Manifolds Notes

Describe a vector field on a manifold?

A vector field on a manifold $M$ assigns a tangent vector to each point of $M$ smoothly.

Formally, it is a smooth map $$ X: M \to TM $$ such that $\pi \circ X = \mathrm{id}_M$, where $\pi: TM \to M$ is the tangent-bundle projection. So $X(p) \in T_pM$ for every $p \in M$.

Equivalent viewpoint: a vector field is a derivation on smooth functions, $$ X: C^\infty(M) \to C^\infty(M), \quad X(fg)=X(f)g+fX(g). $$

In local coordinates $(x^1,\dots,x^n)$, any vector field looks like $$ X = \sum_{i=1}^n X^i(x),\frac{\partial}{\partial x^i}, $$ where the (X^i) are smooth functions.

Intuition:

$M$: the space (possibly curved) $T_pM$: allowed infinitesimal directions at $p$ $X$: picks one direction/speed at each point, like a flow of “arrows” tangent to $M$ Example on $\mathbb{R}^2$: $$ X(x,y)= -y,\frac{\partial}{\partial x} + x,\frac{\partial}{\partial y}, $$ which generates rotation around the origin.