Differential Geometry – Advanced Topics

{$ \def\pp{\mathbf{p}} \def\uu{\mathbf{u}} \def\vv{\mathbf{v}} \def\FF{\mathbf{F}} \def\GG{\mathbf{G}} \def\ww{\mathbf{w}} \def\RR{\mathbb{R}} \def\st{\; \mid \;} \def\cross{\times} \def\dot{\cdot} \def\grad{\nabla} \def\curl{\grad\cross} \def\div{\grad\dot} \def\tgrad{\text{grad}\;} \def\tcurl{\text{curl}\;} \def\tdiv{\text{div}\;} \def\pdiff#1#2{\frac{\partial #1}{\partial #2}} \def\vvec#1{\left<#1\right>} \def\hstar{{\star}} \newcommand{\la}[1]{\xleftarrow{\ \ #1\ \ }} \newcommand{\ra}[1]{\xrightarrow{\ \ #1\ \ }} \newcommand{\las}[1]{\xleftarrow{\ \ \smash{#1}\ \ }} \newcommand{\ras}[1]{\xrightarrow{\ \ \smash{#1}\ \ }} \newcommand{\da}[1]{\bigg\downarrow\rlap{\scriptstyle#1}} \newcommand{\uda}[1]{\bigg\updownarrow\rlap{\scriptstyle#1}} $}

Big Picture

This section is meant to be read after finishing the exterior derivatives section here.

The de Rham Cohomology and the commutative diagram below are advanced topics in differential geometry, but they offer a nice clean picture of how everything connects together and how to generalize even the Laplace operator to arbitrary dimensions. This is just for intuition, and a few things are specialized for {$\RR^3$} or hidden under the rug to avoid distracting complexity.

In {$\RR^3$}, 0-forms are dual to 3-forms (both scalar-like), and 1-forms are dual to 2-forms (both vector-like). So we can have a dual that takes one to the other defined as follows.

With this dual, we have the following big-picture commutative diagram for {$\RR^3$}

{$ \qquad \begin{array}{c} 0 & \ra{d} & 0\text{-form} & \ra{d} & 1\text{-form} & \ra{d} & 2\text{-form} & \ra{d} & 3\text{-form} & \ra{d} & 0\\& & \uda{\hstar} & & \uda{\hstar} & & \uda{\hstar} & & \uda{\hstar}& &\\0 & \las{d} & 3\text{-form}& \las{d} & 2\text{-form} & \las{d} & 1\text{-form} & \las{d} & 0\text{-form} & \las{d} & 0 \end{array} $}