Can we construct differential geometry without multi-variable calculus?

Question

To define the basic constructions of differential geometry (for example, the partial derivative of a function over a manifold), the general tactic seems to be to convert discussions of objects over the manifold into discussions of objects living in $\mathbb{R}^i$ by using the charts.

An example off the top of my head is something like so:

1. Define things over the manifold directly, such as functions $f: M \rightarrow \mathbb{R}$

2. use the charts $(U\subset M, \phi: U \rightarrow \mathbb{R}^n)$ to locally convert parts of the manifold into $\mathbb{R}^n$

3. now use the chart to construct $f'= f \circ \phi^{-1}$ to convert discussions of $f$ over the manifold into discussions over $f': \mathbb{R}^n \rightarrow \mathbb{R}$.

Now we have a "calculus-able" object $f'$ in our hands, so we proceed to use calculus to define things like partial derivatives of $f'$, and their relationship to $f$.

However, to perform this (and other) constructions, we need to have the integral and differential structure of $\mathbb{R}^n$.

Is there some way to "escape" this, and build these structures into the manifold?

Answer 1

I don't know if the following helps you.

Generally, a manifold is a set $M$ to gether with a collection of subsets $\{U_i\}$ and maps $$\phi_i:U_i\to K^n,$$ where $K$ is a field. Any ultra properties on $\phi_i$s makes different names and applications of the manifold. For example, the ordinary manifolds theory is the case $K=\mathbb{R}$, which can inherit properties of $\mathbb{R}$, like differentiability or even "the order" of $\mathbb{R}$.

For complex manifolds, i.e. the case $K=\mathbb{C}$, we deal with analytic functions $\phi_i$ and other concepts like Riemann surfaces or "Klein surfaces" and so on.

There is also quaternionic manifolds, $K=\mathbb{H}$, requiring $\phi_i$ to be "regular" (satisfying Riemann-Fejer equations) or again power series.

You can even go further and speak of "$p$-adic manifolds" (the case $K=\mathbb{Q}_p$).

What I am trying to say, is that being differentiable is not a special property of a manifold. In fact differentiable manifolds are a special cases in the world of general manifolds.

Answer 2

Manifolds are defined via charts.