Definition:Tangential Projection

Definition

Let $\tilde M$ be a Riemannian manifold.

Let $M \subseteq \tilde M$ be an immersed or embedded Riemannian submanifold.

Let $TM$ be the tangent bundle of $M$.

Let $\bigvalueat {T \tilde M} M$ be the tangent bundle of $\tilde M$, but restricted to $M$.

Let $\pi^\top$ be a smooth bundle homomorphism such that:

$\pi^\top : \bigvalueat {T \tilde M} M \to TM$

Then $\pi^\top$ is known as the tangential projection.

Sources

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