I have read most of Lee's "An Introduction to Topological Manifolds". The presentation of 2-Manifolds is based on Euclidean geometry.
I have also been getting familiar with non-Euclidean geometries.
Is there a way to extend the notion of 2-Manifolds to non-Euclidean geometries? Or maybe euivalently is there a notion of a plane/surface in non-Euclidean geometries?
Kind regards,
Vasily Gal'chin
The notion of equivalence for topological manifolds (the topic of study in the book you're reading) is homeomorphism. For instance, the plane $\mathbb{R}^2$ and the 2-sphere with a deleted north pole $S^2 - \{N\}$ are homeomorphic. From a topological perspective, these spaces are "the same."
But from a geometric perspective, we can tell them apart. One is round while the other is flat. There is no such thing as a two-sided polygon in $\mathbb{R}^2$ but there is on $S^2 - \{N\}$. These geometric differences come down to the fact that we measure distances differently in the two spaces.
Look at it from a different perspective: if you start with the topological manifold $\mathbb{R}^2$, you could define a distance $d((x_1,y_1),(x_2,y_2)) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, which would make it Euclidean, or you could define it some other way, which would make it non-Euclidean. The subject of Riemannian geometry deals with all the ways you can have different geometry on manifolds with the same topology.