The fact that U ⊆ℝ^2 but V∩S⊆ℝ^3. How can there be a homomorphism between these two spaces, the dimensions are different !micromass said:
Jianphys17 said:The fact that U ⊆ℝ^2 but V∩S⊆ℝ^3. How can there be a homomorphism between these two spaces, the dimensions are different !
Sorry, but U⊂ℝ^2 and p∈S⊂ℝ^3, with neighborhood V⊂ℝ^3. But for homeomorphism definition, |U| ≠ |V∩S| as can be ?micromass said:
I had realized that dim(V∩S) = 3.micromass said:
Jianphys17 said:
The definition of a regular surface according to Do Carmo's book, chap2 Regular surfaces, definition 1.2, is a subset of Euclidean space that is locally diffeomorphic to an open subset of a plane.
A surface being locally diffeomorphic means that for every point on the surface, there exists a neighborhood around that point that can be mapped onto a neighborhood on a plane in a smooth and differentiable manner.
A regular surface is a subset of Euclidean space that is locally diffeomorphic to an open subset of a plane, while a smooth surface is a subset of Euclidean space that is continuously differentiable. Essentially, a regular surface is a type of smooth surface that is also locally diffeomorphic.
No, a regular surface cannot have sharp edges or corners. It must be smooth and continuously differentiable in order to be considered a regular surface.
Examples of regular surfaces include spheres, cylinders, planes, tori, and cones. Basically, any surface that can be locally mapped onto a plane in a smooth and differentiable manner can be considered a regular surface.