Definition:Uniformly Normal Subset

Definition

Let $\struct {M, g}$ be a Riemannian manifold.

Let $W \subseteq M$ be a subset.

Let $U_p = \map {\exp_p} {\map {B_\delta^-} 0 }$ be a closed geodesic ball around $p \in M$.

Suppose there exists $\delta \in \R_{> 0}$ such that:

$\forall p \in W : W \subseteq U_p$

Then $W$ is called uniformly normal (or uniformly $\delta$-normal).

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Uniformly Normal Neighborhoods