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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”.
11
votes
Accepted
Gauss-Bonnet invariant Ω: explicit intrinsic expression for Π in Ω=dΠ?
I see that the OP may not be entirely convinced by my comments, so let me try this, which may help. It's understandable that reading the older literature can be confusing; the classical language is o …
- 108k
10
votes
Accepted
Existence of integrals of 1-forms up to multiples
Well, the local condition that $\omega\not=0$ be a nonzero multiple of an closed $1$-form is that $\omega\wedge d\omega = 0$. This is necessary and sufficient for the local existence of functions $f$ …
- 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
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
34
votes
Accepted
On the generalized Gauss-Bonnet theorem
When $E\to M$ is an oriented vector bundle of rank $2n$ over a
compact manifold $M$, it has a well-defined de Rham Euler class $e(E)$
in $H^{2n}_{dR}(M)$, and a representative $2n$-form for $e(E)$
…
- 108k
10
votes
Classification of natural invariants of Riemannian structures
You are really asking a question about invariant theory applied to the curvature tensor and its covariant derivatives. The case of scalar invariants (of any weight) was, in principle, worked out by W …
- 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
23
votes
Accepted
existence of totally geodesic hypersurfaces
You should be aware that, for $n\ge3$, the generic Riemannian metric $(M^n,g)$ has no totally geodesic hypersurfaces at all, even locally. Typically, when they do exist, it is for some geometric reas …
- 108k
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
Formulating the calculus of varations with exterior calculus
There is a large literature on this, and the roots go back more than one hundred years. Some of the modern work along these lines can be found by looking for papers containing the term 'variational b …
- 108k
5
votes
Accepted
On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden
The answer is 'no'. To see why, just take any nondegenerate $2$-form $\omega$ on, say, $\mathbb{R}^6$, that has the property that $d\omega$ is not a multiple of $\omega$. (This will be true for a ge …
- 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
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
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
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