Questions tagged [surfaces]

Question 1

Let surface $S$ be called curvelinear if there are two sets of simple curves, $A$ and $O$, with the properties that: At any point $s \in S$, there exists exactly one curve $a \in A$ and exactly one ...

Question 2

In order to qualify as a homeomorphism, a map between topological spaces must be (1) injective, (2) surjective, (3) continuous, and (4) its inverse must be continuous. I suspect that these four ...

Question 3

We are asked to compute the Gaussian curvature of the surface generated by $F(x, y, z)=0$. I solved the problem using the implicit function theorem, regarding $z$ as a function of $(x, y)$. After a ...

Question 4

For integers $p,q,r\ge 1$ What is the genus of the connected orientable surface$$S_{p,q,r}=\{(x,y,z)\in(\mathbb R/\mathbb Z)^3\mid\cos(2q\pi x)+\cos(2r\pi y)+\cos(2q\pi z)=0\}$$ What is the genus of ...

Question 5

I'm looking for a reference with a proof of the following fact: Two closed connected 1-dimensional submanifolds of the Klein bottle are isotopic if the integer homology classes they represent are the ...

Question 6

The accepted answer at https://math.stackexchange.com/a/3335229/268333 (with 25 upvotes) says that in the case of a smooth surface, if points $A$, $B$ and $C$ are on the surface, then plane $ABC$ ...

Question 7

One possible approach to defining the surface area of a smooth 2D surface embedded into 3D Euclidean space, which is a natural generalization of the idea of calculating the arc length of a 1D curve as ...

Question 8

While reading "Complex Algebraic Surfaces" by Beauville and thinking about Exercise III.24.2. I came across the paper "Curves with high self-intersection on algebraic surfaces" by ...

Question 9

Let $S_1,S_2,S_3,S_4 \subset \mathbb{R}^3$ be four mutually isometric, smooth surfaces of revolution, each with the same constant Gaussian curvature $K>0$ and the same cone angles at their two tips....

Question 10

I need to show the following: Let $M$ be a 4-dimensional space. Let $S\subset M$ be a closed (without boundary) 3-dimensional hypersurface embedded in 4 dimensions. $S$ is simply the boundary of a ...

Question 11

Suppose I have a smooth surface $S$ (compact, connected, without boundary). Suppose further I am given an exceptionally good atlas in the sense that it is a finite cover by embedded open balls $(\...

Question 12

On a Riemannian manifold $M^n$ with nonpositive sectional curvature, let $\partial B_r(p)$ be a geodesic sphere of radius $r$ centered at $p \in M$, and let $q \in \partial B_r(p)$. Are the principal ...

Question 13

I read on a paper that “the second fundamental form of surface $T$ is bounded above, since $T$ is supported from below by balls of radius $r$ at each point.” Here $T \subset M$ is a subset of a ...

Question 14

Consider a double covering $p: X \to \Bbb P^2_k$ (over a fixed base field $k$) branched along a curve $C \subset \Bbb P^2$ (especially étale over complement of $C$). Rmk.: Such can be constructed in ...

Question 15

I have three parametric equations in two variables that give the coordinates of points on a three-dimensional, closed, convex surface. I want to find the volume enclosed by that surface, but I haven't ...

Stack Exchange Network

Questions tagged [surfaces]

A transformation preserves distance along two sets of perpendicular curves. Does it preserve area?

How "close" can we get to a homeomorphism between non-homeomorphic surfaces?

Gaussian curvature of $F(x, y, z)=0.$

Genus of a surface embedded in $(\mathbb R/\mathbb Z)^3$

Simple closed curves on the Klein bottle

Does the triangle connecting three points on a smooth surface approach the tangent plane as two of the corners approach the third?

Does requiring that the triangles in a surface triangulation become small avoid the Schwartz lantern problem?

Elementary transformation of a geometrically ruled surface as a projective bundle

Showing that the union of pairwise intersection curves of four surfaces is a four-regular embedded graph with certain properties

Prove that the integral of a divergence (subject to a condition) over a closed 3D hypersurface in 4D vanishes.

Triangulation of a surface obtained from a good cover

How does sectional curvature control the 2nd fundamental form of geodesic sphere

On a Riemannian manifold, Why is the second fundamental form of $T$ bounded above if T is supported from below by balls

Smoothness of Branched Cover depending on smoothness of Branch Locus

Is it possible to calculate the volume of a general 3D (closed & convex) parametric surface?

Hot Network Questions