All Questions
Tagged with dg.differential-geometry differential-topology
759 questions
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 ...