All Questions
29 questions
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 $S$ ...
- 151k
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 $\...
- 151k
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 ...
- 356
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 ...
- 1,075
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 ...
- 135
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 ...
- 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 ...
- 21.8k
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 $...
- 5,407
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 ...
- 5,038
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)\...
- 1,759
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 ...
- 14.3k
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 ...
- 506
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?
- 6,962
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 ...
- 7,149
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 ...
- 965
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)=\...
- 579
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 ...
- 4,081
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?
- 930
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)\}...
3
votes
0
answers
165
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 $...
- 965
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 ...
- 965
2
votes
1
answer
741
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 ...
- 1,759
2
votes
1
answer
109
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 ...
- 965
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 $...
- 965
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
$$\begin{align}
\sigma \colon I \...
- 965
1
vote
1
answer
98
views
Polar coordinates of a set with different radius and angle
Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...
- 135
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 ...
- 2,934
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 ...
- 965
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 $\partial U \backslash p$ ...
- 1,149