Tagged Questions
2
votes
1answer
43 views
Is this intuition behind product manifolds correct?
I've been studying differential geometry on Spivak's books and recently I proved that the cartesian product of manifolds is another manifold. Right, however, what's the intuition behind this? I've ...
3
votes
2answers
229 views
What exactly is a manifold?
Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth.
Does this mean that in mathematics a manifold is essentially a representation of something that ...
6
votes
3answers
293 views
Why are we interested in closed geodesics?
There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds).
In the case of geodesics ...
17
votes
3answers
340 views
What's the connection between derivatives and boundaries?
The (second) fundamental theorem of calculus says that
$$\int_a^b f'(x) dx = f(b) - f(a)$$
which can also be stated, if one knows enough about what's coming next, as:
The integral of the ...
7
votes
2answers
143 views
Demonstrating the value of abstracting away from elements/subsets to maps
Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction:
an actual element, e.g. $s\in S$
an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$
an ...
4
votes
2answers
319 views
Intuition for not-so-smooth manifolds
in standard text books on (smooth) manifolds, for example the known series by John M. Lee or Jeffrey Lee, you either deal with continuous manifolds, or with smooth manifolds.
However, neither in ...
