All Questions
Tagged with dg.differential-geometry differential-topology
759 questions
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 ...
- 491
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 ...
- 491
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 $M$ ...
- 2,816
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 ...
- 141
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! ...
- 51
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 $\...
- 1,452
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 ...
- 1,174
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)=\...
- 383
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
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 ...
- 1,374
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 ...
- 356
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}\...
- 11
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 ...
- 356
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 ...
- 491
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 ...
- 35
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 ...
- 35
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 ...
- 1,875
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 ...
- 491
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 ...
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
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 ...
- 4,135
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 ...
- 612
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
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 ...
- 33
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 ...
- 563
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
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 ...
- 2,792
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 $...
- 363
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 ...
- 383
-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 ...
- 612
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 ...
- 620
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 ...
- 491
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 ...
- 491
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 ...
0
votes
0
answers
85
views
Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
1
vote
0
answers
153
views
Poincaré-Hopf Theorem for domains with a point of vanishing curvature
Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes.
I would like to know ...
- 111
5
votes
1
answer
415
views
A question about the existence of spin maps
Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following:
How can we construct a non-constant spin map $f:M\to N$ of degree zero?
Here spin map means that $f$ preserves ...
- 563
0
votes
1
answer
181
views
Formulating a 3D "twist" transformation for a unit circle into a lemniscate while preserving arc length
I'm exploring the transformation of a 2D unit circle into a lemniscate (infinity symbol) by fixing two antipodal points and "twisting" the circle (in the 3rd dimension) such that the ...
- 375
0
votes
1
answer
376
views
Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
- 447
11
votes
1
answer
701
views
Smooth map between oriented manifolds
Let $f: M\rightarrow N$ be a smooth map between smooth closed oriented connected manifolds of same dimension.
Question: is it true that $f$ is smoothly homotopic to some smooth map $g: M\rightarrow N$...
- 223
6
votes
2
answers
401
views
"canonical" framing of 3-manifolds
In Witten's 1989 QFT and Jones polynomial paper, he said
Although the tangent bundle of a three manifold can be trivialized, there is no canonical way to do this.
So if I understand correctly, ...
- 447
1
vote
0
answers
151
views
Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper,
he wrote in eq.2.22 that
Atiyah Patodi Singer theorem says that the combination:
$$
\frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi}
$$
is a ...
- 447
2
votes
0
answers
136
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
- 247
4
votes
1
answer
198
views
Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces
I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
- 6,025
5
votes
0
answers
211
views
$C^1$ manifold with complex structure
Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, ...
- 661
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
21
votes
1
answer
2k
views
Intuition behind manifolds which are homeomorphic but not diffeomorphic
Popular articles on mathematics often explain the difference between homeomorphism and diffeomorphism with statements like - "A rectangle is homeomorphic to the circle but not diffeomorphic to it&...
- 463
1
vote
0
answers
48
views
Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?
Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
0
votes
0
answers
183
views
Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
- 144
2
votes
0
answers
211
views
When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
- 293