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Results tagged with differential-topology
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user 13972
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
4
votes
Accepted
Are there always flat connections?
Just so there'll be an answer: Whether every vector bundle over $G/\Gamma$ admits a flat connection depends on the group $G$ and the subgroup $\Gamma$.
For example, if $G=\mathrm{SU}(2)\simeq S^3$ an …
- 108k
11
votes
Accepted
Exterior differentiation of foliations
I think you might be confusing $L\subset T^*M$ with $L^\perp\subset TM$, since you can't compute the Lie bracket of sections of $L$, which are $1$-forms.
The condition that $L$ define a foliation is t …
- 2,858
12
votes
Accepted
Is it possible to define contact manifolds as manifolds with a G-structure?
A contact structure on $M^{2n+1}$ defines a $G$-structure (actually, it defines more than one, but there is a 'minimal' $G$-structure that is preserved by all contact transformations, and that is the …
- 108k
11
votes
Accepted
Why non closed differential forms do not play important role for the topology of a manifold?
Actually, the non-closed forms on manifolds play an essential role in the definition of Massey products, which are 'higher cohomology' operations.
Another place where they make an essential appearance …
- 108k
8
votes
Accepted
A vector field whose flow has constant singular values
The vector fields $X$ that satisfy this condition are highly constrained, as there exist only a finite-dimensional family of $C^5$ solutions in a neighborhood of any given point in the plane. Here is …
- 108k
7
votes
Accepted
Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group us...
Here is an argument in the orientable case: If $\omega$ is a $g$-parallel $p$-form on an orientable Riemannian $n$-manifold $(M^n,g)$, then it is closed since $\mathrm{d}\omega = \pi_{p+1}(\nabla^g\o …
- 108k
5
votes
Accepted
Reference for non-parallel harmonic $k$-forms
There are too many of these for them to have any particularly interesting structure. For example, consider any metric $g$ on the $3$-torus $\mathbb{T}^3$. By the Hodge theorem, the space of $g$-harm …
- 108k
15
votes
Accepted
Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
The answer is 'no, not necessarily'.
Consider the following example: Let $M=N=\mathbb{CP}^2$, let $(\omega_0,J_0)$ be the standard Fubini-Study Kähler structure on $M$. Now let $f$ be an arbitrary, …
- 108k
8
votes
Accepted
Non-symplectomorphic isometric compact Kähler manifolds
The answer to your first question is 'no' and the answer to your second question is 'yes'.
A simple example, when $n\ge 2$, is to let $M = \mathbb{R}^{2n}/\Lambda$ where $\Lambda\subset \mathbb{R}^{2n …
- 108k
13
votes
Accepted
Which curves are boundary of pseudoholomorphic curves?
The 'moment conditions' that Ben McKay mentions are simply this: A closed curve $C$ in $\mathbb{C}^n$ bounds a compact Riemann surface (which might be singular) if and only if the integral around $C$ …
- 108k
6
votes
Accepted
Underdetermined system of linear PDEs
Is there anything else that you are not telling us about $a$ and $b$? The particulars of these two vector-valued functions have a great influence on what the general solution of the system
$$
a\cdot …
- 108k
7
votes
Accepted
Is every map of rank smaller than r dominated by a constant rank map?
No. The simplest case is $M = S^1$ and $N = \mathbb{R}$. Then any nonconstant map $f:M\to N$ has rank at most 1, but there is no smooth map from $M$ to $N$ that has constant rank $1$.
The questio …
- 108k
25
votes
Accepted
Manifolds with polynomial transition maps
If I remember correctly, this is impossible for any (nonempty) simply-connected compact manifold of positive dimension. In particular, $S^2$ cannot have such an atlas. Off the top of my head, I don' …
- 108k
8
votes
Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?
Your motivating question has a negative answer: Consider the inclusion $\iota:S^m\to\mathbb{R}^{m+1}$, which has rank at most $m$. If there were a smooth extension $f:D^{m+1}\to\mathbb{R}^{m+1}$ of …
- 108k
2
votes
Accepted
Boundary of the image of a compact manifold in the complex plane
Here is one way to prove the claims about the image of the map $\mathrm{tr}:\mathrm{SU}(n)\to\mathbb{C}$. I'll just outline the steps. For simpicity, I'll always assume $n\ge 3$. (For completeness, …
- 108k