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…
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48 results for “Jacobi variety”
Abstract: Study of metrics on line bundles over complex varieties.
problem Chern-Weil and Hilbert-Samuel formulae for singular metrics.
method Theory of b-divisors and multiplier ideal volume function.
result Generalization of a result for elliptic curves to higher degrees.
The paper proves a vanishing identity for twist knots using character varieties.
problem Proving a vanishing identity for adjoint Reidemeister torsions of twist knots.
method Using Jacobi's residue theorem on the SL2(C)-character variety.
result The conjecture about vanishing identities for twist knots is proven.
This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite ty…
New method for constructing frames for vector distributions with specific symbols.
problem Constructing frames for vector distributions with given Jacobi symbols.
method Introducing Jacobi symbols, Optimal Control ideas, explicit construction of canonical frames, algebraic procedure.
result Explicit construction of canonical frames for all distributions with given Jacobi symbols, description of prolongation algebra for most Jacobi symbols.
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…
A natural map from Lawson homology to Deligne cohomology groups for smooth complex projective varieties is constructed by using the Harvey-Lawson spark complexes. We also compare this to Abel-Jacobi type constructions by others.
Developed spectral theory for sinh-Gordon solutions, solving inverse problem.
problem Solving spectral theory for simply periodic solutions of the sinh-Gordon equation.
method Asymptotic estimates and Jacobi variety construction.
result Spectral data defined and inverse problem solved for sinh-Gordon solutions.
An algebraic curvature tensor A is said to be Jacobi-Tsankov if J(x)J(y)=J(y)J(x) for all x,y. This implies J(x)J(x)=0 for all x; necessarily A=0 in the Riemannian setting. Furthermore, this implies J(x)J(y)=0 for all x,y if the dimension is at most 13. We exhibit a 14-dimensional algebraic curvature tensor in signatur…
Hamilton-Jacobi Equations for Nonholonomic Reducible Hamiltonian Systems on a Cotangent Bundlemath.SG
In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems, we first give to various distributional Hamiltonian systems, by analyzing carefully the dynamics and structures of the nonholonomic Hamiltonian systems. Secondly, we derive precisely the geometric constraint conditions of the induced distribu…
Optimizes control of infectious disease spread using stochastic methods.
problem Optimizing control of highly infectious diseases like COVID-19.
method Reformulated Hamilton-Jacobi-Bellman equation as stochastic minimum principle, leading to forward-backward stochastic differential equations.
result Numerous numerical solutions presented under various scenarios.
We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a split…
Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere and give a set of isometry invariants for the integrabilit…
On a discretization of confocal quadrics. II. A geometric approach to general parametrizationsmath.DG
A geometric approach discretizes confocal quadrics, leading to novel discrete nets.
problem Discretization of confocal quadrics.
method Geometric characterization and factorizable orthogonal coordinate systems.
result Explicit computation of coordinate functions for discrete confocal quadrics.
We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce t…
The paper examines the stability of Killing cylinders in hyperbolic space.
problem Stability of Killing cylinders in hyperbolic space.
method Explicit computation of Morse index for Jacobi operator on various support surfaces.
result Delaunay surfaces can be bifurcated from Killing cylinders supported on geodesic planes.
We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providin…
Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
problem No specific problem stated; focuses on new structure definition.
method Definition and properties of Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
result Defines a new structure on Jacobi-left-symmetric algebroids.
Holomorphic Jacobi Manifoldsmath.DG
Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissoniza…
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
problem Generalizing Dirac pairs to Jacobi algebroids.
method Introducing Dirac pairs on Jacobi algebroids and showing their relationship to Lie algebroids.
result Dirac pairs on Jacobi algebroids characterize compatible structures.
We use the supergeometric formalism, more precisely, the so-called "big bracket" (for which brackets and anchors are encoded by functions on some graded symplectic manifold) to address the theory of Jacobi algebroids and bialgebroids (following mainly Iglesias-Marrero and Grabowski-Marmo as a guideline). This formalism…
Defines compatibility between Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
problem Generalizing compatibility between Poisson and pseudo-Riemannian metrics to Jacobi structures.
method Introduces and studies compatibility conditions for Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
result Compatibility conditions are preserved under Poissonization and equivalent to Sasakian structures for contact pseudo-metrics.
