Gauss Lemma

The crux of the proof that geodesics are locally minimizing is the following deceptively simple geometric lemma.

Theorem 6.9 (The Gauss Lemma). Let $(M, g)$ be a Riemannian manifold, let $U$ be a geodesic ball centered at $p \in M$, and let $\partial_{r}$ denote the radial vector field on $U\setminus \{p\}$. Then $\partial_{r}$ is a unit vector field orthogonal to the geodesic spheres in $U\setminus \{p\}$.

For the proof, see Gauss Lemma.

References

1. Lee, John M. Introduction to Riemannian Manifolds. Vol. 176. Graduate Texts in Mathematics. Cham: Springer International Publishing, 2018. https://doi.org/10.1007/978-3-319-91755-9.

---

Backlinks

• Gauss Lemma (Dendron)