Questions tagged [submersions]
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17
questions
4
votes
1answer
103 views
Analytically submersed manifold still immersed
I have asked this before on MSE, but received no answer yet.
Say I have a set in $\mathbb{R}^n$ defined to be the zero set of an analytic function $F:\mathbb{R}^n\to\mathbb{R}^k$, $k<n$. Everywhere,...
5
votes
1answer
88 views
Finding a volume form on a fibre of a submersion between oriented manifolds
Let $f:X\to Y$ be a submersion between orientable smooth manifolds of respective dimensions $n,p$ and let $j:M=f^{-1}(y)\hookrightarrow X$ denote the inclusion of some fibre of $f$.
My naïve (I am ...
1
vote
1answer
150 views
Quotient of $\mathbb{R}^n$ by a subgroup of $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$ Let $\mathcal{M}$ be an open subset of $\mathbb{R}^n$ endowed with the Euclidean metric and $\mathcal{N}$ be a Riemannian manifold. Assume that $G$ is a Lie subgroup of $\...
2
votes
0answers
73 views
Examples of bundles with minimal fibers
There is a result of Chen
https://link.springer.com/article/10.1007/s00605-007-0451-y
A Riemannian submersion $\pi : F \hookrightarrow (E,g) \to B$ with minimal fibers $F$ and such that $g$ has ...
0
votes
0answers
219 views
Density of sections of vector bundles
Let $X$ be a complex manifold, $f:V \to X$ be a (real) vector bundle and $g:V \to \mathbb{R}^n$ a submersion for some $n>0$ (here $n$ is not the same as the dimension of the fiber of the vector ...
2
votes
1answer
114 views
Do submersions induce open maps between spaces of differentiable maps?
Let $X$, $Y$ and $Z$ be smooth manifolds.
Any differentiable map $f \colon Y \rightarrow Z$ induces a continuous map $f_{\ast} \colon C^{\infty}(X, Y) \rightarrow C^{\infty}(X, Z)$ via composition $g \...
2
votes
1answer
160 views
Pushing forward a complex structure by submersion
I have a surjective smooth map with surjective differential between two balls $\phi:B^{2n}\rightarrow B^{2k}$. Fix an integrable almost complex structure $J$ on $B^{2n}$. Assume that $\mathrm{Ker}\:d\...
1
vote
1answer
75 views
Conditions for a pushforward of a involutive vector bundle to be involutive
I know that the following statement is true, but I would like to find a reference for it so I don't have to write the proof. Do you guys have a reference?
Let $\Omega$ and $\Omega'$ be smooth ...
3
votes
0answers
96 views
How restrictive is $[\mathcal H,\mathcal V] =0$, or weaker, $[X,A_XY] = 0$?
Let $\pi :P \to M$ be a Riemannian submersion. Let us denote by $\mathcal H$ the horizontal space of $\pi$ and by $\mathcal V$ its vertical space. I want to know restrictions and consequences of the ...
3
votes
0answers
202 views
A question about a paper of Bismut and Lebeau
Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
9
votes
2answers
348 views
Riemannian submersions from complex hyperbolic space into the hyperbolic space
Is there a (canonical) Riemannian submersion from the complex hyperbolic space $\mathbb C\mathbb H^n$ into the hyperbolic space $\mathbb H^n$?
In the affirmative case, what can we say about the ...
3
votes
0answers
102 views
Smooth submersions - smallest universal subclass of regular epimorphisms?
The smooth category has many problems. One is that pullbacks of regular epimorphisms need not exist.
However, pullbacks along submersions always exist. It also seems that submersions are universal (i....
2
votes
0answers
282 views
If the fibers of a submersion are connected, does it mean that any 2 sections are homotopic (locally on the base)?
Is the following fact known? If yes - what is the reference?
Let $\phi:X\to Y$ be a submersion of smooth manifolds with connected fibers.
Let $s_0,s_1:Y\to X$ be its (smooth) sections. Then, for any $...
3
votes
1answer
115 views
Reference request for a result regarding density of induced probability measure under a submersion
Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map ...
1
vote
1answer
299 views
In what sense is a generically submersive morphism of varieties subermersive over singular points?
Background/Motivation
I'm currently interested in the duality theorem for projective varieties and more specifically in properties of the conormal variety over the dual variety.
Let $V$ be a $k$-...
12
votes
1answer
732 views
What are the smooth manifolds in the topos of sheaves on a smooth manifold?
The category of internal locales in the Grothendieck topos of sheaves on a locale X
is equivalent to the slice category over X.
In other words, internal locales over X are precisely morphisms of ...
6
votes
5answers
2k views
Examples of Riemannian Submersions
Is there any example of a Riemannian submersion, which is no fibration?
As far as I know, a (any) submersion is locally, but not globally, given by a fibration. The converse holds globally. ...
