Learn Submanifold Definition in Differential Geometry

[orion]

I am trying to learn differential geometry on my own, and I'm finally getting serious about learning the subject. I am using several books which I supplement with material I find on the internet. I have found some excellent lecture notes on differential geometry written by Dmitri Zaisev. I like his discussion of tangent vectors from several viewpoints. My question concerns a definition that I have found in those notes.

The definition of a submanifold is given as:

A subset S ⊂ ℝⁿ is an m-dimensional submanifold of ℝⁿ (m < n) of class C^k if ∀ p in S ∃ an open neighborhood V_p ⊂ ℝⁿ, an open set W_p ⊂ ℝ^m, and a homeomorphism f_p: W_p → V_p ∩ S which is C^k and regular in the sense that ∀ a ∈ W_p, the differential d_af_p: ℝ^m → ℝⁿ is injective.

I need help pulling this definition apart. In particular, I am perplexed as to the part the set W_p plays. If we need an open set in S, why isn't V_p ∩ S an open set in S? Why do we need f_p to map from W_p to V_p ∩ S because it seems to me that they are both sets contained in S.

 

[Lebesgue]

Hi! I'm also interested in learning differential geometry on my own, but I think I'll wait till the end of finals to get serious about it. As far as I know, $V_p \cap S$ is an open set in S with the restricted topology inherited from $\mathbb{R}^n$. I do not find clear the second question, but $W_p$ is not contained in S. S lives in $\mathbb{R}^n$ and $W_p$ lives in $\mathbb{R}^m$.

One thing I am curious about: why not just definig the map to be a $C^k$ regular diffeomorphism from $W_p$ to $S\cap V_p$? The condition over the differential of $f_p$ to be injective implies that the inverse of the map is also $C^k$ regular, so it is a $C^k$-diffeomorphism (thanks to the inverse function theorem).

 

[orion]

I do not find clear the second question, but $W_p$ is not contained in S. S lives in $\mathbb{R}^n$ and $W_p$ lives in $\mathbb{R}^m$.

Thanks for your reply. Good luck studying DG on your own.

I see your point, but that confuses me more. What then is the function of W_p in this definition?

 

[orion]

If the point is to establish that S ⊂ ℝⁿ is homeomorphic to ℝ^m, then why does f map from W_p to V_p ∩ S when usually homeomorphisms φ are defined to map from the manifold to an open set in ℝ^l (for some l as the case may be)? I realize that f^-1 is a homeomorphism from V_p ∩ S to W_p, but I'm interested why it is defined in the way that it is.

 

[Lebesgue]

That's the big deal. S is NOT homeomorphic to $\mathbb{R}^m$. The cool thing is that is LOCALLY homeomorphic to $\mathbb{R}^m$, i.e. when you closely enough to a point P in S, you can find that a neighbourhood of P is homeomorphic to an open (yet unknown explicitly) subset of $\mathbb{R}^m$.
Let's look at an example: the sphere in $\mathbb{R}^3$ is a 2 dimensional submanifold of the big manifold $\mathbb{R}^3$ itself. When you look closely enough at a point P in the surface of the sphere, and consider an open neighbourghood of P contained in S, let's call it $V_P = B \cap S$ where B is a three dimensional ball centered at P, you can find that this set $V_P$ is homeomorphic to an open subset $W_P \subset \mathbb{R}^2$. $V_P$ itself actually looks like a 2-dimensional patch-disc but a little bit bent, centered at P and over the surface of the sphere!

The cool thing about manifolds is that they are not globally topologically similar to some $\mathbb{R}^k$ but just locally. There's even a generalization of the definition of manifold which allows a manifold to vary dimensionality depending on the point chosen to look closely at.

I hope this helps you clarify some issues. I hope someone with more knowledge on DG helps out here, since I'm no more than a begginer.

 

[micromass]

One thing I am curious about: why not just definig the map to be a $C^k$ regular diffeomorphism from $W_p$ to $S\cap V_p$? The condition over the differential of $f_p$ to be injective implies that the inverse of the map is also $C^k$ regular, so it is a $C^k$-diffeomorphism (thanks to the inverse function theorem).

The image of $f_p$ is part of $S$. The classical inverse function theorem only applies between two open sets of $\mathbb{R}^n$. Furthermore, smoothness for a morphism $S\rightarrow \mathbb{R}^n$ has not been defined, classically, you only have smoothness for maps between open sets of $\mathbb{R}^n$. This is why your suggestion does not work.

That said, it is possible to extend the notion of smoothness, to extend the inverse function theorem, etc. That is the entire point of differential geometry. Then it is correct that the $f_p$ are diffeomorphisms. But at this stage it is not yet possible to do this. You first need the notion of a submanifold, only then can you extend the classical notions.

 

[micromass]

I need help pulling this definition apart. In particular, I am perplexed as to the part the set W_p plays. If we need an open set in S, why isn't V_p ∩ S an open set in S? Why do we need f_p to map from W_p to V_p ∩ S because it seems to me that they are both sets contained in S.

As already been said, $W_p$ is not a subset of $S$.

You probably are having some misconceptions. To fix these, I would like to ask you to try to formulate the definition without using the sets $W_p$. This will help you see the necessity of $W_p$.

Also:

why isn't V_p ∩ S an open set in S?

$V_p\cap S$ is an open set in $S$ (and all open sets of $S$ have this form. My guess is that the author wanted to avoid using the subspace topology.

 

[orion]

That's the big deal. S is NOT homeomorphic to $\mathbb{R}^m$. The cool thing is that is LOCALLY homeomorphic to $\mathbb{R}^m$, i.e. when you closely enough to a point P in S, you can find that a neighbourhood of P is homeomorphic to an open (yet unknown explicitly) subset of $\mathbb{R}^m$.

