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Results tagged with differential-topology
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user 36688
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”.
2
votes
0
answers
82
views
Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\el …
- 356
5
votes
1
answer
385
views
No analytic surjection $f:M \to N$ when $\dim(M) >\dim(N)$
Inspired by comment discussions in this MO post smooth version of splitting principle we ask:
Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any analyti …
- 2,934
1
vote
0
answers
237
views
Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting p …
- 356
6
votes
2
answers
435
views
The convex hull of a manifold whose cobordism class is trivial
Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex hul …
- 6,367
4
votes
1
answer
451
views
The maximum number of vertical independent vector fields on the tangent bundle
Let $M$ be a differentiable manifold.
Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for th …
- 19.3k
2
votes
1
answer
249
views
Equidistant points on a compact Riemannian manifold
Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:
$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at leas …
- 10.6k
2
votes
0
answers
191
views
Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)
In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a generalization …
- 356
0
votes
1
answer
249
views
Invariant knot for finite group actions on $S^3$
Inspired by the Smith conjecture, is there a finite group action on $S^3$ (by smooth or analytic diffeomorphisms) which possesses an invariant knotted circle?
- 28.2k
0
votes
0
answers
71
views
Quasi Riemannian submersion and retraction
Let $M, N$ be Riemannian manifolds. A smooth submersion $f:M \to N$ is called a quasi Riemannian submersion if for every $x\in M$ the restriction of linear map $Df_x$ to orthogonal complement of $\k …
- 356
3
votes
1
answer
344
views
On functional equation $f\circ \exp=\exp \circ Df$ on a Riemannian manifold or a Lie Group
Let $M$ be a Riemannian manifold or a Lie group whose corresponding exp map (in corresponding context) is denoted by "exp" which is a map $\exp:TM\to M$
We search for the set $\mathcal{H …
- 356
3
votes
1
answer
274
views
Reduction of the structure group of $\mathbb{R}^n$-fiber bundles to a special subgroup of $\...
Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$
In other words, $G$ is the group of all equivariant self-homeomo …
- 9,580
1
vote
1
answer
345
views
A special non vanishing vector field on odd dimensional compact manifolds
Edit: According to the comment of Michael Albanese we revise the question.
Assume that $n$ is an odd integer and $M\subset \mathbb{R}^{2n}$ is a compact orientable $n$ dimensional submanifold.
Does …
- 1,111
1
vote
1
answer
144
views
Classification of all equivariant structure on the Möbius line bundles
Is there a classification of all equivariant structures of the Möbius line bundle $\ell\to S^1$?.
For example the antipodal action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ cannot be lifted to the total …
- 2,062
2
votes
Is this a submanifold?
For the obvious reflection action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ this set $\tilde S $ has two singular points at the zero vectors tangent at points $p=(1,0)$ and $q=(-1,0)$.
Then $\tilde S$ is …
- 356
1
vote
Are there some other notions of "curvature" which measure how space curves?
Assume that $M$ is a Riemannian manifold which is equipped with symplectic structure $\omega$.
Inspired by the definition of "Scalar curvature", one can define the quantity $tr_{\omega} Ric$ where $ …
- 30.3k