All Questions
Tagged with dg.differential-geometry differential-topology
759 questions
4
votes
0
answers
101
views
What are known properties of the boundary curves of J-holomorphic curve with boundary
Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...
- 285
4
votes
0
answers
178
views
The homotopy type of the space of symplectic structures
While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
- 516
4
votes
1
answer
293
views
Can every diffeomorphism be rescaled into a volume preserving one?
This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism.
Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...
- 45k
5
votes
1
answer
213
views
How to formalize this isotopy?
I'm studying the H-Cobordism theorem following the Lectures of John Milnor, and in the proof of the Whitney trick for cancel pairs of self-intersection points I have the next problem with an isotopy ...
- 9,580
4
votes
0
answers
435
views
When is a level set an immersed submanifold?
Let $M$ be a smooth manifold and $f\in C^\infty(M)$. Let $S:=f^{-1}(\{c\})$ for some $c\in \operatorname{Im}(f)\subseteq\mathbb{R}$. When does exist a manifold $N$ with $\dim(N)<\dim(M)$ and a ...
- 243
5
votes
1
answer
465
views
Is it true that every vector bundle over a non compact smooth manifold is trivial at infinity?
Let $M$ be a non compact smooth manifold and suppose that $\pi:E\rightarrow M$ is a vector bundle over it. Is there a compact subset $K\subset M$ such that the restricted bundle $\pi|_U:E|_U\...
- 4,111
11
votes
1
answer
360
views
Does the Lie algebra of vector fields $\mathfrak{X}(M)$ determine the diffeomorphism class of a manifold $M$?
Let $M_1,M_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M_1)\cong \mathfrak{X}(M_2)$ as Lie algebras.
...
- 66.3k
5
votes
0
answers
289
views
A certain kind of proof of the Hairy Ball Theorem
I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature.
From a given vector field $V_1$ on $S^2$, form another, $V_2$, ...
- 28.2k
4
votes
1
answer
334
views
Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief ...
- 4,713
5
votes
2
answers
769
views
Checking that the image of a curve is not contained in a hyperplane
Let $\gamma : [0,1] \to \mathbb R^n$ be a smooth curve, $n \geq 2$. I would like to find an easy to check condition such that the image of $\gamma$ is not contained in an $n-1$ dimensional linear ...
- 45k
94
votes
4
answers
15k
views
Can every manifold be given an analytic structure?
Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy ...
- 4,713
15
votes
2
answers
1k
views
Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?
A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.
Is there a citeable reference for a proof of this result?
For the sake of being definite, let's say that
“citeable” ...
- 45k
5
votes
0
answers
121
views
How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...
- 12.9k
13
votes
1
answer
661
views
Is an inextensible manifold necessarily compact?
Let $M$ be a connected $n$ dimensional boundary-less smooth manifold with the property that for any connected boundary-less $n$ dimensional manifold $\overline{M}$ and any embedding $i:M\rightarrow \...
- 12.9k
0
votes
1
answer
205
views
Smooth mapping from $\mathrm{RP}^2$ to $\mathbb{R}^3$ with nonsingular derivative
I'm trying to find a mapping $f$ from the 2D real projective plane to $\mathbb{R}^3$ which
is smooth
has non-singular directional derivative. That is,
$\forall x, v, \quad v \ne 0 \implies D_v f(x) \...
- 63.9k
5
votes
1
answer
407
views
Making a submanifold transverse to a vector field by an isotopy
Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known ...
- 501
10
votes
1
answer
569
views
Is every retraction homotopic to a smooth retraction?
I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one.
Let $M$ be a smooth $n$-...
- 1,040
2
votes
0
answers
222
views
On "graphs" of foliations
Let $M$ be a smooth manifold and $\mathcal{F}=\{\mathcal{F}_m\}_{m\in M}$ be a (regular) smooth foliation of $M$. The leaves $\mathcal{F}_m$ are smoothly immersed and moreover weakly embedded ...
- 501
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 ...
- 1,149
4
votes
0
answers
113
views
topological and $\mathscr{C}^{\infty}$-circle bundles
Let $M$ be a closed $\mathscr{C}^{\infty}$-manifold. Suppose that the underlying topological space of $M$ has a topological circle bundle structure $S^1\hookrightarrow M\to B$. Does $M$ admit a $\...
- 113
1
vote
1
answer
124
views
Relative equivariant Thom transversality
I'm looking for a reference for the following:
Suppose that $G$ is a finite group, that $M$ is a smooth $G$-manifold, and that $A\subseteq M$ is a closed $G$-invariant subspace of $M$ such that the ...
- 201
3
votes
0
answers
61
views
Mean curvature flow for non-Lipschitz initial data
It is known that mean curvature flow (MCF) can be used to smoothen out Lipschitz initial data. See Ecker and Huisken, Mean curvature evolution of entire graphsm Ann. Math. (2) 130 (1989), No. 3, 453-...
- 2,083
4
votes
0
answers
230
views
Trivial fibration over $S^1$ and closed 1-form
Tischler's theorem says that a closed differentiable manifold $M$ has a nondegenerate real closed 1-form if and only if $M$ is a fiber bundle over the circle $S^1$.
Except for the condition "...
- 41
-2
votes
1
answer
189
views
Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
- 521
1
vote
1
answer
224
views
Decomposing proper map into closed embedding and proper submersion
Suppose that $f: X \to Y$ is a smooth proper map between two smooth manifolds. Is it always possible to represent $f$ as a composition of a closed embedding $g: X \to Z$ with a proper submersion $h: ...
