I’ve set a tentative goal of heading towards the solution of the Dirichlet problem on compact manifoldw-with-boundary and Hodge theory; these will require various preliminaries, since of course it will be more fun to do it this way than to restrict to open sets in $\mathbb{R}^n$.  Today I will go through some of the basics of how the well-known operators from multivariable calculus work more generally on a Riemannian manifold, which will be necessary in the sequel.   (I shamelessly took the title of the post from a book I haven’t read.)

Recall the well-known operator on functions of ${n}$-variables, the Laplacian

$\displaystyle \Delta f := \sum_i \frac{ \partial^2f}{\partial x_i^{2}}.$

The problem is, ${\Delta}$ doesn’t transform nicely with respect to changes in coordinates, and since we want to define the Laplacian on manifolds, this causes a problem. However, it can be done on a Riemannian manifold. The idea is to use the formula

$\displaystyle \Delta = \mathrm{div} (\mathrm{grad} ),$

which is immediate from the definitions on ${\mathbb{R}^n}$. The key point is that ${\mathrm{div} }$ and ${\mathrm{grad}}$ make sense on any Riemannian manifold—from, respectively, vector fields to functions and from functions to vector fields. (more…)

The purpose of this post is to discuss a few basic facts about differentiable manifolds and state the Darboux theorem, which I will prove next time.  (People who are looking for a more ambitious leap into symplectic geometry might want to try lewallen’s two posts over at Concrete Nonsense.)

A symplectic manifold is a smooth manifold ${M}$ equipped with a closed symplectic 2-form ${\omega}$. In other words, ${\omega}$ is alternating and nondegenerate on each tangent space ${T_p(M)}$

The basic example of a symplectic form is

$\displaystyle \sum_i dx^i \wedge d\xi^i$

on ${\mathbb{R}^{2n}}$ with coordinates ${x^i, \xi^i, 1 \leq i \leq n}$

This can also be written in a more invariant form, which will also give an invariant manner of making the cotangent bundle ${T^*M}$ of any manifold ${M}$ into a symplectic manifold. First, we define a 1-form ${\alpha}$ on ${T^*M}$. Let ${p: T^*M \rightarrow M}$ be the projection downwards. Given ${v \in T^*M}$ lying above ${\xi \in T^*M}$, define

$\displaystyle \alpha(v) = \xi( p_*(v)).$

To make this clearer, here is an interpretation in local coordinates. Let ${x^1, \dots, x^n}$ be local coordinates for ${M}$. Then ${x^1, \dots, x^n, \xi^1, \dots, \xi^n}$ coordinates for ${T^*M}$. Then

$\displaystyle \alpha = \sum \xi^i dx^i$

as is easily checked by working through the definitions. So we can define a canonical 2-form ${\omega}$ as ${\omega = - d \alpha}$; this makes ${T^*M}$ into a symplectic manifold. (more…)