# Questions tagged [h-principle]

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### Recognizing sections up to isotopy

Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ be a smooth embedding. I would like to learn what is known about the following Question. When does there ...
275 views

### Making a submanifold transverse to a vector field by an isotopy

Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known ...
322 views

### Can the methods of algebra characterize nonlinear PDE blow-ups?

Consider 2 simple differential equations: $x'(t)=x(t)^2, x(0)=1$ and $x'(t)=-x(t)^2, x(0)=1$. As $t$ goes from $0$ to $\infty$, the first equation ($x(t)=1/(1-t)$) will lead to a finite-time blow-up, ...
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### relationship between "linear approximation" to immersions and formal immersions

I'm reading these notes Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$ If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element ...
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### h-principle on Hilbert manifolds

Gromov's h-principle is a powerful tool in studying various geometric structure on open, finite-dimensional manifolds. Is there any generalization of h-principle to (necessarily open) infinite-...
426 views

So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions ...
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### The relative h-principle and extension problems

As a beginner for h-principles, I want to know why the relative h-principle cannot imply a positive solution to the problems for extending symplectic structures. The following is a relative h-...
980 views

### Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
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### H-principle and PDE's

According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations ...
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