First and second variation of arc length

Builds on Riemannian manifold Geodesic Covariant derivative Riemann curvature tensor Arc length

Statement

Let $(M, g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and curvature $R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$ (see Riemann curvature tensor), and write $\ip{\cdot}{\cdot}$ for $g$ . Let $\gamma \colon [a, b] \to M$ be a smooth curve with $\abs{\gamma'(t)} = 1$ for all $t$ . Let $\Gamma \colon (-\varepsilon, \varepsilon) \times [a, b] \to M$ be a smooth map with $\Gamma(0, t) = \gamma(t)$ , let $V(t) = \partial_s \Gamma(0, t)$ be its variation field, and let $L(s)$ be the length of the curve $t \mapsto \Gamma(s, t)$ . Let $D_t$ and $D_s$ be the covariant derivatives along the curves $t \mapsto \Gamma(s, t)$ and $s \mapsto \Gamma(s, t)$ .

(First variation.) $L'(0) = \Bigl[ \ip{V}{\gamma'} \Bigr]_{t=a}^{t=b} - \int_a^b \ip{V}{D_t \gamma'} \, dt.$
(Second variation.) If $\gamma$ is a geodesic and $V^\perp = V - \ip{V}{\gamma'}\gamma'$ is the normal part of $V$ , then $L''(0) = \int_a^b \Bigl( \abs{D_t V^\perp}^2 - \ip{R(V^\perp, \gamma')\gamma'}{V^\perp} \Bigr) dt + \Bigl[ \ip{D_s \partial_s \Gamma(0, t)}{\gamma'(t)} \Bigr]_{t=a}^{t=b}.$

Remarks

Both formulas come from differentiating $L(s) = \int_a^b \abs{\partial_t \Gamma(s, t)} \, dt$ under the integral sign, using the symmetry $D_s \partial_t \Gamma = D_t \partial_s \Gamma$ and metric compatibility, and integrating by parts. The second formula also uses $D_s D_t W - D_t D_s W = R(\partial_s \Gamma, \partial_t \Gamma) W$ for vector fields $W$ along $\Gamma$ . For a proper variation, one with $\Gamma(s, a) = \gamma(a)$ and $\Gamma(s, b) = \gamma(b)$ for all $s$ , the boundary terms vanish. So a unit-speed curve is a critical point of length for all proper variations if and only if $D_t \gamma' = 0$ , that is, it is a geodesic. Example: assume all sectional curvatures are at least $k > 0$ and the geodesic $\gamma$ has length greater than $\pi / \sqrt{k}$ . Take $V = \sin\bigl(\pi (t - a)/(b - a)\bigr) E$ , with $E$ a parallel unit normal field along $\gamma$ . Then $L''(0) < 0$ , so $\gamma$ is not minimizing; this is the core of Myers's theorem. The second formula needs $\gamma$ to be a geodesic; otherwise extra terms involving $D_t \gamma'$ appear.

Going deeper

For a student

The integrand of the second variation, restricted to proper normal variations, is the index form of $\gamma$ . Its null space consists of the Jacobi fields vanishing at the endpoints, so it is nondegenerate exactly when $\gamma(a)$ and $\gamma(b)$ are not conjugate along $\gamma$ . This is the bridge from curvature to global results such as the theorems of Myers, Synge and Cartan–Hadamard and the Morse index theorem. Many texts state the same formulas for the energy $E = \tfrac{1}{2}\int \abs{\gamma'}^2 dt$ instead of the length; for unit-speed geodesics and proper normal variations the second variations agree.

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Arc length concept
Covariant derivative concept
Geodesic concept
Riemann curvature tensor concept
Riemannian manifold concept
First and second variation of arc length statement

Where this leads

2 pages build on this one.

In the catalogue

Introduction to Riemannian Manifolds, John M. Lee

About this page

Tag	`ZB91TH6FKV`
URI	`m2://ZB91TH6FKV`
Kind	statement, theorem
Writing	W0 stub
MSC	53C22, 58E10
Status	This copy of the site has no build data. Run `m2 status ZB91TH6FKV`.
About	Geodesic
Uses	Riemannian manifold, Geodesic, Covariant derivative, Riemann curvature tensor, Arc length
Changed	2 November 2026

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