What is a manifold in topology?
So I've learned that a manifold is an 'object' that resembles Euclidean space at any point on it.
The analogy was given that you stand on the Earth. We all know that the Earth is spherical, however because of how big it is, we perceive to be on a plane, i.e. a Euclidean space. Obviously, this analogy has its flaws, but the main idea gets across: at any point on the Earth, it looks like a Euclidean space.
This got me thinking, however, obviously 'seeming like a Euclidean space' isn't actually the definition. What is the actual definition (in terms a layperson can understand :P), also: is my general understanding correct?
And does the real definition of a manifold have anything to do with the base topological shape of an object (e.g. a square is topologically equivalent to a sphere) being differentiable at any point? (my thinking for this is that if an object is not differentiable at all points, a tangent plane can't be made at every point, thus the shape can't 'seem like Euclidean space' at every point.)
Thank you!
So I've learned that a manifold is an 'object' that resembles Euclidean space at any point on it.
The analogy was given that you stand on the Earth. We all know that the Earth is spherical, however because of how big it is, we perceive to be on a plane, i.e. a Euclidean space. Obviously, this analogy has its flaws, but the main idea gets across: at any point on the Earth, it looks like a Euclidean space.
This got me thinking, however, obviously 'seeming like a Euclidean space' isn't actually the definition. What is the actual definition (in terms a layperson can understand :P), also: is my general understanding correct?
And does the real definition of a manifold have anything to do with the base topological shape of an object (e.g. a square is topologically equivalent to a sphere) being differentiable at any point? (my thinking for this is that if an object is not differentiable at all points, a tangent plane can't be made at every point, thus the shape can't 'seem like Euclidean space' at every point.)
Thank you!
1 Answer
Well, there's "layman" and "mathematician". But you can't really straddle the line much. The "simple" explanation (mine) is that a sufficiently limited area will eventually resemble "Euclidean space".
Explanation:
The mathematical interest and utility of manifolds is something else - developing useful transformations between systems. But that discussion is really already past the "layman" stage.
Also remember that topology is about the transformations and similarities. It never attempts to equate the actual objects as identical! I.e. it never claims to represent Euclidean Space "at every point".
I'm not sure if your question is inverted, but the "Topology of Manifolds" is discussed very well (and reasonably clear for non-mathematicians) here:
https://www.encyclopediaofmath.org/index.php/Topology_of_manifolds
If you meant the mathematical definition of "Typological Manifold" it is on the first page of the following reference. Even the simple "definition" is really too complex to write out in full here.
More (academic) discussion of Manifolds here:
http://web.stanford.edu/~jchw/WOMPtalk-Manifolds.pdf
