Two questions here, 1) Can someone give me two or three examples to explain diffeomorphism and diffeomorphic surfaces in layman terms? Some objects that we see in day to day life which are diffeomorphic. 2) This might be a bit naive. When isomorphism tells us that two objects are of same structure what is the need of automorphism which is a map from an object to itself? Won’t an object be intuitively same as itself?
2026-03-25 12:30:39.1774441839
Diffeomorphism and Automorphism
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1) In low dimensions (including $2$, i.e. surfaces), the notions diffeomorphism and homeomorphism (between smooth manifolds) coincide. That means it suffices to have some intuition for homeomorphisms. In layman's terms, a homeomorphism is something like a stretching or bending without ripping anything apart, creating/filling holes etc. Moreover, you are not allowed to change the "dimension" of things (this is somehow important to have some different intuition for homeomorphisms and homotopy equivalences). Let us have a look at some of these continuous deformations (more text including real life examples after the picture):
Basically you can imagine that you are baking some christmas cookies (even though it is a bit early for christmas now) and you are playing around with the dough. If you have some figure out of dough that has a hole (for eyes maybe) you do not want to fill that hole (or create more holes) while deforming it and you also do not want to rip anything apart. So your deformations of the dough that you will naturally think of will be homeomorphisms.
2) An object is always isomorphic to itself via the identity. That is true. But maybe there are more operations that do not change the (core) properties of your object. These would be isomorphisms to some other but similar behaving object. If you now consider automorphisms, the question is not whether an object is isomorphic to itself (which it is as I said), but rather what you are allowed to do with your object without changing anything. This will for example tell you about symmetries of your object.