First Variation Formula
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 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:
- $\ds \valueat {\dfrac d {d s} \map {L_g} {\Gamma_s} } {s \mathop = 0} = - \int_a^b \innerprod V {D_ t \gamma'} \d t - \sum_{i \mathop = 1}^{k - 1} \innerprod {\map V {a_i} } {\Delta_i \gamma'} + \innerprod {\map V b} {\map {\gamma'} b} - \innerprod {\map V a} {\map {\gamma'} a}$
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Minimizing Curves Are Geodesics