Skip to main content
5 votes
Accepted

Differential Geometry of Curves and Surfaces from Riemannian Geometry

My personal preference was Lee for the more direct presentation along the lines you’re asking, and Spivak for bedtime reading. The others are essentially treatments of surfaces using differential ...
peek-a-boo's user avatar
  • 57.1k
2 votes
Accepted

Open sets on a surface with locally connected boundary

To expand my comments. The proof of the Caratheodory-Torhorst Theorem that I know is given in sections 15, 16 of Milnor's notes "Dynamics in one complex variable". The proof is quite long ...
Moishe Kohan's user avatar
  • 99.2k
2 votes
Accepted

How do I calculate the parametrization of a 3D surface given its support function?

If $\vec u$ is a unit vector on the sphere and $h(\vec u)$ is the support function in that direction, then the expression $\vec p(u) = h(\vec u) \vec u + \nabla _S h(u)$ provides a parametrized ...
MathFont's user avatar
  • 4,937
1 vote

A surface passing through two different surfaces

The difference of two surface equations represents another surface passing through their intersection curve. In other words they are concurrent, ( if the word can still be used in this context) ...
Narasimham's user avatar
  • 40.7k
1 vote

Is it true if a face of a graph is not homeomorphic to an open disk, then we may find a noncontractible curve contained in the face?

I'll assume the graph, denoted $G$, is connected, and that the surface $Q$ is closed, connected, and oriented; I believe those are the prerequisites for the definition of genus. It is not true that ...
Lee Mosher's user avatar
  • 122k
1 vote
Accepted

Hessian and the second fundamental form

Regardless of your specific example’s typo, here’s the general relationship: Theorem. Let $(M,g)$ be a Riemannian manifold and $\Sigma$ a (non-empty) regular level set of a smooth function $f:M\to\...
peek-a-boo's user avatar
  • 57.1k
1 vote

How do I calculate the parametrization of a 3D surface given its support function?

Here is one way to do this: The support function $h$ of a convex body $K \subset \mathbb{R}^n$ is a function of $u \in S^{n-1}$. You can extend $h$ to be a homogeneous function $H: \mathbb{R}^n \...
Deane's user avatar
  • 7,777
1 vote

Open sets on a surface with locally connected boundary

Context: Let $\Sigma$ be an orientable surface and $\Omega \subset \Sigma$ a relatively compact open subset that is homeomorphic to the unit open disk $\mathbb{D}$ and whose boundary $\partial \Omega$ ...
Jordan Payette's user avatar
1 vote
Accepted

Curvature of an Embedded Curve

Given an oriented curve in the plane, one can define a signed curvature for it. For an oriented closed curve, the integral of the signed curvature will give $2\pi$ (up to sign). On a general surface, ...
Mikhail Katz's user avatar
  • 43.2k

Only top scored, non community-wiki answers of a minimum length are eligible