Definition:Vector Field on Smooth Manifold

Definition

Let $M$ be a smooth manifold.

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

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

Then by the vector field on $M$ we mean the continuous map $X : M \to TM$ such that:

$\forall p \in M : X_p \in T_p M$

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): $\S 8$: Vector Fields. Vector Fields on Manifolds