Questions tagged [surfaces]
For questions about two-dimensional manifolds.
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A transformation preserves distance along two sets of perpendicular curves. Does it preserve area?
Let surface $S$ be called curvelinear if there are two sets of simple curves, $A$ and $O,ドル with the properties that:
At any point $s \in S,ドル there exists exactly one curve $a \in A$ and exactly one ...
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Gaussian curvature of $F(x, y, z)=0.$
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 ...
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Simple closed curves on the Klein bottle
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 ...
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Does requiring that the triangles in a surface triangulation become small avoid the Schwartz lantern problem?
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 ...
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Showing that the union of pairwise intersection curves of four surfaces is a four-regular embedded graph with certain properties
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....
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Triangulation of a surface obtained from a good cover
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 $(\...
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On a Riemannian manifold, Why is the second fundamental form of $T$ bounded above if T is supported from below by balls
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 ...
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Is it possible to calculate the volume of a general 3D (closed & convex) parametric surface?
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 ...