Integrating over a Manifold

**JG89** · October 17th 2010, 06:42 AM

First let me write out the definition of a manifold given in my book:

Let $k > 0$ . A k-manifold in $\mathbb{R}^n$ of class $C^r$ is a subspace $M$ of $\mathbb{R}^n$ having the following property: For each $p \in M$ , there is an open set $V \subset M$ containing $p$ , a set $U$ that is open in either $\mathbb{R}^k$ or $\mathbb{H}^k$ (upper half space), and a continuous bijection $\alpha : U \rightarrow V$ such that 1) $\alpha$ is of class $C^r$ , 2) $\alpha^{-1} : V \rightarrow U$ is continuous, 3) $D\alpha(x)$ has rank k for each $x \in U$ . The map $\alpha$ is called a coordinate patch on $M$ about $p$ .

In my textbook I am reading the chapter on integrating a scalar map over a compact manifold. My question is this: Suppose M is a compact-manifold. As a subset of $\mathbb{R}^n$ it is bounded. So instead of going through all the mess of defining a manifold and defining the integral of a continuous function f over a manifold, why not just integrate f over M as one usually would? Using Riemann sums in $\mathbb{R}^n$ ?

Surely this would give the same result as integrating f over M using the definition of integral over a manifold. So what's so special about using a manifold M for integration when we could just consider M as a regular bounded subset of Euclidean space and integrate it how we usually would?

In case you're wondering, here is the definition of the integral of a scalar map over a compact manifold M:

Let M be a compact k-manifold in $\mathbb{R}^n$ , of class $C^r$ . Let $f: M \rightarrow \mathbb{R}$ be a continuous function. Suppose that $\alpha_i: A_i \rightarrow M_i$ , for i = 1, ..., N, is a coordinate patch on M, such that $A_i$ is open in $\mathbb{R}^k$ and M is the disjoint union of the open sets $M_1, M_2, ..., M_N$ of M and a set K of measure zero in M. Then $\int_M f dV = \sum_{i = 1}^N \int_{A_i} f(\alpha_i) V(D \alpha_i)$ .

Note that $dV$ represents the integral with respect to volume and $V(D \alpha_i) = \sqrt{det[(D\alpha_i)^{tr} D\alpha_i]}$

**xxp9** · October 17th 2010, 05:39 PM

Manifolds can be defined abstractly without being embedded into an ambient space. Also integral on a manifold can be defined without a metric, while a differential structure is enough. Read more on this topic you'll understand.

**JG89** · October 17th 2010, 06:38 PM

I forgot to mention that I am only talking about manifolds that are subspaces of Euclidean space. I haven't learned about abstract manifolds yet.

Does the answer that I want require you to speak about abstract manifolds? Or can you properly answer my question while only referring to Euclidean space?

**JG89** · October 17th 2010, 07:23 PM

Also, what are useful (in physics, computer science, or even other parts of math) applications of abstract manifolds? I can see how sub-manifolds of Euclidean space would be useful, though at the moment they seem "redundant" to me.

**JG89** · October 18th 2010, 05:30 PM

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