Geodesic in Normal Neighborhood

Theorem

Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $U_p$ be the normal neighborhood for $p \in M$.

Let $\struct {U_p, \tuple {x^i}}$ be a normal coordinate chart.

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

Let $v = v^i \valueat{\dfrac \partial {\partial x^i}} p \in T_p M$.

Let $I \subseteq \R$ be a real interval.

Let $\map {\gamma_v} t : I \to M$ be the geodesic such that:

$\map {\gamma_v} 0 = p$

$\map {\gamma_v'} 0 = v$

where $\gamma_v'$ denotes the velocity of $\gamma_v$.

Suppose:

$t \in I : 0 \in I : \map {\gamma_v} I \subseteq U_p$

Then in normal coordinates:

$\map {\gamma_v} t = \tuple {t v^1, \ldots, t v^n}$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Normal Neighborhoods and Normal Coordinates