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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”.
21
votes
3
answers
2k
views
Manifolds with polynomial transition maps
Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.
What is an example of a m …
- 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
10
votes
1
answer
385
views
A symmetric embedding of manifolds
Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots x_{i},\l …
- 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
7
votes
1
answer
613
views
Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:
By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ whi …
- 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
6
votes
0
answers
291
views
Can we "Curve" a manifold, as much as possible?
Assume that $M$ is a $k$ dimensional manifold which is embedded in $\mathbb{R}^n$. We define the map $\phi_{M,n}: M \to G(k,n)$ with $\phi_{M,n} (x)= T_x M$, the tangent space to $M$ at point $x\in M$ …
- 356
5
votes
1
answer
372
views
A possible characterization of sphere or projective space
Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \qu …
- 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
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
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 …
- 356
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
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 …
- 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
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