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In every book that I have read (Bott and Tu, Absil), Manifolds are defined intrinsically in terms of charts and maximal atlases, after which we show how to "construct" manifolds from level sets of functions by invoking theorems such as constant rank. Where can I find a reference which defines what a manifold is, extrinsically, as a subset of $\mathbb R^n$ that obeys certain properties?

My guess would be that a manifold is a subset $S$ of $\mathbb R^n$ such that for each point $x \in S$, there is a local neighbourhood $x \in U \subseteq S$ such that the embedding map $f: U \hookrightarrow \mathbb R^n$ has a full-rank jacobian. This ought to be enough since we can then import the dot product from the ambient space. Is this correct? If not, what's missing?

Siddharth Bhat
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  • Well - strictly speaking a manifold need not be a subset of $\mathbb{R}^n$ ..., unless I am not understanding your question. – rubikscube09 Apr 23 '21 at 22:36
  • Every Riemannian manifold can be realised as a subset of $\mathbb R^n$, via the Nash embedding? – Siddharth Bhat Apr 23 '21 at 22:38
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    Oh - I see what you're asking now. I think that what you have is in fact correct and was the definition of smooth manifold I learned in an advanced calculus (read - baby manifolds) course when we defined manifolds as subsets of $\mathbb{R}^n$. This was a while ago however so don't quote me. – rubikscube09 Apr 23 '21 at 22:47
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    Differential Topology by Guillemin and Pollack? (Doesn't do much with metrics, but does define manifolds as you describe.) – Andrew D. Hwang Apr 23 '21 at 23:05
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    @SiddharthBhat I see that you are studying General Relativity from MTW' Gravitation (also my favourite), There, MTW justly enphasizes that the intrinsic definition is the best one in the case of spacetime, since we have no knowledge of anything where the universe is immersed in. Also: although we can mathematically define e riemannian metric (ie. positive definite) on any manifold (including our spacetime), this does not mean that the definition makes sense or is physically useful. In newtonian spacetime, for example, no 4D metric (Riemannian or otherwise) makes sense (see MTW chap 12) . – magma Apr 25 '21 at 11:41

1 Answers1

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There are 4 equivalent ways of saying it (the equivalence is given by applying the inverse/implicit function theorems appropriately).

Let $1\leq k \leq n$ be integers, and $r\in \Bbb{N}\cup\{\infty\}$ and $M\subset \Bbb{R}^n$. We say $M$ is a $k$-dimensional (embedded-sub) manifold in $\Bbb{R}^n$ of class $C^r$, if any of the four equivalent conditions is satisfied:

  1. For any point $p\in M$ there is an open $U\subset\Bbb{R}^n$ containing $p$, an open $A\subset \Bbb{R}^k$, a "coordinate permutation" $\sigma:\Bbb{R}^n\to\Bbb{R}^n$ and a $C^r$ mapping $g:A\to \Bbb{R}^{n-k}$ such that $\sigma[M\cap U]=\text{graph}(g)$.

  2. For any point $p\in M$, there is an open $U\subset \Bbb{R}^n$ containing $p$ and a $C^r$ mapping $f:U\to \Bbb{R}^{n-k}$ such that $f(p)=0$ is a regular value for $f$ and $M\cap U=f^{-1}(\{0\})$.

  3. For any point $p\in M$, there is an open $U\subset \Bbb{R}^n$ containing $p$ an open $V\subset\Bbb{R}^n$ and a $C^r$ diffeomorphism $\Phi:U\to V$ such that $\Phi[M\cap U]=V\cap \left(\Bbb{R}^{k}\times \{0_{\Bbb{R}^{n-k}}\}\right)$.

  4. For any point $p\in M$, there is an open $U\subset\Bbb{R}^n$ containing $p$, an open $W\subset\Bbb{R}^k$ and a $C^r$ map $\alpha:W\to \Bbb{R}^n$ such that $\alpha[W]=M\cap U$ and $\alpha$ is an injective immersion which is a homeomorphism onto its image (when it is given the subset topology).

I (and I guess others) call them the "graph definition", "level-set definition", "slice definition" and "parametric definition" respectively. Note of course, that within each definitions, the $U$'s are of course not necessarily the same. In proving the equivalence of the statements, you'll of course have to shrink neighborhoods etc.

In words, here's what they're saying intuitively:

  • After rearranging the coordinates, we can locally express $M$ as the graph of a $C^r$ function.
  • $M$ is locally the level set of a "nice" function.
  • Locally, we can "flatten out" $M$ to make it look like a piece of $\Bbb{R}^k$.
  • $M$ can be locally parametrized by a "nice" function.

In every case, we can take the inclusion map $\iota:M\to \Bbb{R}^n$, and using the usual Riemannian metric on $\Bbb{R}^n$ \begin{align} g=\sum_{i=1}^ndx^i\otimes dx^i, \end{align} we can pull this back $\iota^*g$ to get a Riemannian metric on $M$.

If you can get access to it, I would suggest taking a look at Duistermaat and Kolk's book, particularly the chapter on manifolds (it's about 20 pages, and it contains these four definitions and a bunch of examples, and at the end even talks about Morse's lemma). There are ALOT of exercises in this book so I would highly recommend you try some. Also, @Ted Shifrin has a few lectures on this matter (he mentions definitions 1, 2 and 4 if I remember correctly) and then gives examples in the next lecture (of course he also has his own textbook where this is all proven).

peek-a-boo
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