Length of Velocity of Smooth Curve is Constant iff Covariant Derivative of Velocity along Curve is Orthogonal to Velocity

Theorem
Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.

Let $\nabla$ be a metric connection on $M$.

Let $I \subseteq \R$ be an open real interval.

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

Let $\gamma'$ be the velocity of $\gamma$.

Suppose $D_t$ is the covariant derivative along $\gamma$.

Let $\size {\, \cdot \,}$ be the Riemannian or pseudo-Riemannian inner product norm.

Then $\size {\map {\gamma'} t}$ is constant $D_t \map {\gamma'} t$ is orthogonal to $\map {\gamma'} t$ for all $t \in I$.