First Variation Formula for Proper Variation

Theorem
Let $\struct {M, g}$ be a Riemannian manifold.

Let $I = \closedint a b$ be a closed real interval.

Let $J$ be an open real interval.

Suppose $\gamma : I \to M$ is a unit-speed admissible curve.

Let $\Gamma : J \times I \to M$ be a proper variation of $\gamma$.

Let $T_p M$ be the tangent space at $p \in M$.

Let $V : M \to T_\gamma M$ be the variation field of $\Gamma$.

Let $L_g$ be the Riemannian length of an admissible curve.

Let $\gamma'$ be the velocity of $\gamma$.

Let $D_t$ be the covariant derivative along $\gamma$.

Let $\tuple {a_0, \ldots, a_k}$ be an admissible subdivision for $\gamma$.

For all $i \in \N_{> 0} : i \le k - 1$ let $\Delta_i \gamma' = \map {\gamma'} {a_i^+} - \map {\gamma'} {a_i^-}$.

Let $\innerprod \cdot \cdot$ be the inner product induced by the Riemannian metric.

Then for all $s \in J$ the mapping $\map {L_g} {\Gamma_s}$ is a smooth function of $s$ and:


 * $\ds \valueat {\dfrac d {d s} \map {L_g} {\Gamma_s} } {s \mathop = 0} = - \int_a^b \innerprod V {D_ t \gamma'} \d t - \sum_{i \mathop = 1}^{k - 1} \innerprod {\map V {a_i} } {\Delta_i \gamma'}$