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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”.
45
votes
Accepted
Are there some other notions of "curvature" which measure how space curves?
in addition to these excellent examples of non-local curvature quantities and their extensions to the non-smooth setting (which I am not sure the OP was anticipating), I might add the 'original' non-l …
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
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
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
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
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
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
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
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
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 …
- 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
10
votes
Accepted
Is the space of real conics with a singular point an orientable manifold?
Yes, it is a smooth manifold. No, it is not orientable.
For the first, just think geometrically, i.e., without bases: Fix a $3$-dimensional vector space $V$ and consider the homogeneous quadratic p …
- 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
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
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