Definition:Geodesic Equation

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Let $M$ be an $n$-dimensional smooth manifold with or without boundary.

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

Let $\gamma : I \to M$ be a smooth curve.

Let $\tuple {x^i}$ be smooth coordintes.

Suppose the component functions of $\gamma$ are written as:

$\map \gamma t = \tuple {\map {x^1} t, \ldots, \map {x^n} t}$

Let $\nabla$ be the connection on $M$.

Let $\set {\Gamma^k_{ij}}$ be connection coefficients of $\nabla$.

Then the following system of second-order ordinary differential equations is called the geodesic equation:

$\dfrac {\d^2 \map {x^k} t} {\d t^2} + \dfrac {\d \map {x^i} t}{\d t} \dfrac {\d \map {x^j} t}{\d t} \map {\Gamma^k_{ij}} {\map \gamma t} = 0$

where $k \in \N_{> 0} : k \le n$.