Questions tagged [surfaces]

For questions about two-dimensional manifolds.

Formally, a surface is a two-dimensional topological manifold. Some examples of surfaces are the plane, the cylinder, the sphere, and the graph of a real-valued function of two variables.

More generally, the term "hypersurface" can be used to denote an $(n-1)$-dimensional submanifold of an $n$-dimensional manifold.

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What are all topological spaces obtained by gluing the edges of a triangle?

I am currently learning about polygonal presentations of surfaces. In the notation I'm using (following Lee's "Topological Manifolds"), $\langle a, b \ |\ aba^{-1}b^{-1}\rangle$ is a presentation of the torus $\mathbb{T}^2$, and $\langle a,b\ |\…
Jesse Madnick
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Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic symplectic structures on $ S^2$? If yes, is there a…
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Generalization of ellipse equation to higher dimensional surfaces

This question was motivated by Definition of an ellipsoid based on its focal points . I'd like to avoid terms like ellipsoid, so I'll use terms like one-dimensional ellipse (a normal ellipse in the plane) and two-dimensional ellipse (an ellipsoid…
2'5 9'2
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isolated non-normal surface singularity

I am looking for an isolated non-normal singularity on an algebraic surface. One obvious example occurs to me: the union of two $2$-dimensional affine subspaces of $\mathbb{A}^4$ which meet in a point. But this seems like "cheating." Can someone…
Justin Campbell
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isometries of the sphere

There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere. What is known about isometric deformations of the sphere, when the smoothness…
user7530
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Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows (mostly copied from the question) : "Let $X$ be a…
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Loch Ness monster and Jacob's Ladder Surfaces are NOT homeomorphic

I am looking for proof of the fact that $X($Jacob's Ladder surface$)$ is not homeomorphic to $Y($Loch Ness monster surface$)$. Certainly, fundamentals groups do not help as the fundamental groups of both spaces are free groups of countably many…
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Electric field and curvature

My physic teacher said that In a conductor the electric field, which is non-zero only on the surface, is stronger where the curvature is bigger*. But he did not provide a mathematical proof for this. Furthermore, I don't know the correct statement…
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Fixed points in mapping from Möbius strip to disk [Explanation or reference needed]

One of the most elegant demonstrations in topology is the proof of the inscribed rectangle problem (a solved variant of the unsolved inscribed square problem) which states that for any plain, closed continuous closed curve $\Gamma$ in…
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Deriving the surface area of a sphere from the volume

I am a high school student, so I know how to derive the volume $V=\dfrac{4}{3}\pi r^3$ using calculus, but I am unable to derive its surface area. However, I notice that we can approximate the surface area of a sphere by imaging 'the shell' of the…
John Smith
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Differential geometry: restriction of differentiable map to regular surface is differentiable

From Do Carmo: Let $S_1$, $S_2$ be regular surfaces. Suppose $S_1\subset V\subset \mathbb{R}^3$ and $\varphi:V\rightarrow \mathbb{R}^3$ is a differentiable map such that $\varphi(S_1)\subset S_2$. Then the…
beedge89
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Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to prove Proposition 2.7. The only resource where I…
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Are these definitions of intersection multiplicity equivalent?

I am pretty sure the answer is yes. I normally work over $\mathbb{C}$ so i will do so here as well, to prevent myself from making silly mistakes. In projective space, one has Serre's famous definition of the intersection multiplicity. If we are…
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Projection of ellipsoid

Find the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in advance!
Giiovanna
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Prove that any shape 1 unit area can be placed on a tiled surface

Given a surface of equal square tiles where each tile side is 1 unit long. Prove that a single area A, of any shape, but just less than 1 unit square in area can be placed on the surface without touching a vertex of any tiled area? The Shape A may…
perhapx
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