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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: $$\...
Ali Taghavi's user avatar
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?
Sayoojya's user avatar
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 ...
Learning math's user avatar
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 ...
Zhenhua Liu's user avatar
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 ...
Leo's user avatar
  • 395
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 ...
Radeha Longa's user avatar
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-...
John McManus's user avatar
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 ...
Radeha Longa's user avatar
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 ...
eulershi's user avatar
  • 241
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 ...
No-one's user avatar
  • 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 ...
Radeha Longa's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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(\...
Ian Gershon Teixeira's user avatar
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 ...
Carlos_Petterson's user avatar
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 ...
Radeha Longa's user avatar
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 ...
user473085's user avatar
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 $. ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 $P$ ...
A random mathematician's user avatar
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\...
Radeha Longa's user avatar
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}\ ...
Radeha Longa's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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 \...
Phillip's user avatar
  • 131
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 ...
En Poverty's user avatar
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 ...
Radeha Longa's user avatar
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 ...
Radeha Longa's user avatar
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), ...
Louis Esperet's user avatar
27 votes
2 answers
3k views

Is there a Chern-Gauss-Bonnet theorem for orbifolds?

There's a Gauss-Bonnet theorem for compact 2-orbifolds(due to Satake, I think), which gives a relation between the curvature of a Riemannian orbifold and the orbifold topology(i.e. taking into account ...
Gordon Craig's user avatar
  • 1,665
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 $\mathbb E^n$. ...
Tim Campion's user avatar
  • 63.9k
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 ...
Yuhang Liu's user avatar
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\...
GSM's user avatar
  • 223
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/...
管山林's user avatar
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 $(n-1)$-...
Shijie Gu's user avatar
  • 2,083
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 ...
J. Doee's user avatar
  • 61
3 votes
1 answer
424 views

Riemann Hurwitz vs Gauss Bonnet

The Gauss-Bonnet theorem implies the Riemann Hurwitz theorem http://sma.epfl.ch/~troyanov/Papers/Prescribing.pdf Prop 1 => Cor 2 In what sense is the Gauss- Bonnet theorem stronger? Are these ...
Arielle Leitner's user avatar
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....
Nuxil's user avatar
  • 73
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:...
Will Chen's user avatar
  • 10.7k
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 $Y$ ...
user44651's user avatar
  • 1,069
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 ...
asv's user avatar
  • 21.8k
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 ...
Carl's user avatar
  • 619
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, \...
Ali Taghavi's user avatar
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 ...
Jess Boling's user avatar
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 ...
user82786's user avatar
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 $x^n>0$ ...
Ali's user avatar
  • 4,135
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....
Guangbo Xu's user avatar
  • 1,207
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 ...
aceituna's user avatar
  • 121
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 ...
user60933's user avatar
  • 481