Skip to main content

All Questions

Filter by
Sorted by
Tagged with
8 votes
0 answers
232 views
+300

Maps with small fibers between manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold. Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
7 votes
1 answer
300 views

Fibers of generic smooth maps between manifolds of equal dimension

I have heard that the following is a "well-known" Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
11 votes
2 answers
303 views
+500

Cohomology of foliations and closed forms along the leaves

Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ ...
3 votes
2 answers
247 views

Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
4 votes
0 answers
177 views

Basis of topology on space of properly embedded smooth manifolds

In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
5 votes
0 answers
122 views

Explicit parallelization of an exotic sphere

I asked this question on MathStackExchange a week ago (see here), but, despite a few upvotes, I received no comments or answers. Ideally, I would love a detailed answer, but a yes/no would do the job! ...
1 vote
1 answer
94 views

Linear combinations of smooth sections

I am dealing with smoothness issues for which I do not even know a successful approach, so any help or reference would be welcome. Let $M$ be a manifold, $E\to M$ a smooth vector bundle, and let $\...
15 votes
1 answer
2k views

When does a leaf space admit a (non-Hausdorff) manifold structure?

If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a ...
20 votes
4 answers
8k views

What is an immersed submanifold?

An immersed submanifold is by definition the image of a smooth immersion. I know some examples but I lack general understanding of what immersed submanifolds look like. For example, can one ...
10 votes
1 answer
695 views

Does the continuous mapping space between topological manifolds always admit a Banach manifold structure?

Let $M$ and $N$ be smooth, i.e. $C^\infty$, manifolds. Suppose that $M$ is compact. Then for every $k \geq 0$ it is well known that $$C^k(M,N)$$ admits the structure of a smooth Banach manifold. I am ...
7 votes
7 answers
503 views

Theorems similar to Tischler fibering theorem

Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other differential topology ...
3 votes
2 answers
199 views

Effect of a Lutz twist on Euler number

I already asked this question on the Math Stack Exchange but did not get an answer. I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be found here, and am trying to ...
7 votes
1 answer
201 views

Lipschitz bounds and homotopy groups of diffeomorphism groups

Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz ...
0 votes
0 answers
42 views

Fiber-wise mappings composed with projection map $\pi$

Let $M^2=(0,1)^2$. Recall that a chart is a diffeomorphism $\varphi:M^2 \to M^2$. Given a chart $\varphi:(M^2,g_0)\to (M^2,g_0)$ for $g_0$ the Euclidean metric, consider the curves $\varphi^{-1}(u,t)=\...
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: $$\...
3 votes
1 answer
203 views

Cohomology of the complex of differential forms with Schwartz coefficients

Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
5 votes
1 answer
393 views

No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$

Inspired by comment discussions in this MO post smooth version of splitting principle we ask: Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
1 vote
0 answers
87 views

A counterexample to the extendibility property for submanifolds

Let $(M,g)$ be a Riemannian manifold. We say that a bounded embedded submanifold $S\subset M$ of class $C^k$ has the extendibility property if there exists a larger embedded submanifold $\tilde{S}\...
1 vote
0 answers
240 views

Smooth version of the splitting principle

Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
5 votes
3 answers
1k views

Representation of fundamental group and flat connections

I read Differential Geometry Of Complex Vector Bundles by Kobayashi, and he says there that a vector bundle $E$ has flat connection is equivalent to $E$ being defined by a representation of $\pi_1$. ...
4 votes
0 answers
116 views

Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...
0 votes
0 answers
133 views

What do associated fibre bundles have in common?

Two fibre bundles are said associated if they have isomorphic associated principal bundles. I understand that this means they are defined by the same transition functions, but still is there some more ...
5 votes
1 answer
298 views

Schoenflies problem in PL setting

What is the status of the Schoenflies problem in the PL category? In other words, given an injective PL map $f:S^{n-1} \hookrightarrow S^n$, is it always PL equivalent to the equatorial inclusion? (I ...
0 votes
0 answers
232 views

What is the adjoint bundle of groups $P\times_{G}G$?

It is said that G acts on itself by conjugation. I am familiar with another type of adjoint bundle in which a representation of G on a vector space is given. Can someone explain the differences and ...
39 votes
4 answers
9k views

How to tackle the smooth Poincaré conjecture

The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^...
3 votes
0 answers
87 views

Are these contact structures on the open solid torus diffeomorphic?

