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I'm interested in the great amount of mathematics involved in the study of low-dimensional manifolds, above all the interaction between topology, geometry and algebra (and physics?).


Aug
21
comment Doubt about local flatness of low dimensional embeddings
Ok this should follow from Thm 13 in chapter 10: if a subset of $\mathbb{R}^2$ is abstractly triangulable, then there is a homeomorphism $f$ of the plane taking it to a polyhedron of $\mathbb{R}^2$, which is of course locally flat. Thanks for the reference.
Aug
19
awarded  Popular Question
Aug
13
revised Doubt about local flatness of low dimensional embeddings
added 94 characters in body
Aug
13
asked Doubt about local flatness of low dimensional embeddings
Aug
12
comment Question on “up to isotopy” when attaching two spaces
Do you know if there is any version of this theorem for complex manifolds?
Aug
8
asked Existence of orientation-reversing automorphisms
Aug
7
comment Is there any difference between a flat manifold and an affine space?
If you want to do affine combinations on differentiable manifold, then you have to require that the local charts are such that the change of charts are affine maps. This is what is called an affine manifold (see the wiki page for detailed discussion). Without this additional structure, the only "affine" aspects of a differentiable manifold is the lack of a preferred point, I guess.
Aug
7
comment Is there any difference between a flat manifold and an affine space?
If we agree that the main feature of an affine space is the availability of affine combinations, then I would say no. I agree that on a Riemannian manifold you can pick a chart around a point and perform affine combinations, but what if you pick a different chart around the same point? then the affine combination will not be preserved, because the general change of variable is not going to be an affine map. Tu put it in another way, affine combinations are not intrinsecally/well defined on a general Riemannian manifold.
Aug
6
comment Is there any difference between a flat manifold and an affine space?
..on an affine space is the underlying vector space, which gives you the ability to add vectors to points and to perform affine combinations; this is something not available on a general Riemannian manifold. I do agree that you have a way to turn an affine space into a Riemannian manifold (by means of non canonical choices). The examples above show that, conversely, you cannot turn every Riemannian manifold into an affine space.
Aug
6
comment Is there any difference between a flat manifold and an affine space?
Of course you are right: Riemannian manifolds don't have a preferred point. [By Euclidean space I usually mean a real vector space endowed with a scalar product, so it would have an origin; but this is just a matter of terminology]. Anyway I guess the main feature of a structure is what it has, not what it lacks: I wouldn't consider an affine space similar to a Riemannian manifold "because" both of them lack the choice of a special point, since this property is shared by a lot of structures; for example a general topological space comes without the choice of a base point. And what you have...
Aug
6
accepted Topology of space of continuous functions
Aug
6
accepted A 3-manifold with fundamental group isomorphic to a surface group.
Aug
6
accepted Riemann removable singularity theorem for annuli
Aug
6
accepted Boundary behaviour of holomorphic function on unit disk
Aug
6
comment Is there any difference between a flat manifold and an affine space?
so you can consider Euclidean vector spaces as infinitesimal versions of Riemannian manifolds. Affine spaces arises, roughly speaking, whenever you have some object where you could define some structure of vector space, but you have no canonical choice of an origin. These are quite ubiquitous. To stay in the realm of manifolds, the space of connections on a differentiable manifold carries the structure of an affine space (you have no preferred connection on a differentiable manifold). Another important example is given by the space of solutions of an inhomogeneous linear differential equation.
Aug
6
comment Is there any difference between a flat manifold and an affine space?
imho I wouldn't regard affine spaces as precursors to Riemannian manifolds, essentially because the essence of Riemannian geometry, i.e. the possibility to measure the length of tangent vectors (hence of curves) is not available on affine spaces. To have such a feature you have to turn your affine space into a Euclidean space: for example fix a point and a scalar product; this would allow you to measure angles and length. I would consider this as a precursor to Riemannian geometry. Actually, the tangent space to a point of a Riemannian manifold carries the structure of Euclidean vector space..
Aug
5
answered Is there any difference between a flat manifold and an affine space?
Jul
25
asked Easy solution to Yamabe problem for surfaces
Jul
23
asked Boundary behaviour of holomorphic function on unit disk
Jul
22
comment Riemann removable singularity theorem for annuli
Great. Any reference for this result?