$\| \gamma'(t) \|$ = constant for all $t$, if and only if $\gamma''(t)$ is normal to the tangent vector space for all $t$.

Question

Let $\gamma = \sigma \circ \mu$ be a regular parametrized curve on a surface $\sigma$. Prove that $\gamma$ is a geodesic and $\| \gamma'(t) \|$ = constant for all $t$, if and only if $\gamma''(t)$ is normal to the tangent vector space for all $t$.

Require Hints for the problem.

Answer 1

$||\gamma'(t)||^2 = \langle \gamma' (t), \gamma'(t)\rangle$

Differentiate this and use what you know about inner products and differentiation.