Questions tagged [transversality]
The transversality tag has no summary.
26 questions
14
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2
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How to prove that a generic Riemannian metric does not admit nontrivial local isometries
Definition: Let $(M,g)$ be a Riemannian manifold. We say that two points $p,q\in M$ are locally isometric iff there exists an open set $U$ around $p$, an open set $V$ around $q$, and an isometry $\phi:...
- 1,441
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82
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Smooth vs. topological: foliation into closures of orbits
Consider a (partial) map $f\colon X → X$ and the maximal closures of orbits of $f$ (i.e., closures of orbits which are not contained in larger closures of orbits). Assume that $X$ is foliated into ...
- 499
4
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157
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Relative Thom transversality and the D-topology
Suppose we are given a smooth manifold $M$ and, for the sake of simplicity, some compact submanifold $L\subseteq M$ of the same dimension, as well as $f\in C^{\infty}(M,N)$ and some submanifold $V\...
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A question regarding how Thom-Boardman strata sit in their closures
I just started learning about these things, so there is a chance I might have misunderstood some things. My apologies if that is the case.
Some context. Suppose that we are given a differentiable map $...
- 151
2
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0
answers
111
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On bounded solutions of a given fourth-order linear ODE
Consider the fourth-order linear ODE
$$
\label{eq1}
v^{(4)} + \frac{-C_2 - 2\alpha \phi}{C_4}v'' + \frac{4\alpha \phi'}{C_4}v' + \frac{k_1 + 3k_3\phi^2 -2\alpha \phi''}{C_4}v = 0.
$$
Without getting ...
- 121
3
votes
0
answers
321
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The boundary of the transversal pre-image of a submanifold with boundary
A similar question on MSE without answer.
Let $M, N$ be smooth manifolds such that $\partial N=\varnothing$. Let $A$ be a smoothly embedded submanifold of $N$ such that $\partial A\neq \varnothing$. ...
- 1,732
1
vote
1
answer
186
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Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood
A similar post on MSE without answer.
Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...
- 1,732
5
votes
1
answer
245
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Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures
Recently I have been trying to get a grip on transversality results in Floer homology. That is suppose we the section $\partial_{J,H}: W^{1,p}(u^*(TM))\rightarrow L^{p}(u^*(TM))$ and we want to prove ...
- 811
2
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0
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233
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Genericity of an induced projection map
I am cross-posting a question asked on Math Stackexchange that has not been answered, in which I am still interested in.
Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\...
- 31
1
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59
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Existence of closed transversals for taut foliations in arbitrary codimension
There are several different definitions of "tautness" for foliations, the most widely know is probably topological tautness, which is specific to codimension one and means that the foliation ...
- 343
1
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0
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269
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Transversality theorem for maps between fiber bundles
I am looking for a possible generalization of the standard Trasversality Theorem which roughly says that transverse maps are generic. For example, see the version below:
From page 74, Theorem 2.1 in ...
- 11
3
votes
1
answer
152
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Almost geodesic on non complete manifolds
Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\...
- 2,212
1
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96
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Stratification of the space of maps transverse to another given one
If $M,N,S$ are manifolds ($M$ and $S$ compact) and $g:S\to N$ is a smooth map, then if we endow $C^\infty(M,N)$ with the strong topology we get that
$$\pitchfork(M,N;g):=\{f\in C^\infty(M,N)\,;\,f\...
- 31
9
votes
1
answer
505
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Every immersion can be deformed to have only transverse self-intersections
I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it.
Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true ...
- 2,127
3
votes
1
answer
391
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Finite-dimensional argument for Morse-Smale pairs?
Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...
- 1,377