What is manifold in Geometry?

Question

What is manifold in geometry? WE always use this word like non-manifold geometry but I was wondering what is manifold in the first place. I got some definition online but couldn't understand.

A manifold is a topological space that is locally Euclidean

can anyone explain it to me please
thanks in advance

Answer 1

There are different kind of manifolds. It's one is being used for specific theories.

For example in topology (like algebraic topology) we need the space to be Hausdorff, second-countable and locally Euclidean in order to be a topological manifold. There you can build charts from the manifold to the Euclidean space via homeomorphisms and eventually get an atlas in your manifold. For example if $M$ is a top.manifold and for every $p\in M$ there is an open neighbourhood of $p$ say $U$ and a homeomorphism $\phi:U\to \phi(U)\subset \mathbb R^n$ with $\phi(U)$ open.

What you want is get info about an area of your manifold. So you "throw" this area in $\mathbb R^n$ where you know how the things work there and then you get the info back with $\phi^{-1}$.

Also if you allow more restrictions in the definition of the manifold like that we want diffeomorphisms in the place of homeomorphisms you get differential or smooth manifolds.

The $S^2,\mathbb RP^2$ are top. manifolds (and diff. manifolds but this is another story,you need to define much more things) in $\mathbb R^3$(embedded) of dimension $2$ where $S^1$ is of dimension $1$.

If you need more strict(mathematical) details tell me.

Answer 2

Consider a two-dimensional manifold embedded in three-dimensional space. Such manifold would look locally like a sheet of rubber: it might be dirtorted, bent, but it must not be cut. It must not have a border, so it might either be infinite or bend back onto itself. You can't have it tapering to a one-dimensional string, because it has to look two-dimensional everywhere.

But the surface doesn't have to be embedded into some higher-dimensional space, or it might do so only with self-intersection. So although visualizing manifolds as embedded, you should try to assume the viewpoint of some insect living on that surface, taking note of your two-dimensional surroundings but not any properties of the ambient space.

This might give you some intuiton; if you need stricter definitions, you'll likely have to cope with the definitions given in literature, e.g. Wikipedia.