All Questions
59 questions
1
vote
0
answers
119
views
Orbits and indices of vector fields
I'm afraid this might be an exercise in differential topology (in which case a reference to a book where it is would be very much appreciated); apologies in advance. Given an analytic vector field (in ...
- 4,432
2
votes
1
answer
551
views
Heisenberg group: research themes
I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...
- 31
2
votes
0
answers
260
views
Perturbation of Morse function at a critical point
I recently learned from a knowledgeable person that for a Morse function $f: M \to R$ with a critical point $x_0$, one can perturb $f$ in such a fashion that the new function has the same critical ...
- 1,211
36
votes
2
answers
5k
views
Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
- 151k
3
votes
3
answers
2k
views
How do we use an Ehresmann connection to define a semispray?
Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
- 8,512
15
votes
1
answer
2k
views
Good introduction to Morse-Novikov theory?
Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...
- 22.1k
8
votes
3
answers
1k
views
Higher derivatives than Jacobi fields
The first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
- 516
4
votes
0
answers
162
views
Question about density of $C^{\infty}(M,N)$ in $W^{1,p}(M,N)$ with $N$ not compact
Hi!
Let $M$ be a compact manifold possibly with boundary with $\dim(M)=m$, let $N$ be a non compact manifold with $\dim(N)=n$. Let me recall the definition of the sobolev space $W^{1,p}(M,N)$. ...
- 1,727
33
votes
4
answers
7k
views
Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
- 33.3k