All Questions
Tagged with dg.differential-geometry differential-topology
759 questions
2
votes
0
answers
39
views
Can a multivariable mapping that is linear in each variable separately have a local extrema?
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$, $m<n$ be a mapping that is linear in each variable separately (i.e., in each of the functions $f_i(x_1,\cdots,x_n)$, $1\leq i\leq m$, the degree of ...
- 503
7
votes
2
answers
562
views
Is the union of a compact and the relatively compact components of its complementary in a manifold compact?
I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is ...
- 10.6k
2
votes
0
answers
74
views
Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
- 6,962
1
vote
0
answers
93
views
A question about Homotopy equivalence (II)
I posted a similar but different question before in the link
https://math.stackexchange.com/questions/4311982/why-does-x-0-times-s1-simeq-x-x-0/4312530?noredirect=1#comment8987557_4312530.
Now, my new ...
- 563
0
votes
1
answer
154
views
Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 563
8
votes
1
answer
426
views
Orbifolds are Thom-Mather stratified spaces
Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?
edit: after some search, I found the proof should be contained in either
GIBSON, C....
- 803
4
votes
1
answer
305
views
Classification of functorial smooth vector fiber bundles
Let $\mathrm{Bundle}$ be the category whose objects are smooth vector fiber bundles over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let ...
- 6,962
4
votes
0
answers
893
views
Submersion and fiber bundle
It is known by Ehresmann's result, that proper surjective submersions are fiber bundles. The properness of a map somehow relates to the compactness of the fibers or the level sets. So my question is ...
- 1,177
3
votes
2
answers
604
views
Calculation of the top Chern class of spinor bundle over $S^{2n}$
It's well known that for a complex vector bundle $E$, we have
$$c_n(E)=e_n(E_\mathbb{R}) $$
But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
- 563
7
votes
1
answer
246
views
Currents in sub-Riemannian geometry
Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The ...
- 215
6
votes
0
answers
302
views
Are there are any surprising diffeomorphisms?
Two smooth manifolds are often viewed to be equivalent if there is a diffeomorphism between them. Are there examples of two manifolds that one would not expect to be equivalent (in this sense), but in ...
- 1,379
1
vote
0
answers
194
views
Existence of Morse function on suspension
Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...
- 61
6
votes
0
answers
516
views
Yang–Mills existence and mass gap official statement on Euclidean $\mathbb{R}^4$, why not Minkowski $ \mathbb{R}^{3,1}$?
Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute:
Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial ...
- 10.5k
1
vote
1
answer
69
views
Generalisation of conservative covector fields
The following theorem is well-known:
Theorem. Let $M$ be a smooth manifold. A smooth covector field $\omega \in \Omega^1(M)$ is conservative, that is, $$\int_{\mathbb{S}^1}f^*\omega = 0 \qquad \...
- 548
5
votes
1
answer
408
views
A question related to fiber bundle
Let $f:\mathbb{C}^3 \to \mathbb{C}$ be a morphism of varieties such that it is a smooth fiber bundle. Can I say that the fiber is $\mathbb{C}^2$?
- 1,177
3
votes
6
answers
2k
views
The purpose of connections in differential geometry [closed]
I am currently reading through differential geometry as a mathematics graduate.
Can somebody give me a brief explainer on the purpose of connections?
I could also use explainers on differential forms. ...
4
votes
1
answer
364
views
When is a smooth field's flow map volume preserving diffeomorphism
Let $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a $C^{\infty}$ vector field. Fix a (single) real number $d$ such that
$$
1\leq d\leq n
.
$$
Under what conditions is the flow map $\Phi_V$ defined as ...
- 43
4
votes
1
answer
304
views
Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for $n=2k+1$
Atiyah, Bott, and Shapiro paper on Clifford Modules around page 10 shows two facts.
1 - There is a lift $U(n) \to Spin^c(2n)$ from $U(n) \to SO(2n)\times U(1)$. Also an embedding (injective group ...
- 317
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 $C^2$-...
- 91
0
votes
1
answer
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
0
votes
0
answers
81
views
Continuous transformation of a connected set
Let $1\leq m\leq n$ and $\mathcal{O}$ is an open set of $\mathbb{R}^n$. Let $f\in\mathcal{C}^1(\mathcal{O},\mathbb{R}^m)$, such that the differential of $f$ at any $x\in\mathcal{O}$ is surjective.
...
- 449
3
votes
1
answer
3k
views
What's the point of differential geometry? [closed]
I've been self studying differential geometry for a little while now (4-6 months). I am learning from Lee's Introduction to Smooth Manifolds, and I just don't quite get the point of the subject. Why ...
- 349
1
vote
1
answer
649
views
Local diffeomorphisms, covering maps and smooth path lifting
Let $f: M\to N$ be a surjective local diffeomorphism of noncompact smooth manifolds.
Suppose that every smooth path is liftable, that is, for any smooth path $\gamma: [0,1]\to N$ and any point $p\in f^...
- 2,934
8
votes
1
answer
176
views
Analog of Cerf theory in PL
Is there an analog of Cerf theory in PL?
More specifically, given two handle decompositions of a PL (relative) cobordism $W$, is it always possible to go from one handle decomposition to the other via ...
- 186
6
votes
0
answers
167
views
Elliptic operators with with same index but non homotopic symbols
Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$.
Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold.
In Atiyah-Singer "the index of ...
- 2,533
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 ...
