V)$ . Endpoint evaluation is linear, so $X_{0} (γ)$ is a vector subspace and the restriction remains symmetric bilinear.

2.1

For each fixed-endpoint two-parameter variation with central $γ$ , its mixed energy derivative equals the index form. [F1, step 1.1, step 1.2, step 1.3] If a two-parameter variation fixes both endpoints, its variation fields vanish at $a$ and $b$ . In [F1] the mixed endpoint acceleration therefore vanishes, leaving exactly the integral that defines $I_{γ} (V, W)$ ; this proves the stated fixed-endpoint energy-Hessian interpretation for the variation fields in question.

3.1

The interval, endpoint, empty, dimension, constant-curve, zero-field and choice cases are as follows. [A1, F2, F4, step 1.1, step 1.2, step 1.3, step 2.1] The condition $a < b$ excludes the singleton interval, where $D_{t}$ is not defined. At included endpoints the stipulated one-sided derivatives apply. If $M$ is empty, no given geodesic exists; in dimension zero all fields and the form are zero. In dimension one the curvature term vanishes by first-pair skewness in [F4]. A constant geodesic has $\dot{γ} = 0$ , so its curvature term is zero while $D_{t} V$ and the derivative-product integral may remain nonzero. If either field is zero, bilinearity makes the form zero. Assumption [A1] is used only for the curvature symmetry in step 1.3; the finite refinement and integral operations use no choice. This is a definition and identity, not an iff claim. $□$

Depends on

Used by

Index form in constant curvature Example
Integration by parts for the index form Lemma
At a conjugate endpoint the index form is degenerate Proposition
Jacobi fields are the null solutions of the index form with fixed endpoints Proposition
The Morse index theorem for geodesics Remark
A geodesic does not minimize past its first conjugate point Theorem
Index lemma Theorem

Dependency tree · two levels

57 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), Definition 21.2.1 (standard reference, not scraped)
John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Equation (10.15) (standard reference, not scraped)