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…)