Questions tagged [surfaces]
For questions about two-dimensional manifolds.
3,409 questions
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What does Keisler mean by "touch" in his discussion about tangent planes?
I am reading a short section on p. 666 of the third edition of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler about tangent planes. He defines a tangent plane to a smooth function ...
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Push-forward of a big divisor is big
Consider a finite surjective morphism $f:X\to Y$ between surfaces. Here by a surface we mean a complex projective surface. Suppose $D$ is a big divisor on $X$. Is $f_{*}(D)$ also a big divisor on $Y$?
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A single-sided infinite surface, does that exist? [closed]
I just read a question about the surface of a sphere, and it just hit me:
The surface of a sphere is infinite: in every direction you choose, you can go on forever. On the other hand, the surface of ...
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A non valid triangulation on the torus
Consider the unit square and the canonical way of constructing the torus identifying opposite sides. A triangulation on the surface is a partition of it as subsets (which I'll call triangles) ...
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Surface area of $3$-dimensional convex bodies via spray paint
Let $K \subset \mathbb{R}^3$ be a (real-world) convex body, with uniform density and mass $m_K$. I’d like to find the surface area of $K$, $S(K)$.
Assume that all measurements are perfectly accurate, ...
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The surface $x + y + z = \sin xyz$: Can it be visualized?
The surface $$x + y + z = \sin xyz$$ is very hard to visualize:
What can be said about it, geometrically or visually?
Results so far
Start with a similar, but simpler, curve: $$x + y = \sin xy.$$ It ...
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Proving hermite conditions of a polygonal patch
In parametric lines, constructing the standard hermite basis is trivial.
We have two points $p_1, p_2$ and two tangent vectors $t_1, t_2$. Thus we have 4 unknowns that will be sampled, for that we ...
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Preimage of curves in a double cover of surface
Let $S$ be a smooth complex projective surface. We consider the double cover $\pi:X\to S$ branched along $B\equiv 2L$(suppose $B$ is smooth). Now let $C\subset X$ be an irreducible curve.
If $C$ is ...
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How to plot harmonic vector fields on a genus-2 surface?
On a closed oriented Riemannian surface $M$, a vector field $X$ has divergence and curl defined via the metric and Hodge star.
If $\alpha = X^\flat$ is the metric dual 1-form, then
$$
\operatorname{...
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Explicit free-group automorphism between the "palindromic" relator and the standard surface relator
Let $g \ge 1$ and consider the free group
$$
F_{2g} = \langle x_1,\dots,x_{2g} \rangle.
$$
On the one hand, I have the "palindromic" relator
$$
R_{\mathrm{pal}} := x_1 x_2 \cdots x_{2g} \; ...
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Descend constant vector fields on Euclidean $4g$-gon via opposite-side identification to vector fields with only one singularity on genus-$g$ surface
For the case of genus $g=2$, to construct a genus 2 surface we can identify the diametrically opposed edges of an octagon:
The Construction:
Consider the regular octagon in the complex plane with the ...
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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 $O$, with the properties that:
At any point $s \in S$, there exists exactly one curve $a \in A$ and exactly one ...
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How "close" can we get to a homeomorphism between non-homeomorphic surfaces?
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 ...
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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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Genus of a surface embedded in $(\mathbb R/\mathbb Z)^3$
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 ...
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