# Questions tagged [curves-and-surfaces]

A surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line

301 questions
Filter by
Sorted by
Tagged with
90 views

112 views

103 views

### Representing relative homology classes orientable surfaces with boundary

Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies ...
151 views

### On cycloids and other roulettes

It is well known that the cycloid is the curve traced by a point on a circle as the circle rolls along a line without slipping. Consider wheels with smooth convex shapes (not necessarily circular) ...
73 views

2k views

### Conceptual proof of classification of surfaces?

Every compact surface is diffeomorphic to $S^2$, $\underbrace{T^2\#\ldots \#T^2}_n$, or $\underbrace{RP^2\#\ldots \#RP^2}_n$ for some $n\ge 1$. Is there a conceptual proof of this classification ...
156 views

### Area of a surface confined by a sphere II

[A followup on two related posts: Area of a surface confined by a sphere Area of a elliptic surface confined by a sphere . Thanks to all the inputs so far.] Let $S$ be a surface enclosed inside the ...
151 views

### Area of a elliptic surface confined by a sphere

Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If every point of S is elliptic, then must $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?