Admissible Family of Curves Symmetry Lemma

Theorem

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

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

Let $J$ is an open real interval.

Let $\Gamma : J \times I \to M$ be an admissible family of curves.

Let $\tuple {a_0, a_1, a_2, \ldots, a_{n - 1}, a_n}$ be a finite subdivision of $I$.

Then:

$\forall i \in \N_{> 0} : i \le n : \forall s \in J : \forall t \in \closedint {a_{i - 1} }{a_i} : D_s \partial_t \Gamma = D_t \partial_s \Gamma$

where $D_t$ denotes the covariant derivative along the main curve, and $D_s$ denotes the covariant derivative along the transverse curve.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics and Minimizing Curves