Proof of the First and Second Variation Formulas

The variation formulas compute how the energy of a curve changes under deformation, establishing geodesics as critical points and providing the tools for analyzing their stability.

Setup

Definition8.9Variation of a curve

A variation of a curve $\gamma: [a,b] \to M$ is a smooth map $\Gamma: [a,b] \times (-\epsilon, \epsilon) \to M$ with $\Gamma(t, 0) = \gamma(t)$ . The variation vector field is $V(t) = \frac{\partial \Gamma}{\partial s}(t, 0)$ . The variation is proper (fixed-endpoint) if $\Gamma(a, s) = \gamma(a)$ and $\Gamma(b, s) = \gamma(b)$ for all $s$ .

First Variation Formula

Theorem8.8First variation of energy

Let $\gamma: [a,b] \to M$ be a smooth curve and $\Gamma$ a proper variation with variation vector field $V$ . Then

\frac{dE}{ds}\Big|_{s=0} = -\int_a^b g(\nabla_{\gamma'}\gamma', V) \, dt.

Proof

Write $T = \frac{\partial \Gamma}{\partial t}$ and $V = \frac{\partial \Gamma}{\partial s}$ . Then:

\frac{dE}{ds} = \frac{1}{2}\frac{d}{ds}\int_a^b g(T, T) \, dt = \int_a^b g(\nabla_V T, T) \, dt.

By the symmetry lemma: $\nabla_V T = \nabla_T V$ (since $[T, V] = [\partial_t, \partial_s] = 0$ ). So:

\frac{dE}{ds} = \int_a^b g(\nabla_T V, T) \, dt = \int_a^b \left(\frac{d}{dt}g(V, T) - g(V, \nabla_T T)\right) dt.

The first term integrates to $g(V, T)\Big|_a^b = 0$ (proper variation: $V(a) = V(b) = 0$ ). Thus:

\frac{dE}{ds}\Big|_{s=0} = -\int_a^b g(V, \nabla_{\gamma'}\gamma') \, dt. \quad \blacksquare

■

RemarkGeodesics as critical points

The first variation vanishes for all proper variations if and only if $\nabla_{\gamma'}\gamma' = 0$ , i.e., $\gamma$ is a geodesic. This is the Euler-Lagrange equation for the energy functional.

Second Variation Formula

Theorem8.9Second variation of energy

Let $\gamma: [a,b] \to M$ be a geodesic and $\Gamma$ a proper variation with variation vector field $V$ . Then

\frac{d^2 E}{ds^2}\Big|_{s=0} = \int_a^b \left(g(\nabla_{\gamma'} V, \nabla_{\gamma'} V) - g(R(V, \gamma')\gamma', V)\right) dt = \int_a^b \left(|V'|^2 - K(V, \gamma')|V \wedge \gamma'|^2\right) dt,

where $V' = \nabla_{\gamma'} V$ and $K$ is the sectional curvature.

Proof

Differentiating $\frac{dE}{ds} = \int g(\nabla_T V, T) \, dt$ once more:

\frac{d^2E}{ds^2} = \int_a^b \left(g(\nabla_V \nabla_T V, T) + g(\nabla_T V, \nabla_V T)\right) dt.

For the first term, use $\nabla_V \nabla_T V = \nabla_T \nabla_V V + R(V, T)V$ (definition of curvature and $[T,V] = 0$ ). At $s = 0$ , $\nabla_T T = 0$ (geodesic), so integrating by parts:

\int g(\nabla_T \nabla_V V, T) \, dt = g(\nabla_V V, T)\Big|_a^b - \int g(\nabla_V V, \nabla_T T) \, dt = 0.

The curvature term gives $\int g(R(V, T)V, T) \, dt = -\int g(R(V, T)T, V) \, dt$ . The second term in the original sum is $\int |\nabla_T V|^2 \, dt$ . Combining at $s = 0$ :

\frac{d^2E}{ds^2}\Big|_{s=0} = \int_a^b \left(|\nabla_{\gamma'} V|^2 - g(R(V, \gamma')\gamma', V)\right) dt. \quad \blacksquare

■

The Index Form

Definition8.10Index form

The index form along a geodesic $\gamma$ is the bilinear form on vector fields along $\gamma$ vanishing at endpoints:

I(V, W) = \int_a^b \left(g(\nabla_{\gamma'} V, \nabla_{\gamma'} W) - g(R(V, \gamma')\gamma', W)\right) dt.

The second variation equals $I(V, V)$ . The geodesic is a local minimum if and only if $I$ is positive definite, which holds precisely when there are no conjugate points in $(a, b)$ .