Riemannian Geodesics and Completeness

Riemannian Geodesics

Geodesic

A geodesic is a curve which locally minimizes distance and whose velocity is constant, which means $\left|\frac{d\gamma }{dt}\right|:=\sqrt{g_{\gamma (t)}\left(\frac{d\gamma }{dt},\frac{d\gamma }{dt}\right)}$ is independent of time $t$.

Geodesically Convex

A smooth curve joining $x$ to $y$ is a minimizing geodesic if its arc-length is the riemannian distance $d(x,y)$. A riemannian manifold $(X,g)$ is geodesically convex if every point $x$ is joined to every other point $y$ by a unique (up to reparametrization) minimizing geodesic.