Definition Checking. Manifold and Diffeos.

[caffeinemachine]

Hello MHB,
I have been reading some literature on manifolds, I recently changed the text book I was reading from and because of that things have gotten pretty messy. I have tried my best to extract the definitions from the text I am reading. Please tell me which ones I have got right and which ones I have not.
Thanks.

Definition 1.
Let $f:\mathbb R^n\to \mathbb R^m$ be a function, $x$ be a point in $\mathbb R^n$ and $r>0$ be an integer. Then $f$ is said to be of class $C^r$ at $x$ if all the partial derivatives of $f$ of order $r$ exist and are continuous at $x$. If $f$ if of class $C^r$ at each point in a subset $A\subseteq \mathbb R^n$ then we say that $f$ is of class $C^r$ in $A$. A continuous function is also said to be a $C^0$-function.

Definition 2.
Let $U$ and $V$ be open sets in $\mathbb R^n$. A function $h:U\to V$ is said to be a diffeomorphism, or a $C^0$-diffeomorphism, if $h$ is differentiable and has a differentiable inverse. For $r>0$, a diffeomorphism $h:U\to V$ is said to be of class $C^r$ if both $h$ and $h^{-1}$ are of class $C^r$.

Definition 3.
A subset $M$ of $\mathbb R^n$ is said to be a $k$-dimensional manifold of class $C^r$ in $\mathbb R^n$ if for all $x\in M$ there exist open sets $U$ and $V$ in $\mathbb R^n$ and a diffeomorphism $h:U\to V$ of class $C^r$ such that $x\in U$ and $$h(U\cap M)=V\cap (\mathbb R^k\times\{0\})$$

 

[mathbalarka]

I have skimmed through the definitions; everything seems alright to me. I haven't ever formally tried to understand the definitions so certainly there could be something small missing in these which I haven't noticed.**

** As I happen to be a Number Theorist, I never tried that much hard to grasp topology to research level.

Balarka
.

 

[caffeinemachine]

I have skimmed through the definitions; everything seems alright to me. I haven't ever formally tried to understand the definitions so certainly there could be something small missing in these which I haven't noticed.**

** As I happen to be a Number Theorist, I never tried that much hard to grasp topology to research level.

Balarka
.

Thank you mathbalarka for the heads up.

I guess number theory has benefited from topological ideas too. May be I am wrong. Wiles' proof of Fermat's Last theorem relied on Algebraic geometry. There's also a branch of math called the 'geometry of numbers' which is about number theoretic results using geometrical ideas. Since topology and geometry are like brothers and sisters, I guess topology should have something to contribute to number theory.

 

[mathbalarka]

I guess number theory has benefited from topological ideas too.

That is true indeed. Arithmetic topology is just the thing you mentioned, a mixture of algebraic NT and Topology but unfortunately, not my fields of research in general. I usually study analytic number theory.

Wiles' proof of Fermat's Last theorem relied on Algebraic geometry.

Algebraic geometry of elliptic curves-like structures (which has been used by Wiles) has very less to do with topology as I understood it. Maybe topological algebraic geometry is something you might refer to?

There's also a branch of math called the 'geometry of numbers' which is about number theoretic results using geometrical ideas.

Yes, yes, I am familiar with that branch. Indeed, there are several important results on that area which relies on basic results from Topology.

PS Perhaps we should discuss these in another thread?

Balarka
.

 

[Klaas van Aarsen]

Hello MHB,
I have been reading some literature on manifolds, I recently changed the text book I was reading from and because of that things have gotten pretty messy. I have tried my best to extract the definitions from the text I am reading. Please tell me which ones I have got right and which ones I have not.
Thanks.

Definition 1.
Let $f:\mathbb R^n\to \mathbb R^m$ be a function, $x$ be a point in $\mathbb R^n$ and $r>0$ be an integer. Then $f$ is said to be of class $C^r$ at $x$ if all the partial derivatives of $f$ of order $r$ exist and are continuous at $x$. If $f$ if of class $C^r$ at each point in a subset $A\subseteq \mathbb R^n$ then we say that $f$ is of class $C^r$ in $A$. A continuous function is also said to be a $C^0$-function.

Definition 2.
Let $U$ and $V$ be open sets in $\mathbb R^n$. A function $h:U\to V$ is said to be a diffeomorphism, or a $C^0$-diffeomorphism, if $h$ is differentiable and has a differentiable inverse. For $r>0$, a diffeomorphism $h:U\to V$ is said to be of class $C^r$ if both $h$ and $h^{-1}$ are of class $C^r$.

Definition 3.
A subset $M$ of $\mathbb R^n$ is said to be a $k$-dimensional manifold of class $C^r$ in $\mathbb R^n$ if for all $x\in M$ there exist open sets $U$ and $V$ in $\mathbb R^n$ and a diffeomorphism $h:U\to V$ of class $C^r$ such that $x\in U$ and $$h(U\cap M)=V\cap (\mathbb R^k\times\{0\})$$

Hi caffeinemachine,

Your definitions look fine to me.

However, I do consider your definition 3 rather limited, although it is correct within its restrictions.
I am used myself that a manifold is defined by a set of charts (called an atlas) that map $\mathbb R^k$ to some unspecified set $\mathcal M$ which is the manifold. The charts have to be homeomorphisms.

In your definition you are restricting the manifold (unnecessarily) to $\mathbb R^n$ with furthermore the artificial restriction that the domain and image of the charts are subsets of that same $\mathbb R^n$.

Visually, what I mean looks like this:

Where the charts overlap, there needs to be a diffeomorphism between the charts.

 

[caffeinemachine]

Hi caffeinemachine,

Your definitions look fine to me.

However, I do consider your definition 3 rather limited, although it is correct within its restrictions.
I am used myself that a manifold is defined by a set of charts (called an atlas) that map $\mathbb R^k$ to some unspecified set $\mathcal M$ which is the manifold. The charts have to be homeomorphisms.

In your definition you are restricting the manifold (unnecessarily) to $\mathbb R^n$ with furthermore the artificial restriction that the domain and image of the charts are subsets of that same $\mathbb R^n$.

Visually, what I mean looks like this:

Where the charts overlap, there needs to be a diffeomorphism between the charts.

Thank you ILS for this great help.

I am reading Munkres' Analysis on Manifolds and there the author doesn't discuss arbitrary manifolds. I guess it is better for me to stick with this for the time being and later when I use Lee'd book to study manifolds then I can go for the more abstract and more general approach to study manifolds.

 

[Fantini]

I would like to point out that if it is a diffeomorphism then $r \geq 1$ in your definition 2. Otherwise it is going to be a homeomorphism.