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Results tagged with smooth-manifolds
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user 4213
Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
11
votes
Accepted
On the $\mathbb R$-algebra structure on $C^\infty(M)$.
If $M$ is connected then one can determine $\mathbb{R}$
(the constant functions) within
$C^\infty(M)$ (the ring of smooth real-valued functions on $M$).
One can certainly determine $\mathbb{Q}$ withi …
- 20.8k
9
votes
Accepted
The ring $C^{\infty}(M)$?
You are correct: $C^\infty(M)$ does contain all the geometry and topology
of $M$ (at least when it is considered as an $\mathbb{R}$-algebra).
For example when $M$ is compact the points of $M$ correspo …
- 20.8k
5
votes
Which manifolds admit a diffeomorphism of order $n$?
Here's an example with $g=2$ and $n=5$. Consider the hyperelliptic
curve defined by the equation
$$y^2=x^5-1$$
or to be more precise the corresponding desingularized projective curve.
Now this is a Ri …
- 20.8k
4
votes
Accepted
Ideals in the ring of smooth endomorphisms of the real line
This is all fairly well-known stuff. In Q1, the ideal $I_p^n$ is the
set of functions for which the derivatives up to order $n-1$
at $p$ vanish in a local coordinate system centred at $p$.
I'll just …
- 20.8k