I'm sorry, I misspoke. I meant locally isomorphic to ℝ^m. My question is still why is the homeomorphism f_p map W_p to V_p ∩ S?

 

[orion]

As already been said, $W_p$ is not a subset of $S$.
You probably are having some misconceptions. To fix these, I would like to ask you to try to formulate the definition without using the sets $W_p$. This will help you see the necessity of $W_p$.

Thank you for your reply.

I think I'm finally clear on the necessity of W_p but I still am wondering why f_p is defined as going from W_p to V_p ∩ S.

 

[micromass]

Thank you for your reply.

I think I'm finally clear on the necessity of W_p but I still am wondering why f_p is defined as going from W_p to V_p ∩ S.

Again, what else do you propose? Does your intuition understand what a manifold is (or should be)?

 

[micromass]

I'm sorry, I misspoke. I meant locally isomorphic to ℝ^m.

Do you mean locally homeomorphic?

 

[WWGD]

I am not sure if this is what you are looking for, but the definition of submanifold I know of is that $S \subset M$ is a k-dimensional submanifold of $M^m$ ; $k<m$ iff (def.) $S$ is topologically embedded in $M$ in the same way that $\mathbb R^k$ is "standardly" embedded in $\mathbb R^m$, i.e., as $(x_1, x_2,...,x_k, 0,0,...,0)$. I think this is what the author may be "modeling".

 

[orion]

Again, what else do you propose? Does your intuition understand what a manifold is (or should be)?

My intuition says that f should be defined as f_p: V_p ∩ S → W_p in much the same way as when we have a manifold M with a chart U,φ we say φ: U_p → ℝⁿ. In other words, the homeomorphism maps an open set in the manifold to an open set in ℝⁿ. I realize that f is a homeomorphism so I guess it doesn't really matter since I can view f^-1 like this, but I am still wondering why he did it that way (and he's not the only author I've seen have this definition).

And, yes, I meant homeomorphism earlier when I wrote isomorphism.

 

[orion]

I am not sure if this is what you are looking for, but the definition of submanifold I know of is that $S \subset M$ is a k-dimensional submanifold of $M^m$ ; $k<m$ iff (def.) $S$ is topologically embedded in $M$ in the same way that $\mathbb R^k$ is "standardly" embedded in $\mathbb R^m$, i.e., as $(x_1, x_2,...,x_k, 0,0,...,0)$. I think this is what the author may be "modeling".

Thanks. The author presents submanifolds before he gets into embeddings so you might be right and that's the reason for defining a submanifold this way.

 

[micromass]

My intuition says that f should be defined as f_p: V_p ∩ S → W_p in much the same way as when we have a manifold M with a chart U,φ we say φ: U_p → ℝⁿ. In other words, the homeomorphism maps an open set in the manifold to an open set in ℝⁿ. I realize that f is a homeomorphism so I guess it doesn't really matter since I can view f^-1 like this, but I am still wondering why he did it that way (and he's not the only author I've seen have this definition).

They are equivalent situations as you noticed, so it doesn't matter. Often, when things are done classically (so no abstract manifolds, but everything is a subset of $\mathbb{R}^n$), then the maps are $\mathbb{R}^n\rightarrow S$ (or between open subsets of those). This is to illustrate that a submanifold of $\mathbb{R}^n$ is locally the surface of a graph. When you look at more modern approaches (ie abstract manifolds), then for some reason the situation reverses and the maps are $S\rightarrow \mathbb{R}^n$. Which one you prefer is a matter of taste, it is not really important.

 

[orion]

They are equivalent situations as you noticed, so it doesn't matter. Often, when things are done classically (so no abstract manifolds, but everything is a subset of $\mathbb{R}^n$), then the maps are $\mathbb{R}^n\rightarrow S$ (or between open subsets of those). This is to illustrate that a submanifold of $\mathbb{R}^n$ is locally the surface of a graph. When you look at more modern approaches (ie abstract manifolds), then for some reason the situation reverses and the maps are $S\rightarrow \mathbb{R}^n$. Which one you prefer is a matter of taste, it is not really important.

Thank you. That makes a lot of sense.

 

[orion]

I just want to say thanks again to all who replied. I'm pretty much flying solo here and it definitely helps to have someone to talk this over with.

 

[lavinia]

While the definition in your book is correct, I think it is somewhat confusing.

First of all, in differential geometry today, a manifold is almost always assumed to be smooth. This means that f has continuous partial derivatives of all orders. Your book assumes that f is C^k which then brings up the question of the differences between manifolds of different k's. There are theorems about when a C^k manifold is C^j for j greater than k but this is almost always a side subject that no one worries about.

Second, your book says that f is a homeomorphism whose differential is non-singular at every point, that is, df(x) = 0 only if the vector x is zero. With work one can show that f has a differentiable inverse - using the Inverse Function Theorem - but this seems indirect and too complicated . One could just say that f is a diffeomorphism from the start.

An m dimensional submanifold of R^n is a subset of R^n that is locally diffeomorphic to R^m.

Expanding this definition out in detail just says that around each point in S there is an open neighborhood ( in the subspace topology) that is diffeomorphic to an open subset in R^m. By the definition of subspace topology, this open set must be the intersection of S with an open set V in R^n. That is why your definition uses Vp ∩ S.

Finally, one must define what it means for f^-1 to be smooth. This is because f^-1 is not defined on an open subset of R^n. There is more than one way to do this, but perhaps the simplest is just to says that f^-1 can be extended to a smooth map from an open set in R^n into R^m. This definition is also general and works for arbitrary subsets of R^n even very weird ones.

If you do not already have Milnor's Topology from the Differentiable Viewpoint, get it. It is simple and clear and fascinating and will get you past elementary calculus on manifolds.