All Questions
Tagged with dg.differential-geometry differential-topology
759 questions
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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 ...
- 501
8
votes
1
answer
306
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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 ...
- 6,581
11
votes
2
answers
306
views
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 $M$ ...
- 2,792
3
votes
2
answers
247
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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 $C^2$-...
- 716
4
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0
answers
177
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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 ...
- 141
5
votes
0
answers
122
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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! ...
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1
vote
1
answer
94
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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 $\...
- 148k
15
votes
1
answer
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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 ...
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20
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4
answers
8k
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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 ...
- 12.9k
10
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1
answer
695
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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 ...
CommunityBot
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7
votes
7
answers
503
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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 ...
- 2,858
3
votes
2
answers
199
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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 ...
- 221
7
votes
1
answer
202
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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 ...
- 44.4k
0
votes
0
answers
42
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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)=\...
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2
votes
0
answers
82
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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
3
votes
1
answer
203
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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 ...
- 21.5k
5
votes
1
answer
393
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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 ...
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1
vote
0
answers
87
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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}\...
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1
vote
0
answers
240
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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 ...
- 356
5
votes
3
answers
1k
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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$. ...
- 176
4
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0
answers
116
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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 ...
- 501
0
votes
0
answers
133
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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 ...
- 12.9k
5
votes
1
answer
298
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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 ...
- 1,875
0
votes
0
answers
232
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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 ...
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39
votes
4
answers
9k
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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^...
CommunityBot
- 1
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 ...
- 501
2
votes
1
answer
241
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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 ...
- 12.8k
6
votes
0
answers
129
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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
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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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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 ...
- 383
0
votes
1
answer
91
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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 ...
- 39k
10
votes
1
answer
862
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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 ...
- 612
10
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2
answers
3k
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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(...
- 28k
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 ...
- 6,367
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?
- 29.1k
10
votes
2
answers
1k
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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 ...
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2
votes
1
answer
122
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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 ...
- 19.3k
2
votes
1
answer
108
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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 ...
- 16.2k
9
votes
2
answers
2k
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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 $\phi(x)= y$ ?
- 63.9k
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?
- 35
0
votes
1
answer
155
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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 ...
- 2,792
2
votes
0
answers
46
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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 $...
- 363
0
votes
0
answers
77
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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 ...
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1
answer
328
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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 ...
- 39k
3
votes
1
answer
200
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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 ...
- 5,596
2
votes
1
answer
320
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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 ...
- 44.4k
2
votes
1
answer
125
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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}...
- 173
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 ...
- 501
1
vote
1
answer
85
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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 ...
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