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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”.
4
votes
2
answers
165
views
functions which covers(good covers) manifolds
Let $M$ be a (not necessarily compact)) smooth manifold.
1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U_{\alpha}$ such that each $f^{-1}(U_{\alpha})$ i …
- 356
2
votes
1
answer
246
views
A polynomial function on $\mathbb{R}^3$ whose all level sets are mutually non isometric Riem...
Is there a polynomial function $P: \mathbb{R}^3 \to \mathbb{R}$ with the following property?:
P does not have any critical value and for all $c \neq c'$, $f^{-1}(c)$ and $f^{-1}(c')$ are non …
- 356
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 …
- 356
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 …
- 356
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
11
votes
Accepted
Are there vector fields which are gradients with respect to one metric but not another?
Consider the vector field $$X=(y-10x)\partial_x-x\partial_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient …
- 356
4
votes
0
answers
149
views
Higher tangent bundles of manifolds with non integer dimension
One way to define the tangent space of a manifold at a point $p\in M$ is the following: We define an equivalent relation on the space of curves passing $p$ as follows: Two curves $\alpha, \beta$ are …
- 356
5
votes
2
answers
448
views
Is $TS^n$ diffeomorphic to an open subset of $\mathbb{R}^{2n}$
For what values of $n \neq 1,3,7$ is the tangent bundle $TS^n$ of the $n$-sphere diffeomorphic to an open subset of $\mathbb{R}^{2n}$?
- 356
10
votes
2
answers
347
views
A quantity associated with a smooth groupoid
Assume that $(G,G^0,r,s)$ is a smooth groupoid such that $G$ is a compact connected manifold.
The graph of "source" and "range" maps $s, r: G \to G^0$ are compact submanifolds $S$ and $R$ of $G\times …
- 356
2
votes
1
answer
655
views
Non vanishing vector field on torus in the form of $ad_X (Y)$
Assume that $X$ is a non vanishing vector field on torus $\mathbb{T}^2$.
Is there a vector field $Y$ such that $[X,Y]$ does not vanish on torus?
What about if we replace torus by an arbitrary compac …
- 356
5
votes
2
answers
384
views
Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces
I apologize in advance if this question has an obvious answer.
Let $(M,g)$ be a Riemannian manifold.
Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g …
- 356
3
votes
1
answer
137
views
A geometric property about certain polynomials in two variables
Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$
where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last …
- 356
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
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 …
- 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 …
- 356