Exponential map on the n-sphere

Question

I would like to calculate the exponential map in the n-sphere, however, i don't know how to get started. Someone could give me a tip or bibliographic reference?

Answer 1

$\newcommand{\Vec}[1]{\mathbf{#1}}$If $\Vec{v}$ is a non-zero tangent vector to the $n$-sphere at a point $\Vec{p}$, the geodesic starting at $\Vec{p}$ with initial velocity $\Vec{v}$ is a circle of speed $\|\Vec{v}\|$ lying in the plane spanned by $\Vec{p}$ and $\Vec{v}$: $$ \gamma(t) = \cos(\|\Vec{v}\|t) \Vec{p} + \sin(\|\Vec{v}\|t) \frac{\Vec{v}}{\|\Vec{v}\|}. $$