- 55.9k
13
votes
1
answer
1k
views
Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks?
In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of generalized smooth spaces ...
- 37.8k
6
votes
1
answer
961
views
Can a smooth manifold be realised as the image of a smooth function?
Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$?
$X$ may have points at which the ...
- 10.6k
5
votes
2
answers
249
views
Patching up two trivial fibre bundles induces homology equivalence
I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.
Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...
- 959
5
votes
0
answers
133
views
On the vertical cohomology of a fibered manifold
Let $\pi:Y\rightarrow X$ be a $C^\infty$ fibered manifold (all constructions, unless otherwise stated are over the smooth manifold category) with $\Omega_k$ the sheaf of (smooth) $k$-forms on $Y$.
...
- 2,792
8
votes
1
answer
386
views
Is it possible to define contact manifolds as manifolds with a G-structure?
Many geometries (Riemannian, symplectic, complex, Kähler, Calabi-Yau) can be defined as categories of G-structures on manifolds with the first integrability condition (zeroing of torsion of G-...
- 6,962
7
votes
0
answers
270
views
The Todd class and Weyl's character formula
Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...
- 1,379
10
votes
2
answers
2k
views
Parallelizability of 3-manifolds
Robert Bryant's answer here ( https://mathoverflow.net/a/149496/85500 ) states that any orientable 3-manifold is parallelizable.
Previously I was under the impression that only closed (compact & ...
- 3,146
2
votes
0
answers
45
views
Mean curvature flow starting from a wildly embedded 2-sphere
Let $\Sigma$ be a wildly embedded 2-sphere in 3-sphere $S^3$. For simplicity, we may assume that $\Sigma$ is the Alexander horned sphere.
Question. Can we define the mean curvature flow (MCF) ...
- 2,083
3
votes
1
answer
118
views
Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)
Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the ...
- 6,367
0
votes
0
answers
56
views
Embedding and compactness of thickened family of graphs
Let $\{G_n\}_\mathbb{N}$ be a family of graphs $G_n=(V_n,E_n)$ for which $|V_n|$ and $|E_n|$ tend to infinity. I would like to know if the family of objects $\{M_n\}_\mathbb{N}$ obtained by thickening ...
1
vote
0
answers
64
views
dimension of fibre of a generic point in an intersection of two sets
Let $M_m := (f_1, \cdots, f_m )$ be an algebraic map from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $f_1^2,...,f_m^2$ are homogeneous polynomials of the same degree in $Q[x_1,...,x_n]$ . Similarly define $...
- 263
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 ...
- 27.5k
6
votes
3
answers
603
views
Flat regions on surfaces of genus greater than 1
Here is a polygonal disk + gluing scheme model of a surface we are attempting to construct. We want the regions of the surface bounded by the two vertical, dotted lines $\alpha,\beta$ to have zero ...
- 2,821
1
vote
0
answers
139
views
Cartesian product of spin manifolds [closed]
Is it true that the Cartesian product of two spin manifolds is spin?
4
votes
1
answer
131
views
Smooth circle actions on Riemannian manifolds and harmonicity of quotient map
Let $(M, g)$ be a compact Riemannian manifold. Suppose $M$ admits a smooth free circle action (Denote the circle group by $G$. The action $G$ on $M$ is not necessarily isometric) and the orbit space $...
- 7,583
9
votes
1
answer
517
views
Is the diffeomorphism group of the Euclidean space generated by two simpler subgroups?
Consider the group $\mathrm{Diff}(\mathbb R^n)$ of smooth diffeomorphisms. It has two interesting subgroups:
the orthogonal group $O(n)$,
the group of "diffeomorphisms applied along each axis&...
- 251
1
vote
0
answers
70
views
What *piecewise* smooth curves/surfaces/hypersurfaces give rise to forward-invariant regions of dynamical systems?
Consider a set $\mathcal{B}\subset \mathbb{R}^n$ that is homeomorphic to a closed n-dimensional ball, and denote its boundary by $\mathcal{H}$. Assume that $\mathcal{H}$ is a "piecewise smooth&...
- 111
6
votes
4
answers
3k
views
Does every smooth manifold of infinite topological type admit a complete Riemannian metric?
To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric ...
- 2,821
14
votes
2
answers
2k
views
Pairing used in Lefschetz duality
I am thinking about the precise formulation of the Lefschetz duality for the relative cohomology. If I understand this Wikipedia article correctly, there is an isomorphism between $H^k(M, \partial M)$...
- 1,803
14
votes
1
answer
681
views
When does an open manifold admit two linearly independent vector fields?
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$\DeclareMathOperator{\co}{H}$
$\newcommand{\kk}{\mathbb{F}}$
$\newcommand{\qq}{\mathbb{Q}}$
$\newcommand{\zz}{\mathbb{Z}}$
$\newcommand{\rr}{\mathbb{R}}$
$\...
- 1,726
41
votes
3
answers
3k
views
Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?
First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ ...
- 4,713
8
votes
0
answers
409
views
What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?
In his paper [2], Paul Ehrlich write
In [1], Aubin stated a theorem which implied as a corollary that if a manifold
$M$ admits a Riemannian metric with nonnegative Ricci curvature and
all Ricci ...
- 4,195
1
vote
0
answers
105
views
Codimension of cusp singularities in the space of 2-jets
In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem.
For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\...
- 2,533
9
votes
1
answer
481
views
Existence of a vector field with a finite number of limit cycles.
The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
- 1,446
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 ...
- 26.3k