- Published 2010

Given a complete embedded submanifold N ⊂ M, we can parametrize M locally near N by some neighborhood of N in N × Rn−k, via the normal exponential map. This requires identifying N × Rn−k with the normal tangent bundle T⊥N , which can be done locally. Namely fixing a point p ∈ N , one identifies T⊥ q N with T⊥ p N by parallel transport in the normal bundle, whenever there is a unique geodesic from p to q. Putting coordinates {x, . . . , x} on N near p and coordinates {x, . . . , x} on Rn−k, we can express the metric tensor gij and the volume form dV ol = √ det gijdx ∧ · · · ∧ dx in components.

@inproceedings{2010Lecture4,
title={Lecture 4 - Convergence Theorems February 9 , 2010 1 Volume comparison and the Heintze - Karcher theo - rem},
author={},
year={2010}
}