All Questions
76 questions
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
7
votes
2
answers
614
views
Locally conformally flat
Is there any example of a locally conformally flat manifold that is neither a space form nor a product of space forms?
- 71
3
votes
2
answers
265
views
For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
- 1,467
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 ...
- 587
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
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
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
1
vote
2
answers
148
views
Construct a hypersurface with fixed principal curvatures at a point
I'm reading Eschenburg's paper Local convexity and nonnegative curvature —
Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to ...
- 241
2
votes
1
answer
119
views
Density of smooth bi-Lipschitz maps in smooth maps
Setup/Motivation:
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
- 195
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}}$$
where $x,y,z$ is ...
- 121
2
votes
0
answers
108
views
Questions about symmetric spaces
I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
- 563
5
votes
0
answers
132
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 4,081
3
votes
1
answer
425
views
3 dimensional solvmanifolds and Thurston geometries
Does every three dimensional compact solvmanifold admit either Euclidean, nil, or sol geometry?
definitions/motivation/background:
A solvmanifold is a manifold $ M $ admitting a transitive action by a ...
- 4,081
2
votes
1
answer
484
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 4,081
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
8
votes
1
answer
599
views
Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
- 4,081
2
votes
1
answer
138
views
noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous
Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive ($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant ...
- 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
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
0
votes
1
answer
154
views
Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 563
3
votes
2
answers
604
views
Calculation of the top Chern class of spinor bundle over $S^{2n}$
It's well known that for a complex vector bundle $E$, we have
$$c_n(E)=e_n(E_\mathbb{R}) $$
But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
- 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
1
vote
0
answers
236
views
Books and References on Geometry of Submanifold [closed]
In this semester I want to study Geometry of Submanifolds. I know Chen Bang Yen's book: Geometry of submanifolds, but it is too hard to read since its strange print. Can people recommend textbooks and/...
- 47
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
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
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
2
votes
1
answer
888
views
vector field on the torus [closed]
Is the following statement true?
Let $T$ be diffeomorphic to the solid torus. Let $v$ be a vector field such that $v$ and $curl(v)$ are both tangent to $\partial T$ everywhere and $|v|$ is constant ...
- 61
5
votes
1
answer
389
views
Lengths of closed geodesics on a flat vs hyperbolic punctured torus
Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$.
Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f:...
- 10.7k
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
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
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
8
votes
1
answer
311
views
Laplacian spectrum asymptotics in neck stretching
Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
- 1,207
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
1
answer
233
views
Flat solvmanifolds?
I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...
- 481
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
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 $P$ ...
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
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, \...
- 356
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
36
votes
7
answers
5k
views
Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?
I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free.
The idea is to use ...
- 395
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
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
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
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