Related papers: Jacobi fields along harmonic 2-spheres in ${\bf C}…

We determine the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces for (i) arbitrary constant magnetic fields on the two-sphere and (ii) strong constant magnetic fields for higher genus surfaces. On $S^2$ we further…

The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence…

Jacobi sigma models are two-dimensional topological non-linear field theories which are associated with Jacobi structures. The latter can be considered as a generalization of Poisson structures. After reviewing the main properties and…

We show that trace functions on modules of topological N=2 super vertex algebras give rise to conformal blocks on elliptic supercurves. We show that they satisfy a system of linear partial differential equations with respect to the modular…

We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field,…

In this paper, we study the existence of harmonic and bi-harmonic maps into Riemannian manifolds admitting a conformal vector field, or a nontrivial Ricci solitons.

The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both…

Answering an old question, we find a domain X in the complex projective plane CP^2 which admits a strongly plurisubharmonic function, but such that every holomorphic function on X is constant. The domain X can be chosen diffeomorphic to an…

We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in…

We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that…

In recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus…

The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray $S:TM\to TTM$. The flow lines of the vector field $\ka_{TM}\o TS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called…

This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories - recently introduced by the authors - which share and generalise relevant…

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a…

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their…

The purpose of this paper is to generalize a theorem of Segal from [Seg79] proving that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps…

This paper describes the theory of Jacobi curves, a far reaching extension of the spaces of Jacobi fields along Riemannian geodesics, developed by Agrachev and Zelenko. Jacobi curves are curves in the Lagrangian Grassmannian of a symplectic…

A geometric approach to integrability and reduction of dynamical system is developed from a modern perspective. The main ingredients in such analysis are the infinitesimal symmetries and the tensor fields that are invariant under the given…