Definition:Normal Neighborhood

Definition
Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold without boundary.

Let $T_p M$ be the tangent space of $M$ at $p \in M$.

Let $\EE$ be the domain of the exponential map.

Let $\EE_p = \EE \cap T_p M$.

Let $\exp_p : \EE_p \to M$ be the restricted exponential map.

Let $U_p \subseteq M$ be a neighborhood of $p \in M$.

Suppose $U_p$ is the diffeomorphic image of $\exp_p$ of a star-shaped neighborhood of $0 \in T_p M$.

Then $U_p$ is called a normal neighborhood of $p$.