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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How to distinguish between walking on a sphere and walking on a torus?

Imagine that you're a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can't see the difference from this point of view. If you are interested, this question arose while I was…
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How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A few ideas that might lead somewhere, maybe: (1) For…
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What Rubik's Twist configuration has the lowest visible surface area?

The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that they can be twisted, but not separated. By being…
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Is there such thing as a "3-dimensional surface"?

The reason I'm asking this question: I work at the National Museum of Mathematics and, amidst my sundry duties (which generally have nothing to do with the exhibits), I do have the authority to alter some of the text in the exhibit descriptions,…
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When does a space admit a flat metric?

Once upon a time I was told that the torus is flat. This was supposed to be surprising, since the ordinary picture of a torus we have in our heads looks inherently curved. However, thinking instead of a torus as a square in the plane with opposite…
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Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any two points $a,b\in X$ can be joined by a curve…
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Exponential map on the ellipsoid.

Consider the ellipsoid $M \subset \mathbb{R}^3$ defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$ where $0 < a < b < c$, equipped with the usual Riemannian induced metric from $\mathbb{R}^3$. Let $p \in M$ be a non-umbilic…
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Interesting implicit surfaces in $\mathbb{R}^3$

I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement! Some that I've implemented are: Basic surfaces like…
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this question that the Gauss-Bonnet theorem actually…
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Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? More formally, let $\mathbb{D}$ be the unit disk in…
Jim Belk
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"Self-sliding" surfaces

There are surfaces which you can "slide" onto themselves so that you see no change. In particular, the plane, the cylindre, the surfaces of revolution (circular cone, sphere, torus...), the helicoid. The sliding motion can be a combination of…
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First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second fundamental forms are could be good. Then, what…
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Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner…
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How to determine that a surface is symmetric

Given a surface $f(x,y,z)=0$, how could you determine that it's symmetric about some plane, and, if so, how would you find this plane. The special case where $f$ is a polynomial is of some interest. The question is somewhat related to this one: How…
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Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Suppose $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ $c_2(V\otimes L)=c_2(V)+(r-1)c_1(V).c_1(L)+{r \choose 2}c_1(L)^2…
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