All Questions
31 questions
2
votes
0
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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:
$$\...
- 356
4
votes
0
answers
182
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Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.8k
2
votes
0
answers
65
views
Connection between a function and its usage in geometry [closed]
I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...
- 121
4
votes
1
answer
230
views
Generalizing a result about hyperbolic 2-folds to hyperbolic 3-folds
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ \Sigma_g $ be a compact orientable surface of genus $ g $. Let the subgroup $ \pi_1(\Sigma) $ of $ \SL_2(\...
- 4,081
5
votes
1
answer
394
views
Embedding round manifolds into low dimensional spheres
Robert Bryant's answer to Isometric embedding of SO(3) into an euclidean space mentions that there is an isometric embedding of the round tetrahedral space $ SO_3/A_4 $ into the round sphere $ S^6 $.
...
- 4,081
9
votes
0
answers
336
views
Nash embedding for 3 manifolds
The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$ (edit: 14 is a better bound for compact 3 manifolds thanks @mme). What ...
- 4,081
1
vote
1
answer
175
views
A question about Gromov-Lawson construction
We all know that if we consider the connected sum $S^n\# S^n$ of two spheres $S^n$ for $n\geq 3$, then by Gromov-Lawson construction(cf. Gromov, Mikhael; Lawson, H.Blaine Jun., The classification of ...
- 563
4
votes
1
answer
381
views
Injectivity of map of fundamental groups from totally geodesic hypersurface
Let $X$ be a compact manifold of non-positive sectional curvature which carries a connected totally geodesic hypersurface $X_0\subset X$. Let $K$ be any compact subset of $X-X_0$. That's to say we ...
- 563
2
votes
0
answers
177
views
Structure of hyperbolic manifolds of finite volume
Let $X$ be a hyperbolic manifold of finite volume.
I want to prove that $X$ has ends of the form $N\times \mathbb{R}$ where $N$ has a finite covering by a nilmanifold and $\pi_1N\to \pi_1 X$ is ...
- 563
7
votes
1
answer
759
views
Complete geodesics on hyperbolic a pair of pants
I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:
I am trying to understand the article by Maryam ...
- 231
4
votes
2
answers
327
views
Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension?
Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-...
- 2,083
6
votes
2
answers
317
views
Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 935
10
votes
2
answers
751
views
On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$
In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
$$\...
- 469
4
votes
3
answers
3k
views
Covariant derivative of determinant of the metric tensor
Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
- 131
21
votes
6
answers
3k
views
Smooth functions on sphere
Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...
5
votes
1
answer
375
views
A possible characterization of sphere or projective space
Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \...
- 356
8
votes
1
answer
412
views
Homeomorphism/ homotopy types of non-negatively curved manifolds
A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative ...
- 21.8k
4
votes
1
answer
400
views
Canonical Metrics on 3- and 4-Manifolds
From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique ...
- 989
3
votes
0
answers
70
views
Does the orbital function divided by the volume of a ball decrease?
Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...
user50806
4
votes
0
answers
207
views
Integral of second fundamental form
Let us have Riemannian manifold $M$ with boundary $N.$ Let $F$ be an immersion, such that $F:N\to M$ and $B$ be a second fundamental form on $N$ relative to $F.$ And let $f$ be a function on $N.$
...
- 41
1
vote
1
answer
273
views
Regularity of a generalized polar coordinate metric with two angles
Flat space in polar coordinates takes the form
$$ds^2=dr^2+r^2d\phi^2$$
To avoid a conical singularity at the origin, we must impose that $\phi$ is periodic with period $2\pi$.
Now consider the ...
- 293
5
votes
1
answer
1k
views
Does every smooth manifold admit a metric with bounded geometry and uniform growth?
Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...
- 541
7
votes
2
answers
436
views
Are square tiled surfaces dense in the moduli space of translation surfaces?
I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
- 73
22
votes
1
answer
1k
views
Just how close can two manifolds be in the Gromov-Hausdorff distance?
Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...
- 646
2
votes
1
answer
215
views
Lamination as limit of arcs
I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
- 23
14
votes
2
answers
786
views
For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?
Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ ...
- 1,069
8
votes
1
answer
696
views
Geodesics on manifolds with boundary
Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
- 131
15
votes
1
answer
350
views
Closed geodesics avoiding points in hyperbolic surfaces
Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...
- 380
12
votes
3
answers
371
views
Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?
The geometrization theorem tells us:
Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a complete ...
- 647
14
votes
1
answer
3k
views
How metric is Riemannian geometry
Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by
$$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, dt\,,$$...
- 5,824
7
votes
2
answers
1k
views
G-spaces and manifolds
In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
The space is metric
The space is finitely compact, i.e., a ...
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