All Questions
Tagged with riemannian-geometry gt.geometric-topology
32 questions with no upvoted or accepted answers
40
votes
0
answers
3k
views
Minimal volume of 4-manifolds
This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
- 68.9k
34
votes
0
answers
724
views
Metrics on the 3-sphere with knotted geodesics
According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic (...
- 10.9k
20
votes
0
answers
540
views
Homeomorphisms of the sphere mapping a geodesic triangulation to another one
Consider the standard Riemannian 2-sphere $S$, equipped with a geodesic triangulation $T$. Let $L(S,T)$ be the space of homeomorphisms of $S$ which map
$T$ to a geodesic triangulation. What is the ...
16
votes
0
answers
425
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
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12
votes
0
answers
381
views
Two ways a manifold can have little symmetry
Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $...
- 11.9k
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
6
votes
0
answers
341
views
When exponential map is 1-1 from vector fields to diffeomorphisms
Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
- 5,405
5
votes
0
answers
132
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 4,081
5
votes
0
answers
315
views
Gromov Hausdorff limit and Ricci flow
Let $M$ be a compact, smooth manifold and $\{g(t)\}$ be a family of Riemannian metrics on $M$ evolving under Ricci flow. Suppose the maximal existence time $T$ is finite. To what extent the following ...
- 789
5
votes
0
answers
148
views
configuration space of Riemannian manifolds with a parameter on the distance of distinct points
Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\...
- 2,223
4
votes
0
answers
182
views
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
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
4
votes
0
answers
143
views
Does uniform convergence of (Riemannian) distances implies convergence of conformal structures?
I don't know much about the Teichmüller space, so maybe the question I ask is well known; still I can not find the answer by myself...
Let $\Sigma$ be a closed surface. Let $g_m$ be a sequence of (...
- 353
4
votes
0
answers
74
views
Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]
Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
- 3,051
3
votes
0
answers
608
views
Show that continuous maps between smooth manifolds can be approximated by smooth maps WITHOUT using Whitney's embedding theorem
As it is well-known (and for example this question shows) each continuous map between smooth manifolds is homotopically equivalent to a smooth map that can be constructed using the Whitney embedding ...
- 1,149
3
votes
0
answers
150
views
A question about index of Dirac operator
Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...
- 563
3
votes
0
answers
216
views
Is there a measure for the space of submanifolds?
Let $(M,\mu)$ be a pair of a manifold $M$ ($C^\infty$ or Riemann if you like) and a probability measure $\mu$ on $M$. Is there a sensible way to put a probability measure on the space of submanifolds ...
- 549
3
votes
0
answers
79
views
Virtually abelian fundamental groups equivalent to nonnegative curvature
This is a follow up question inspired by
Fundamental groups of compact manifolds with non-negative Ricci curvature.
In dimensions 3 and 2 (and 1) a manifold has a virtually abelian fundamental group ...
- 4,081
3
votes
0
answers
334
views
Action on a torus
I was asked the following question: suppose that $M= T^{2n}$ a torus of dimension 2n. And let $\mathbf{Z}/2\mathbf{Z} \subset \mathrm{Homeo(M)}$ such that the space of fixed points $N=M^{\mathbf{Z}/2\...
- 223
3
votes
0
answers
214
views
If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non-positive sectional curvature?
If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature?
I wish to use the result about the question and find Leeb's work 3-...
- 1,799
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
3
votes
0
answers
75
views
Two questions regarding flat fibre bundles and the corresponding group action on the fibre
Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...
- 404
2
votes
0
answers
82
views
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
2
votes
0
answers
265
views
A Question about an article by Birman, Series
Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON
SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
- 231
2
votes
0
answers
127
views
Foliation of $X$ by once punctured planes without any singularities
Let $n=3.$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...
- 383
2
votes
0
answers
108
views
Questions about symmetric spaces
I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
- 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
2
votes
0
answers
205
views
Can a non-compact manifold become compact by cutting it?
I'm trying to understand a step in a proof, where one starts with a non-compact manifold $V$ containing a trapped (2-sided, closed) surface $\Sigma$ that's non-separating. In order to complete the ...
- 121
2
votes
0
answers
157
views
Ricci flow with surgery without the "no locally separating $\Bbb RP^2$" assumption
In many places, Ricci flow with surgery is done with orientable manifolds. Morgan and Tian do not require orientability, but instead they impose the condition that $M^3$ have no embedded $\Bbb RP^2$ ...
- 738
1
vote
0
answers
97
views
about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
1
vote
0
answers
93
views
A question about Homotopy equivalence (II)
I posted a similar but different question before in the link
https://math.stackexchange.com/questions/4311982/why-does-x-0-times-s1-simeq-x-x-0/4312530?noredirect=1#comment8987557_4312530.
Now, my new ...
- 563
1
vote
0
answers
85
views
characterizing the singularity for a geometric flow
Suppose that $(M,g)$ is a complete Riemannian manifold and let $\Gamma_0$ be a closed hypersurface in $M$. Let $(x^n,x')$ denote the normal coordinate system on $M$ about $\Gamma_0$ with $x^n>0$ ...
- 4,135