Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)u^H+\beta \left(h\left(u\right)\right)u^V,$$ where $u^V$ is the vertical lift of $u\in T_{x}M$, $u^H$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta$ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma }_1\left(h\left(u\right)\right)\Lambda (g,K)+{\gamma }_2\left(h\left(u\right)\right)u^H\wedge u^V,$$ where ${\gamma }_1$, ${\gamma }_2$ are smooth real functions defined...

In this paper we present new examples of $(2n+1)$-dimensional compact cosymplectic manifolds which are not topologically equivalent to the canonical examples, i.e., to the product of the $(2m+1)$-dimensional real torus and the $r$-dimensional complex projective space, with $m,r\ge 0$ and $m+r=n.$ These new examples are compact solvmanifolds and they are constructed as suspensions with fibre the $2n$-dimensional real torus. In the particular case $n=1,$ using the examples obtained, we conclude that a $3$-dimensional compact flat orientable...

Recently Entov and Polterovich asked if the Grubb measure was the only symplectic topological measure on the torus. Much to our surprise we discovered a whole new class of intrinsic simple topological measures on the torus, many of which were symplectic.

We describe a connection between Nielsen fixed point theory and symplectic Floer homology for surfaces. A new asymptotic invariant of symplectic origin is defined.

We extend here results for escapes in any given direction of the configuration space of a mechanical system with a non singular bounded at infinity homogeneus potential of degree -1, when the energy is positive. We use geometrical methods for analyzing the parallel and asymptotic escapes of this type of systems. By using Riemannian geometry methods we prove under suitable conditions on the potential that all the orbits escaping in a given direction are asymptotically parallel among themselves. We...

Let $A$ be a dg algebra over ${𝔽}_2$ and let $M$ be a dg $A$-bimodule. We show that under certain technical hypotheses on $A$, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M{\otimes }_A^{L}M$ and converges to the Hochschild homology of $M$. We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

It is well-known that the Fundamental Identity (FI) implies that Nambu brackets are decomposable, i.e. given by a determinantal formula. We find a weaker alternative to the FI that allows for non-decomposable Nambu brackets, but still yields a Darboux-like Theorem via a Nambu-type generalization of Weinstein’s splitting principle for Poisson manifolds.

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

Let $𝒳\to S$ be a smooth proper family of complex curves (i.e. family of Riemann surfaces), and $ℱ$ a $G$-bundle over $𝒳$ with connection along the fibres $𝒳\to S$. We construct a line bundle with connection $({ℒ}_{ℱ},{\nabla }_{ℱ})$ on $S$ (also in cases when the connection on $ℱ$ has regular singularities). We discuss the resulting $({ℒ}_{ℱ},{\nabla }_{ℱ})$ mainly in the case $G={ℂ}^{*}$. For instance when $S$ is the moduli space of line bundles with connection over a Riemann surface $X$, $𝒳=X\times S$, and $ℱ$ is the Poincaré bundle over $𝒳$, we show that $({ℒ}_{ℱ},{\nabla }_{ℱ})$ provides a prequantization of $S$.

We extend the constructions and results of Damian to get topological obstructions to the existence of closed monotone Lagrangian embeddings into the cotangent bundle of a space which is the total space of a fibration over the circle.

Two symplectic structures on a manifold $M$ determine a (1,1)-tensor field on $M$. In this paper we study some properties of this field. Conversely, if $A$ is (1,1)-tensor field on a symplectic manifold $(M,\omega )$ then using the natural lift theory we find conditions under which ${\omega }^A,{\omega }^A(X,Y)=\omega (AX,Y)$, is symplectic.

This note gives a generalization of spherical twists, and describe the autoequivalences associated to certain non-spherical objects. Typically these are obtained by deforming the structure sheaves of $(0,-2)$-curves on threefolds, or deforming $\mathbb{P}$-objects introduced by D.Huybrechts and R.Thomas.