Let $M=S^1\times \mathbb{R}^2$ and $\alpha_1, \alpha_2$ be a pair of contact one-forms on $M$ such that the restrictions $\alpha_1|_{S^1\times \{0\}}$, $\alpha_2|_{S^1\times \{0\}}$ coincide and ...
2 votes
1 answer
241 views

Leaf holonomy of Reeb foliation on Möbius strip

I am trying to understand the leaf holonomy of the Reeb foliation on the Möbius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...
6 votes
0 answers
129 views

Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum? I would be happy ...
4 votes
1 answer
179 views

Convex hull and least area discs in Riemannian 3-manifolds

$\DeclareMathOperator\Conv{Conv}$Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all ...
1 vote
1 answer
248 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
0 votes
0 answers
27 views

Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$

It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
0 votes
1 answer
91 views

On nontrapping manifolds

Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary. Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...
10 votes
1 answer
862 views

On the topology induced by a Lorentzian metric

Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread: Lorentzian distance induced topology(a.k.a. Interval topology) physicist @ValterMoretti ...
10 votes
2 answers
3k views

Gluing two diffeomorphisms together

A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have $\psi(...
3 votes
1 answer
367 views

Topology and local isometry, spinning cosmic string

Suppose one is given the spacetime $(M,g)$ where $M$ is a fixed differentiable manifold and $g$ is a Lorentzian metric whose local expression is: $$g= -(dt + a \, d \phi)^2 + d\rho^2 + \kappa^2 \rho^2 ...
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?
10 votes
2 answers
1k views

Odd differential forms

In de Rham's classical book "Variétés Différentiables" de Rham, Georges, Variétés différentiables. Formes, courants, formes harmoniques. 3e éd. revue et augmentée, Publications de l’Institut de ...
2 votes
1 answer
122 views

Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonnegative curvature operator

Let $S^m$ be a standard sphere of dimension $m=n+4k$, and let $M$ be any closed Riemaninan manifold of dimension $n$ with nonnegative curvature operator. My question: Is there always a smooth spin map ...
2 votes
1 answer
108 views

Finite group extensions of lattices

I'm currently reading the proof of Geroch's conjecture in Lawson-Michelsohn's Spin Geometry book and in the proof of Proposition IV.5.8 that every Ricci-flat enlargeable manifold is flat the following ...
9 votes
2 answers
2k views

Transitivity of automorphism group of smooth manifolds

Suppose $M$ is a connected smooth manifold and $x,y \in M$ are two points. Is there always a diffeomorphism $\phi: M \rightarrow M$ with $\phi(x)= y$ ?
2 votes
0 answers
96 views

Differential operators and iterations of tangent bundle

Is there a relationship between higher order differential operators and higher tangent bundle viewed as bundle on the base manifold?
0 votes
1 answer
155 views

Vector bundles over a homotopy-equivalent fibration

I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here. Let $\pi:N\rightarrow M$ be a smooth ...
2 votes
0 answers
46 views

Under what conditions principal directions define an integrable distribution?

Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
0 votes
0 answers
77 views

Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
-4 votes
1 answer
328 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
3 votes
1 answer
200 views

Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
2 votes
1 answer
320 views

How to chart tubes around manifolds with boundary/corners?

Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
2 votes
1 answer
125 views

Minimal dimension for immersions to be dense in the continuous function space

Let $f:[0,1]^n \rightarrow \mathbb{R}^m$ be an arbitrary continuous function. My question is, under what conditions for $m$, there exists an immersion $g:(-\epsilon,1+\epsilon)^n \rightarrow \mathbb{R}...
2 votes
0 answers
208 views

Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
1 vote
1 answer
85 views

When is the real Abel-Jacobi-Albanese map injective?

$ \def\tMA{\tilde{M}_a} \def\tomega{\tilde{\omega}} \def\tx{{\tilde{x}}} \def\tzeta{\tilde{\zeta}} \def\T{{\mathbb T}} \def\R{{\mathbb R}} \def\Z{{\mathbb Z}} \def\raw{\rightarrow} $ I want to work ...

1
2 3 4 5
16