Functions on a manifold

Question

I am learning about diff geometry and I have a question about ways of defining a function on a manifold. So if we think about manifolds in general we can define a chart on it. Then, through composition of a function and a chart we can analyze this function. My question is this. How can we define a function in the first place since it is on points P which need not be numbers. Dont we need to define the function on a chart in order for it to be defined on a manifold? And dont we then need to define a manifold through charts also? But in order for it to be defined properly we have to look for chart independent ptoperties which are read off the charts... I am confused.

Answer 1

How can we define a function in the first place since it is on points P which need not be numbers.

Functions need not be defined only on "numbers", whatever that means. Functions can be defined on arbitrary sets, in particular manifolds. For instance, given a manifold $M$, you can consider $f:M\to\mathbb R$ defined as $f(p)=0$ for every $p\in M$. Also notice that charts are actually functions $M\supset U\to\mathbb R^d$, so I don't understand why you are ok with them but not with other functions.

Answer 2

I like your question, since I believe it is connected to one of the core topics of differential geometry. Being not an expert, I can only give you my humble opinion on it, which is better expressed by a few questions, rather than long answers...

Abstract vs. concrete functions...

For sets $X, Y$, a function $f: X \to Y$ assigns a value $f(x)$ for each $x \in X$. But if $X$ is large, we may want to use efficient ways to express $f(x)$ in terms of $x$.

To define a function on a manifold, we just need to assignment a value $f(p)$ for all $p \in \mathcal M$. (Which maybe also satisfied other conditions, e.g. smoothness.) If such an assignment is given, it will be automatically coordinate-free, since we did never toughed the charts to define it. That's the reason why we like this approach for abstract purposes.

But since our human brain has only finite memory, this method is a bit unpractical if we want to do calculations etc.

How can we practically define functions on manifolds

1. Gobal chart: If we have a global chart $h: \mathcal M \to \mathbb R^n$, then we can define a function $f = \tilde f \circ h$, with $\tilde f:\mathbb R^n \to \mathbb R$ being a smooth function.

   Similary, we might use embeddings $e: \mathcal M \to \mathbb R^N$ to define a function for example on submanifolds of $\mathbb R^N$. This is for example commonly used to define maps on submanifolds ($\mathrm{GL}(n)$, $\mathrm{SO}(n)$, ...) of the manifold of matrices $\mathrm{Mat}(n\times n)$.

2. Finitely many charts: For two charts $h_1: \mathcal U \to \mathbb R^n$ and $h_2: \mathcal V \to \mathbb R^n$, we may define $f_1 : \mathcal U \to \mathbb R$ and $f_2 : \mathcal V \to \mathbb R$ as above, but then $f_1 = f_2$ must be assured on the intersection of the chart domains, i.e. for all $p \in \mathcal U \cap \mathcal V$.

3. Implicit definitions: It is sometimes still possible to calculate on manifolds, without having an explicit formula for a function.

   1. Solutions of (partial) differential equations: Assume, we can define a differential equation on a manifold.

      For example the Laplace equation $\Delta u = 0$ (with some boundary conditions, for example constant values on 'some' boundaries). Then we can define a map $u(p)$, by claiming that it is the unique solution of the partial differential equation. In particular, it is sufficient to provide a finite amount of data, to define the function! If the manifold has two boundaries, we just need to pick two numbers and $u$ is defined (if the PDE is well-defined, of course)! With this function $u$, we can continue to work, since we have (implicit) information about the function $u$ as well, which is sufficient for many tasks.

      But the examples don't stop here: We have the geodesic equation (and ODE), or Hamiltons equations etc...

      Of course there is a hidden trap: To define a differential equation, say $\dot x(t) = F(x(t))$ on a manifold, we need a vector field $F$, which is again a function $F: \mathcal M \to T \mathcal M$.

      But in important special cases, we know a little bit about $T \mathcal M$, which allows us to define, maybe implicitly, interesting functions of that type. (For example if $\mathcal M$ is a Lie group, then we can define one value $F(1) \in T_1 \mathcal M$. Now Lie groups are super cool and for each point $p$ there is a meaningful map $T \lambda_p : T_1 \mathcal M \to T_p \mathcal M$. This allows us to define $F(p) = D \lambda_p F(1)$. Expressed in coordinates, this map may not look like a constant map, and the solutions $x(t)$ are also somehow interesting.)

At the end... It is good to ask these equations! And be patient with the theory. Differential geometry need some time to feel natural, but after a while you will see that numbers, charts and coordinates are sometimes false friends. Spaces are often quite nonlinear!

EDIT:

Concrete Example:

We use a function $f: \mathbb R^3 \to \mathbb R: (x,y,z) \mapsto x^2+y^4+z^2$.

Then we define $\mathcal M := f^{-1}(\{1\})$. This is indeed a manifold by the constant rank theorem, since $\mathrm{rk}(\mathrm{D} f(p)) = 1$, for all points $p \in \mathcal M$.

But we also have an embedding $\mathcal M \to \mathbb R^3: p=(x,y,z) \mapsto (x,y,z)$.

Therefore I can choose a function on $\mathbb R^3$, say $$ \tilde f(x,y,z) = \sin(x) \cdot \exp(y)^{z^2} $$ and define $$ f(p) = \tilde f \circ e = \tilde f(x,y,z). $$

As you see, we did not directly use the charts of $\mathcal M$, but only implicit information about $\mathcal M$.