Definition:Bounded Subset of Connected Riemannian Manifold

Definition

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

Let $A \subseteq M$ be a subset.

Let $p,q \in M$ be points.

Let $d_g$ be the Riemannian distance.

Let $C \in \R$ be a constant.

Suppose:

$\exists C \in \R : \forall p, q \in A : \map {d_g} {p, q} \le C$

Then $A$ is said to be bounded.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances