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3 votes
1 answer
257 views

Asymptotic parametrization for negatively curved surfaces

Let $S$ be a complete simply connected negatively curved surface immersed in Euclidean space $\textbf{R}^3$. Does there exist a parametrization $f\colon\textbf{R}^2\to\textbf{R}^3$ for $S$ such that ...
Mohammad Ghomi's user avatar
3 votes
0 answers
166 views

A higher-dimensional "line of curvature"?

Let $M$ be a submanifold of a Riemannian manifold $Q$, and let $\Gamma$ be a $d$-dimensional submanifold of $M$, i.e., $\Gamma^{d} \subset M \subset Q$. Suppose that, for all (unit) normal vectors of $...
Matteo Raffaelli's user avatar
2 votes
1 answer
110 views

Parametrization of $k$-ruled submanifold: can we choose the base to be orthogonal to the rulings?

A smooth $m$-dimensional submanifold of $\mathbb{R}^{d}$ is said to be $k$-ruled if it is foliated by $k$-dimensional planes, called rulings. Let $M$ be a $k$-ruled submanifold. Then $M$ can be ...
Matteo Raffaelli's user avatar
3 votes
1 answer
255 views

Planar curves in $M^{m}$ vs curves in $M^{2}$

Following Anton Petrunin’s suggestion, I revise the question to make it less vague. Let $M^{m}$ be an $m$-dimensional Riemannian manifold, and let $\gamma$ be a unit-speed curve $I \to M^{m}$. We say ...
Matteo Raffaelli's user avatar
8 votes
0 answers
295 views

Intuition for the volume form - combinatorial definition?

I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
Sprotte's user avatar
  • 1,075
1 vote
1 answer
76 views

Cross product of two infinitesimal bendings

Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$. Let $Z_1,Z_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential ...
Dmitrii Korshunov's user avatar
1 vote
1 answer
137 views

Smoothness of the asymptotic parametrization of a ruled surface

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form $$\begin{align} \sigma \colon I \...
Matteo Raffaelli's user avatar
3 votes
0 answers
189 views

The classification of developable surfaces: Are these statements equivalent?

This is a cross-post from MSE (https://math.stackexchange.com/q/4330772/242708). I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some ...
Matteo Raffaelli's user avatar
3 votes
1 answer
470 views

Is the Moebius strip Riemannian homogeneous?

Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively? My ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
327 views

Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$?

Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?
MAS's user avatar
  • 930
2 votes
1 answer
742 views

Continuity of the perimeter of level sets w.r.t. level function

Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
Bogdan's user avatar
  • 1,759
5 votes
2 answers
1k views

Continuity of Hausdorff measure on level sets

Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that: $\bullet$ $\phi^{-1}(0)\neq\emptyset$; $\bullet$ $\nabla\phi(x)\...
Bogdan's user avatar
  • 1,759
6 votes
1 answer
604 views

When is the cut locus a finite tree?

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
Leo Moos's user avatar
  • 5,038
6 votes
2 answers
428 views

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
Alex M.'s user avatar
  • 5,407
2 votes
0 answers
134 views

Hypersurfaces whose unit normal $N$ satisfies $[N,X] =0$ for every tangent vector field $X$

Let $M$ be a hypersurface of a Riemannian manifold, and assume that $M$ satisfies the following property: For each $p \in M$, given a unit normal vector field $N$ defined in a neighborhood $U$ of $...
Matteo Raffaelli's user avatar
1 vote
1 answer
210 views

Definition of first normal space

Given an immersed submanifold $M$ of a Riemannian manifold $\overline{M}$, the first normal space of $M$ at a point $p \in M$ is defined as the linear subspace $N_{p}^{1}M$ of $N_{p}M$ spanned by the ...
Matteo Raffaelli's user avatar
31 votes
6 answers
2k views

If a triangle can be displaced without distortion, must the surface have constant curvature?

Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$). Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics. If $T$ can be moved around arbitrarily on $S$ ...
Joseph O'Rourke's user avatar
3 votes
1 answer
274 views

Symmetry of functions on $S^2$

Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}...
A random mathematician's user avatar
7 votes
1 answer
162 views

Estimate of number of boundary components of a compact Riemannian 2-surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
asv's user avatar
  • 21.8k
7 votes
1 answer
231 views

Estimate of area of 2-dimensional surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
asv's user avatar
  • 21.8k
4 votes
1 answer
139 views

Convex hull of a connected subset on a complete surface of non-positive curvature

Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
aglearner's user avatar
  • 14.3k
1 vote
1 answer
98 views

Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<...
Sammyy Delbrin's user avatar
7 votes
2 answers
348 views

Most general version for the Gauss-Bonnet theorem for polygons

Suppose $M$ is a 2-dimensional smooth Riemannian manifold and $P\subset M$ is an open and connected subset with compact closure and a piecewise geodesic boundary. My question is: What further ...
Sammyy Delbrin's user avatar
4 votes
1 answer
215 views

Finding the shortest curve that is at distance $\epsilon$ of every point of a surface

Let $M$ be a compact connected (smooth) surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant. Is there (and if there's not, what conditions on ($M$, $\epsilon$) should ...
LCO's user avatar
  • 506
17 votes
2 answers
1k views

Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$. Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
Ali Taghavi's user avatar
3 votes
4 answers
1k views

Intrinsic definition of arc length [closed]

Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?
Felix Goldberg's user avatar
0 votes
0 answers
121 views

Positive curvature of the boundary away from a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact: Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...
foliations's user avatar
  • 1,149
22 votes
4 answers
3k views

What is the analog of the "Fundamental Theorem of Space Curves," for surfaces, and beyond?

The "Fundamental Theorem of Space Curves" (Wikipedia link; MathWorld link) states that there is a unique (up to congruence) curve in space that simultaneously realizes given continuous curvature $\...
Joseph O'Rourke's user avatar
3 votes
3 answers
2k views

Gaussian curvature radius

In the paper Surface sampling and the intrinsic Voronoi diagram (2008), Ramsay Dyer defines the Gaussian curvature radius at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where $K(x)=\...
Dror Atariah's user avatar