All Questions
Tagged with dg.differential-geometry mg.metric-geometry
617 questions
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Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 965
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32
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
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152
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Isoperimetric inequality for Kähler manifolds
I am interested in the following form of isoperimetric inequality for Kähler Manifolds (for example unit ball $B^n\subset \mathbb{C}^n$ with Bergman metric). It should say something like this: if $F$ ...
- 37
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125
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How does a conformal transformation affect the frame bundle metric of that manifold?
Suppose I have a metric $g_{\mu\nu}$ over an n-dimensional smooth orientable Riemannian manifold $M$. We then utilize Cartans repere mobile (moving frames) to define oriented orthonormal frames $e^{a}=...
- 250
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78
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Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
- 51
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160
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Spanning curves by flat surfaces
Given a smooth closed connected curve $\gamma$ in $\mathbb R^3$, is there an immersed surface $S$ with boundary, such that its Gaussian curvature is equal to zero and $\partial S=\gamma$?
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82
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Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...
- 356
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38
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Constructing an $n$-simplex at the border of a $n$-ball by orthogonal hyperplanes
I want to construct an $n$-simplex the following way:
Choose $n$ vectors in the boundary of an $n$ dimensional ball, which are forming an $(n-1)$-simplex together.
Place the orthogonal affine $n-1$-...
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1
answer
118
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Contraction and consensus on Hadamard manifolds
Let $\mathcal M$ be a Hadamard manifold and $\{x_i\}_{i=1}^n\subseteq{\cal M}$ be $n$ points. Define $\{y_i\}_{i=1}^n$ as the weighted Fréchet means:
$$
y_i=\arg\min_{y\in\mathcal M}\sum_jw_{ij}d^2(y,...
- 125
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37
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Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?
Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
- 151
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44
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Under what conditions do distances from pivot points uniquely identify a point on a manifold?
Let $X$ be a smooth manifold of dimension $n$ equipped with a Riemannian metric.
Suppose that $x_1, \dots, x_m$ are pivot points on that manifold. We consider the distance functions
$$
f_i(x) = d( x_i,...
- 5,327
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86
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Exhaustion function with uniform controls of level sets on universal covers of compact manifolds
recently I encountered the following problem in my research. Roughly speaking, it asks if, on the universal covers of a closed Riemannian manifold, one can find exhaustion functions with uniformly ...
- 587
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72
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About planar curves on a manifold
I recently came upon the following situation (think of $\mathbb{R}^3$ to simplify): let $S$ be a compact smooth surface with $K>0$ everywhere and define
$$Q=\frac{\sup_{p}\lambda_{1}(p)}{\inf_{p}\...
- 11
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Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
- 175
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154
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Continuity of the volume function
Consider a continuous map $F:(a,b)\times\mathbb{S}^n\to\mathbb{R}^{n+1}$ such that for any $t\in(a,b)$, the map $F(t,\cdot)=F_t:\mathbb{S}^n\to\mathbb{R}^{n+1}$ is Lipschitz continuous. The $n$-...
- 73
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197
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What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
- 24.9k
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207
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
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172
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Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
- 1,467
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245
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Convergence of metric and eigenvalues on a tubular neighbourhood
Background:
Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
- 537
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2
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228
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A triangle comparison in CAT(0) spaces
Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
- 7,149
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1
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139
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Characterization of convexity by connectedness of hyperplane sections
Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.
Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
- 2,934
2
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1
answer
183
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Osculating sphere at point of maximal curvature lies to one side
I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
- 337
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0
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88
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Can the volume of a neighborhood of the cut locus be arbitrarily small?
Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus. It's known that $C_p$ has Hausdorff dimension $\...
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94
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Bounding the area of the image of a set by product of maximum of lengths
Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$.
My question feels ...
- 502
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1
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202
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How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature
Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
- 1,219
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125
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Naming convention for different type of triangulations
When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...
- 183
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157
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Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...
- 969
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1
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155
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Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
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0
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110
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Moser iteration epsilon-regularity for non-linear system in general dimension
I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...
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1
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255
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
- 1,467
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186
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Convex hull of 3 points in Cartan-Hadamard manifolds
Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth?
A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...
- 7,149
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2
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265
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Convergence of metric spaces of increasing dimension
Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and ...
- 41
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107
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Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
- 7,149
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183
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Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
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1
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155
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Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?
A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
- 23
3
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0
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786
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Is this set a manifold?
Take a general spacetime that is not strongly causal.
Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
- 612
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53
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The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
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210
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Bi-Lipschitz embeddings of compact doubling spaces
Suppose that $(X,\rho)$ is a compact doubling metric space. Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x_i\}_{i=1}^n\subseteq X$ such that the map
$$
\begin{...
- 5,405
2
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74
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Asymptotic volume of intersection of n-ball and a cube
Let $B$ be the unit ball in $\mathbb{R}^n$, and let $c\in(0,1)$ be a constant. I'm trying to find the asymptotics for the volume of the intersection $[\frac{c}{\sqrt{n}},1]^n\cap B$ as $n\rightarrow\...
- 21
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530
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Geodesic distance on $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
- 364
3
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120
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Understanding $\kappa$-cones
I recently came across the concept of a $\kappa$-cones of a metric space (Chapter I.5.2) of Bridson and Haefliger's book. In their Proposition 5.8, the provide some intuition of $\kappa$-cones by ...
- 169
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2
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489
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Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,149
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60
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Isometric embedding of 4-element metric spaces into Riemannian manifolds and the curvature
I came across this question Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension. In one of the answers it was stated that it is always possible to isometrically ...
- 820
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262
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Rolling a sphere on a fractal curve
Given a rectifiable curve C : [0, 1] → R2 in the plane, it makes sense to roll a unit sphere S2 on the plane along that curve and ask what is its net rotation in SO(3).
I wonder if this also makes ...
- 2,858
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70
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Orbit projection geometry
Background:
As shown in [1] and [2], for a closed smooth submanifold $M$ of $\mathbb R^d$, the domain $D_M$ of the projection map $P_M:D_M\rightarrow M$ has a dense interior $\Omega_M$ over which $P_M|...
- 71
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75
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non-negative curvature condition for polyhedral manifolds
A polyhedral manifold P, i.e, a topological manifold with a triangulation where each simplex is isometric to a simplex in Euclidean space (other constant curvature spaces are allowed), is said to have ...
- 85
2
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0
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126
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Conformal changes of metric and normal coordinates
Suppose that $(M,g)$ is a smooth Riemannian manifold of dimension $n\geq 2$. Let $p\in M^{\textrm{int}}$. Does there exist a small $\delta>0$ and a smooth function $c>0$ such that for the ...
- 4,135
1
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1
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57
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Isolated maxima for sum of distances of points on a manifold
Let $X$ be a closed Riemannian manifold and consider the function $f_n : X \times \cdots \times X \to \mathbb{R}$ where the domain of $f_n$ is the $n$-fold cartesian product of $X$ and where $f_n(p_1,....
- 1,075
4
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2
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527
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On diffeomorphisms that preserve the metric
Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that
$$ F: \Omega \to \Omega,$$
is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|_{\partial \Omega}$ is ...
- 4,135
1
vote
1
answer
266
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Local Lipschitz constant of exponential map for Hadamard manifolds
Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$. Let $r>0$ and fix any point $p\in M$. Is there a bound on the local ...
- 5,405