Can anyone recommend me some nice references about lengh minimizing properties of geodesics?
I'm looking for a treatment in the case of surfaces, but more general viewpoints will also be welcome.
More precisely I'd like to find a clear proof of the following fact:
Theorem: For each point $q$ in a normal neighbourhood of a given point $q$ the radial segment from $p$ to $q$ uniquely minimizes arc length.
Thanks
M. do Carmo "Riemannian Geometry", Chapter 3.