arXiv research

Defines Jacobi fields in nonholonomic mechanics.

A new definition for vector fields extends the Jacobi set concept.

The connection between Jacobi fields and odular structures of affine manifold is established. It is shown that the Jacobi fields generate the natural geoodular structure of affinely connected manifolds.

Geometrically describes Jacobi equations for field theories with dissipation.

We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.

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 $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called the Jacobi flow.

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can…

The paper studies Jacobi fields and conjugate points in projective sprays.

We show that Jacobi fields along harmonic maps between suitable spaces preserve conformality, holomorphicity, real isotropy and complex isotropy to first order; this last being one of the key tools in the proof by Lemaire and the author of integrability of Jacobi fields along harmonic maps from the 2-sphere to the comp…

I briefly review my proposal about how to extend the geometric Hamilton-Jacobi theory to higher derivative field theories on fiber bundles.

We formulate extensions of Wilking's Jacobi field splitting theorem to uniformly positive sectional curvature and also to positive and nonnegative intermediate Ricci curvatures.

We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold $Σ$ of a Riemannian manifold $\mathcal{M}$. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of …

In this article, we investigate the stability of leaves of minimal foliations of arbitrary codimension. We also study relations between Jacobi fields and vector fields which preserves a foliation and we use these results to Killing fields.

Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.

Unified Jacobi coupling construction for various geometric settings.

The most general Jacobi brackets in $\mathbb{R}^3$ are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed.

We prove that a normal homogeneous space with the property that every Jacobi field along a geodesic vanishing at two points is the restriction of a Killing field along that geodesic is a globally symmetric space.

This paper extends Jacobi field theory to Jacobi curves and their curvatures.

Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.

In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.

It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible.…

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.

Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton-Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton-Jacobi theory.

Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds are reviewed. We present both in a unified approach using the representation of the Lie algebra of functions on itself by means of the hamiltonian vector fields. The use of the associated Lie algebroid allows to prove that the Lichnerowicz-Jacobi cohomolog…

The study refines stability results for Yang-Mills fields and harmonic maps.

In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but…

We extend the geometric Hamilton-Jacobi formalism for hamiltonian mechanics to higher order field theories with regular lagrangian density. We also investigate the dependence of the formalism on the lagrangian density in the class of those yelding the same Euler-Lagrange equations.

In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which are explicitly constructed and shown to give a complete description of the enti…

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure an…

We reformulate the notion of a Jacobi algebroid in terms of weighted odd Jacobi brackets. We then show how a Jacobi algebroid can be understood in terms of a kind of curved Q-manifold. In particular the homological condition on the odd vector field is deformed in a very specific way. This leads to the notion of a quasi…

As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.

Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.

A unified framework for Poisson and Jacobi structures from 2-covariant tensors

Introduces a new bracket for multicontact geometry and applies it to field theories.

In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sp…

In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…

Study conjugate locus in convex 3-manifolds using Jacobi fields.

In this paper, we study submanifolds with constant $r$th mean curvature $S_r$. We investigate, the stability of such submanifolds in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros - Sousa and Alías - Colares, concerning conformal fields, to an arbitrary manifol…

Twistor methods provide a powerful tool in the study of harmonic maps and harmonic morphisms. Indeed, their use has enabled us to produce a variety of examples of harmonic morphisms defined on 4-dimensional manifolds, and a complete classification in some cases. In the first part of this work, we generalize those const…

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 non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence …

Proves properties of sub-Riemannian exponential map, showing it's not injective.

In this paper, we derive the first and the second variation of the energy functional for a pseudo-Finsler metric using the family of affine connections associated to the Chern connection. This opens the possibility to accomplish computations with coordinate-free methods. Using the second variation formula, we introduce…

This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.

It is proved that all invariant functions of a complex Finsler manifold can be totally recovered from the torsion and curvature of the connection introduced by Kobayashi for holomorphic vector bundles with complex Finsler structures. Equations of the geodesics and Jacobi fields of a generic complex Finsler manifold, ex…

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 vanishes on both points. Based on arguments that involve the eigendistributions of t…

We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization p…

Global Morse index theorem applied to Jacobi fields on CMC surfaces.

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…