Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...
For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...
Let $M$ be a complete Riemannian manifold. For a fixed point $p$ in $M$, the Riemannian distance to $p$ is denoted by $d_p$. Fix a strongly convex geodesic ball $B(o,R)$ in $M$ and some disjoint ...