Let $M$ be a manifold and $\gamma$ a curve in $M$. Geodesic curves are curves, for which $\ddot y(t)$ is orthogonal to $T_{y(t)}$
Proof that $\vert\dot \gamma \vert =\text{constant}$
Would appreciate any help.
Geodesics on a manifold
-4
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differential-geometry
manifolds
geodesic
1 Answers
4
Hint: given a Riemannian metric $\langle \cdot, \cdot \rangle$ on a manifold $M$, and letting $\frac{D}{dt}$ be the covariant derivative (i.e. parallel transport) with respect to a curve $\gamma: (-\epsilon, \epsilon) \to M$, then we have that
$$ \frac{d}{dt} \langle V, W \rangle \;\; =\;\; \left \langle \frac{DV}{dt}, W\right \rangle + \left \langle V, \frac{DW}{dt} \right \rangle. $$
If $\gamma$ is a geodesic, what do you know about $\frac{D\dot{\gamma}}{dt}$?