# Thread: Integrating over a Manifold

1. ## Integrating over a Manifold

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

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

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 $\displaystyle \mathbb{R}^n$ 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 $\displaystyle \mathbb{R}^n$?

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

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

2. 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.

3. 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?

4. 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.

5. Bump...