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Francisco-Javier Turiel

Publications and source records attributed to Francisco-Javier Turiel.

8 recordsLinked to original sources
arXiv2022

On the minimal dimension of the orbits of a $\mathbb R^n$-action

Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a consequence, one shows that if there is a non-zero element of the ring of Pontrjagin classes of $M$ of degree $4\ell\geq 4$, then there exists an orbit of the action of dimension $\leq m-\ell-1$.

math.DS↗
arXiv2018

Primary singularities of vector fields on surfaces

Unless another thing is stated one works in the $C^\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb R$. A subset $K$ of the zero set ${\mathsf Z}(X)$ is an essential block for $X$ if it is non-empty, compact, open in ${\mathsf Z}(X)$ and its Poincaré-Hopf index does not vanishes. One says that $X$ is non-flat at $p$ if its $\infty$-jet at $p$ is non-trivial. A point $p$ of ${\mathsf Z}(X)$ is called a primary singularity of $X$ if any vector field defined about $p$ and tracking $X$ vanishes at $p$. This is our main result: Consider an essential block $K$ of a vector field $X$ defined on a surface $M$. Assume that $X$ is non-flat at every point of $K$. Then $K$ contains a primary singularity of $X$. As a consequence, if $M$ is a compact surface with non-zero characteristic and $X$ is nowhere flat, then there exists a primary singularity of $X$.

math.DS↗
arXiv2015

Flatness of generic Poisson pairs in odd dimension

Given a $(m-2)$-form $\zw$ and a volume form $\zW$ on a $m$-manifold one defines a bi-vector $\zL$ by setting $\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$ for any $1$-forms $\za,\zb$. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, $(\zL,\zL_1 )$ is always represented by a couple of $(m-2)$-forms $\zw,\zw_1$ and a volume form $\zW$. Here one shows that, for $m\zmai 5$ and odd and $(\zL,\zL_1 )$ generic, $(\zL,\zL_1 )$ is flat if and only if there exists a $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$.

math.DG↗
arXiv2011

The Local Product Theorem for bihamiltonian structures

In this work one proves that, around each point of a dense open set (regular points), a real analytic or holomorphic bihamiltonian structure decomposes into a product of a Kronecker bihamiltonian structure and a symplectic one if a necessary condition on the characteristic polynomial of the symplectic factor holds. Moreover we give an example of bihamiltonian structure for showing that this result does not extend to the $C^\infty$-category. Thus a classical problem on the geometric theory of bihamiltonian structures is solved at almost every point.

math.SG↗
arXiv2010

On the local theory of Veronese webs

This work is an introduction to the local geometric theory of Veronese webs developed in the last twenty years. Among the different possible approach, here one has chosen the point of view of differential forms. Moreover, in order to make its reading easier, this text is self-contained in which directly regards Veronese webs.

math.DG↗
arXiv2007

Complex Polynomial Representation of $π_{n+1}(s^{n})$ and $π_{n+2}(s^{n})$

The complex affine quadric $Q^{m}=\{z\in {\Bbb C}^{m+1}\mid z_{1}^{2}+...+z_{m+1}^{2}=1\}$ deforms by retraction onto $S^{m}$; this allows us to identify $[Q^{k},Q^{n}]$ and $[S^{k},S^{n}]=π_{k}(S^{n})$. Thus one will say that an element of $π_{k}(S^{n})$ is complex representable if there exists a complex polynomial map from $Q^{k}$ to $Q^{n}$ corresponding to this class. In this Note we show that $π_{n+1}(S^{n})$ and $π_{n+2}(S^{n})$ are complex representable.

math.AT↗