All Questions
76 questions
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
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, \...
- 576
16
votes
2
answers
605
views
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
- 63.9k
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 ...
- 15.4k
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) \...
- 856
16
votes
3
answers
1k
views
SO(3) action on (simply connected) 6 manifold with discrete fixed point
If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
- 6,249
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, \...
- 13.6k
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
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)\}...
- 26.3k
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
14
votes
2
answers
787
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$ ...
- 68.9k
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 ...
- 96.4k
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 ...
- 26.3k
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 ...
- 31.5k
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 ...
- 25k
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 ...
CommunityBot
- 1
1
vote
2
answers
427
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 ($...
CommunityBot
- 1
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$?
- 96.4k
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 ...
CommunityBot
- 1
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$. ...
- 108k
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, ...
- 12.8k
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 ...
- 44.8k
14
votes
2
answers
1k
views
Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?
I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...
- 8,717
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.
- 29.1k
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\,,$$...
- 25.3k
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 ...
- 10.6k