All Questions
76 questions
6
votes
2
answers
256
views
Generalization of Bieberbach's second theorem
Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...
- 419
7
votes
1
answer
423
views
3-manifolds with all geodesics closed
A theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \...
- 1,170
7
votes
1
answer
376
views
Does any surface of constant curvature admit a cocompact group action?
Suppose $S$ is a non-compact and complete surface (2 dimensional smooth Riemannian manifold) of constant curvature. I am wondering if there exists a group $G$ which acts by isometries and properly ...
- 107
15
votes
1
answer
1k
views
Thurston geometries in dimension 4
In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3.
Question: How many different geometries (in the sense of Thurston) do we have in ...
- 1,607
2
votes
1
answer
152
views
How to define "interior" for the unit arc? [closed]
Let the unit arc be,
$$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$
There is something I found curious about the unit arc which is that,
It has an empty interior viewed as a ...
- 179
13
votes
1
answer
306
views
Were 3-manifolds with $\sec>0$ known to be space forms before Ricci flow?
It is well known that R. Hamilton (JDG 1982) used Ricci flow to show that a closed $3$-manifold with positive Ricci curvature must be diffeomorphic to a spherical space form $S^3/\Gamma$, since such ...
- 5,891
11
votes
1
answer
584
views
Curvature of maximum of two riemannian metrics
Consider $g_1$ and $g_2$ two Riemannian metrics on a differentiable manifold $M$ of dimension $n\ge 4$. Suppose locally $g_i=f_i\sum_{j=1}^ndx_j^2$, where $f_i:M\rightarrow \mathbb{R}$ are non ...
user99087
11
votes
2
answers
2k
views
Retraction of a Riemannian manifold with boundary to its cut locus
This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus.
Recall that the cut locus of a set $S$ in a ...
- 28.9k
3
votes
1
answer
178
views
Sheaves on solenoids
Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...
user133532
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
4
votes
2
answers
410
views
Can we convert any non-vanishing vector field into geodesic field by changing metric?
If $M$ is a smooth closed manifold together with a non-vanishing (maybe unit) vector field $X$. In what condition can we construct a Riemannian metric on $M$ s.t $X$ be the geodesic field of on $TM$?
- 3,751
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
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
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
2
votes
1
answer
188
views
Extending metrics from $M =\mathbb{T}^2 \times (-\pi , \pi)$ to $ \mathbb{T}^3$
I would like to know of a similar result for the below but for the torus:
$\textbf{Cylinder to sphere rule:}$ Let $0< w \leq \infty$, and let $g$ be a metric on the topological
cylinder $(-w, w) \...
- 121
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)\}...
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
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
9
votes
2
answers
367
views
Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?
Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...
- 2,623
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
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
1
vote
2
answers
828
views
Handle body of 3-manifold with boundary
We know from Morse theory that smooth manifold(with or without boundary) is a handlebody.
However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, ...
- 481
1
vote
2
answers
425
views
Generalizations of Hopf-Rinow theorem
Let $(M,g)$ be a connected Riemannian manifold of dimension $n>1$. Then the Hopf-Rinow theorem states that $(M,g)$ is geodesically complete if and only if $(M,d_g)$ is complete as a metric space ($...
user99087
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
5
votes
1
answer
299
views
Can an open manifold with positive Ricci curvature be non simply connected at infinity?
The question is in the title, I haven't been able to locate a discussion of these kind of properties.
- 4,123