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Submersion function from a product space

Let $\Phi(x,y) \colon U_N \times U_M \to \mathbb{C}^n$ be a submersion, where $U_N \subset \mathbb{C}^N$ and $U_M \subset \mathbb{C}^M$. Under which condition on $\Phi$ can I find some $s \in \...
Serge the Toaster's user avatar
9 votes
1 answer

Submersion vs fiber bundle

If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is ...
piper1967's user avatar
  • 1,059
5 votes
2 answers

When is a real-analytic variety a union of non-singular subvarieties?

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,...
nicrot000's user avatar
  • 212
5 votes
1 answer

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 ...
Georges Elencwajg's user avatar
1 vote
1 answer

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 $\...
Chevallier's user avatar
2 votes
0 answers

Examples of bundles with minimal fibers

There is a result of Chen A Riemannian submersion $\pi : F \hookrightarrow (E,g) \to B$ with minimal fibers $F$ and such that $g$ has ...
L.F. Cavenaghi's user avatar
0 votes
0 answers

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 ...
Ron's user avatar
  • 2,126
2 votes
1 answer

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 \...
Rahmpilz's user avatar
  • 165
2 votes
1 answer

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\...
lolo's user avatar
  • 23
1 vote
1 answer

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 ...
Max Reinhold Jahnke's user avatar
3 votes
0 answers

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 ...
L.F. Cavenaghi's user avatar
3 votes
0 answers

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 $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
DLIN's user avatar
  • 1,915
9 votes
2 answers

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 ...
Marcos Petrúcio Cavalcante's user avatar
3 votes
0 answers

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....
Arrow's user avatar
  • 10.3k
2 votes
0 answers

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 $...
Rami's user avatar
  • 2,591
3 votes
1 answer

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 ...
John Jiang's user avatar
  • 4,354
13 votes
3 answers

What is the characteristic property of surjective submersions?

In Lee's Introduction to smooth manifolds he states that given smooth manifolds $X,Y$ and a surjective submersion, $f:X \rightarrow Y$, then $f$ is a smoothly final map, that is for any further smooth ...
Mozibur Ullah's user avatar
1 vote
1 answer

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 $k$-...
Rainer Sinn's user avatar
17 votes
1 answer

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 ...
Dmitri Pavlov's user avatar
6 votes
5 answers

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. ...
Henry Wegener's user avatar