Definition:Metric Induced by Smooth Immersion

Definition

Let $\struct {\tilde M, \tilde g}$, $\struct {M, g}$ be Riemannian manifolds.

Let $F : M \to \tilde M$ be a smooth immersion.

Suppose, the metric $g$ is such that $g = F^* \tilde g$.

Then $g$ is called the metric induced by $F$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics