Jonginn Yun

Differential Forms

A differential $k$-form is a section of

$$\Lambda^kT^*M.$$

It eats $k$ tangent vectors and returns a number, alternating sign when two arguments are exchanged. The exterior derivative

$$d:\Omega^k(M)\to \Omega^{k+1}(M)$$

satisfies $d^2=0$ and generalizes gradient, curl, and divergence in a coordinate-independent way.

Pullbacks

If $F:M\to N$ is smooth, then forms on $N$ can be pulled back to forms on $M$:

$$F^*:\Omega^k(N)\to \Omega^k(M).$$

Pullback is the natural direction because forms evaluate on tangent vectors, and $F$ pushes tangent vectors forward. This is also the correct language for change of variables and coordinate invariance.

Flows

A vector field $X$ generates a local flow $\Phi_t$, a family of diffeomorphisms satisfying

$$\frac{d}{dt}\Phi_t(p)=X_{\Phi_t(p)},\qquad \Phi_0(p)=p.$$

The flow tells us how geometric objects move when points are transported along the vector field.

Lie Derivative

The Lie derivative measures the infinitesimal change of a tensor field along a vector field. For a function,

$$\mathcal L_X f=X(f).$$

For a differential form $\omega$, Cartan’s formula gives

$$\mathcal L_X\omega=i_Xd\omega+d(i_X\omega),$$

where $i_X$ is contraction with the vector field.

The Lie derivative should not be confused with a connection. It differentiates along a flow that moves points of the manifold. A connection differentiates sections by comparing nearby fibers.