4
votes
0
answers
121
views
planar linkage isomorphic to an exotic sphere
I recently came across this paper, which showed that any compact smooth manifold is diffeomorphic to a connected component of the moduli space of a planar linkage.
Briefly, if we have an undirected ...
- 840
2
votes
2
answers
523
views
Orthogonal smooth vector field on a Riemannian manifold
Consider a compact Riemannian manifold $M$ with a smooth metric, and a smooth vector field $X$ on $M$. My question is, when can we construct another smooth vector field $Y$ on $M$ such that $Y$ is ...
- 1,407
4
votes
1
answer
899
views
The Yang-Mills Higgs Lagrangian
Let's say we have a principal bundle $(P,B,\pi;G)$ and associated bundle $E=P \times_{(G,\rho)}V$and $Ad(P)=P\times_{(G,Ad)} \mathfrak{g}$ the adjoint bundle. The Yang-Mills-Higgs action (without ...
- 247
2
votes
1
answer
371
views
Smooth submanifold of a complex manifold with invariant tangent space under multiplication by $i$
Let $M$ be a complex manifold, $N$ is a smooth immersed submanifold of $M$. If $T_p M$ is invariant under the multiplication by $i$ for any $p\in M$, then can we conclude that $N$ is a complex ...
- 661
2
votes
1
answer
676
views
Why non closed differential forms do not play important role for the topology of a manifold?
Cross-posted from MSE.
I know that De Rham cohomology reveal some properties of the topology of smooth manifolds by finding closed differential $k$-forms $\mathsf{d}\omega=0$ that are not exact $\...
- 4,195
17
votes
8
answers
4k
views
Books in advanced differential topology
I am looking for books or other sources in differential topology that include topics like: vector bundles, fibration, cobordism, and other related topics.
In general, if anyone has recommendation of ...
- 179
4
votes
0
answers
169
views
Compactly supported geometric representatives for Seiberg-Witten invariant
The question is introduced at the end of the second paragraph.
Readers familiar with Seiberg-Witten theory may well skip the first paragraph.
The first paragraph is meant to set up some notation which ...
- 1,040
5
votes
1
answer
1k
views
Curvature of principal bundle
Let $(P,M,G)$ be a principal bundle with connection 1-form $\omega$. In all books I have seen so far, the curvature is defined by
\begin{equation}
F:=D_{\omega}\omega \in \Omega({P,\mathfrak{g}})
\end{...
- 247
12
votes
1
answer
1k
views
Smoothness of distance function to a compact set
Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$.
Suppose that:
$K$ is regular : ...
- 5,405
3
votes
1
answer
507
views
Local coordinates of one form on a principal bundle
I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.
Let's say ...
- 247
3
votes
0
answers
125
views
Restrictions on pointed lifts of isometries
Let $M$ be a (closed) Riemannian manifold and let $f$ be an isometry of $M$ fixing a point $\ast \in M$ that acts trivially on $\Gamma := \pi_1(M,\ast)$.
Then there is a unique isometry $\tilde{f}$ of ...
- 11.9k
2
votes
0
answers
255
views
Extending an embedding with trivial normal bundle
I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...
- 131
19
votes
1
answer
989
views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
- 293
1
vote
0
answers
170
views
Uniqueness of collar neighborhoods for non-compact boundary case in smooth setting
Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times
\partial M \to M$ be two smooth embeddings that are the identity map
on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is
...
- 265
8
votes
1
answer
393
views
Non orientable, closed manifold covered by two contractible charts
This is a follow up of my previous MO question "Non orientable, closed manifold covered by two simply-connected charts." Nick L's nice answer shows that such manifolds actually exist, ...
- 66.3k
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 ...
- 4,195
8
votes
2
answers
775
views
Godbillon–Vey invariant and leaf space of foliations
I recently got to know about the existence of the so-called Godbillon–Vey invariant, and I am interested in its relationship with foliation theory in 3-manifolds. I briefly recall here the definition:
...
- 521
2
votes
1
answer
560
views
Collar neighborhood theorem for manifold with corners
I was reading this wonderful sequence of posts:
nlab: manifold with boundary
and nlab: collar neighbourhood theorem
and I couldn't help but wonder. Is there an extension of the Collar neighborhood ...
- 5,405
4
votes
1
answer
211
views
Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology
Cross-post from MSE.
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. ...
- 4,195
4
votes
0
answers
142
views
Classification of square roots of line bundles and metalinear/metaplectic structures
Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...
1
vote
0
answers
151
views
Density of $G$-invariant morse functions
Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
- 99
0
votes
0
answers
55
views
Antisymmetric tensor coordinates and tensorial spaces
I am currently working on some geometric aspects of higher-spin models for which there appear antisymmetric tensor coordinates
$X^{\mu\nu}=-X^{\nu\mu}$,
with $\mu,\nu=1,...,N$,
which have been ...
- 405
1
vote
1
answer
243
views
Reference for non-parallel harmonic $k$-forms
I want to get some deep understanding on closed orientable Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions:
$$\nabla \omega\neq 0,\quad \Delta\...
- 4,195
3
votes
0
answers
113
views
Spin structures induced on embedded circles and choices of trivialisations
I have a presumably basic question concerning spin structures that has me a bit confused.
Let $C$ be a circle embedded in a spun manifold $X^n$. Given a choice of trivialisation of the normal bundle ...
- 5,349