# Questions tagged [curves-and-surfaces]

A surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line

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### The area of the strip [closed]

Let $\gamma$ be a convex smooth plane curve of length $l$. I need to compute the area of the strip swept by the outer normal segments of length $r$ to $\gamma$ and the length of the outer boundary ...
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### When are principal lines of curvature geodesics?

Let $S$ be a smooth surface embedded in $\mathbb{R}^3$. When are (some of) the principal lines of curvature geodesics on $S$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal ...
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### Is the Frenet frame is independent of the choices of parameters?

I asked this question on StackExchange, but until now there is not any answer or hint. I hope I can get some help here. When I am reading ''A course in differential geometry'' of Klingenberg, I ...
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### Interesting geometric flow of space curves with non-vanishing torsion

Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by \begin{equation} \partial_t \gamma = \tau^{-\frac{1}{2}} n, \end{...
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### Reducing curves in surfaces by Dehn twists

Let $F$ be a compact, oriented surface. A Dehn move, $D_\beta$, on a simple closed curve (scc) $\alpha$ is a Dehn twist applied to $\alpha$ along a scc $\beta$ which intersects $\alpha$ once. Is ...
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### Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...
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Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\... 1answer 708 views ### Hadamard theorem about embedding The following theorem is commonly attributed to Jacques Hadamard. Assume$\Sigma$is a smooth locally convex immersed surface in the Euclidean space. Then$\Sigma$is embedded and bounds a convex ... 1answer 81 views ### Bernstein type theorems for CMC hypersurfaces in$\mathbb{R}^{n+1}$Is there any Bernstein type theorems for CMC hypersurfaces in$\mathbb{R}^{n+1}$in the literature? More precisely I would like to know if there is an answer to the following QUESTION: Let$f : \...
Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look ...
While teaching a course in differential geometry, I came up with the following problem, which I think is cool. Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature. ...