This paper extends Jacobi field theory to Jacobi curves and their curvatures.
problem Characterizing and understanding Jacobi curves and their curvatures.
method Developed a new theory of Jacobi curves and associated curvatures, derived Ricci curvature, and presented a Cartan-like theory.
result Jacobi curves are fully characterized by a family of conformal symplectic invariant curvatures.
The paper proves normal forms for Dirac-Jacobi bundles and splitting theorems for Jacobi structures.
problem Proving normal forms and splitting theorems for Jacobi structures.
method Using recent techniques from Bursztyn, Lima and Meinrenken, the paper proves normal forms for Dirac-Jacobi bundles and splitting theorems for Jacobi pairs.
result The paper provides an alternative proof of the splitting theorem of homogeneous Poisson structures.
Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.
problem Calculating inverse metric matrices on Siegel-Jacobi spaces.
method Inversion of metric matrices on and .
result Explicit calculations of inverse metric matrices for .
Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the …
We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial …
Paper introduces stochastic HJB on Jacobi structures.
problem Stochastic analysis on Jacobi manifolds.
method Global stochastic analysis techniques, extending Bismut and Lázaro-Camí work.
result Proposes a stochastic HJB framework.
Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids.
Jacobi Structures in math-ph
The most general Jacobi brackets in 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.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
Berry phases and connection matrices defined on homogeneous spaces attached to Siegel-Jacobi groupsmath.DG
Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.
problem Understanding Berry phases and connection matrices on Siegel-Jacobi spaces.
method Examines the Siegel-Jacobi disk and upper half-plane, calculates connection matrices and covariant derivatives.
result Calculates the connection matrix and covariant derivatives on the extended Siegel-Jacobi upper half-plane.
New algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an algebra is associated with each Dirac-Jacobi structure.
result There is a one-to-one correspondence between MC elements of the algebra and small deformations of the Dirac-Jacobi structure.
Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.
problem Analyzing Hamilton-Jacobi theory across different geometric backgrounds.
method Geometric review of Hamilton-Jacobi theory, focusing on Jacobi and Leibniz identities.
result Novel Hamilton-Jacobi equation for conformal Hamiltonian vector fields.
New characterization of Osserman tensors using Jacobi-orthogonality.
problem Characterizing Osserman tensors.
method Introducing Jacobi-orthogonality as a new potential characterization.
result Jacobi-orthogonal tensors are Osserman, and all known Osserman tensors are Jacobi-orthogonal.
A new definition for vector fields extends the Jacobi set concept.
problem Describing interactions between vector fields on complex domains.
method Piecewise linear approach for simplicial complexes.
result Generalizes Jacobi set concept to vector fields.
New action functionals for sigma models on Jacobi bundles.
problem Defining action functionals for sigma models on Jacobi bundles.
method Generalizing Jacobi brackets to Jacobi bundles and proposing different approaches.
result Solutions correspond to morphisms of Jacobi algebroids.
The paper explores geometry and arithmetic of the Siegel-Jacobi space.
problem None explicitly stated; focuses on theory.
method Theoretical discussion of geometry and arithmetic aspects.
result Theoretical insights into the geometry and arithmetic of the Siegel-Jacobi space.
The paper outlines key geometry issues in Siegel-Jacobi space.
problem Basic problems in the geometry of the Siegel-Jacobi space.
method Proposes basic problems.
result Outlines key geometry issues in Siegel-Jacobi space.
Abstract: From complex financial models to simpler Hamilton-Jacobi equations.
problem Complex financial models for multi-dimensional Black-Scholes.
method Linked Hamilton-Jacobi equations to simplify financial models.
result Simplified financial models using Hamilton-Jacobi equations.
Defines Jacobi fields in nonholonomic mechanics.
problem No specific problem stated; focuses on definition and properties.
method Characterizes Jacobi fields using Riemannian geometry methods and finds them explicitly. Uses complete lift and curvature/torsion of nonholonomic connection.
result Derives nonholonomic Jacobi equations in two equivalent ways.
We analyze the relationship between the covering of the Jacobi group and the squeezed states. We attach some nonclassical states to the Jacobi group. The matrix elements of the Jacobi group are presented.
The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.
problem Classifying real hypersurfaces with a particular Jacobi operator.
method Introducing and classifying real hypersurfaces with a quadratic Killing structure Jacobi operator.
result A classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator.
Equations of motion for linear Hamiltonians in the real Jacobi group
problem Equations of motion for linear Hamiltonians in the real Jacobi group
method Using the energy function on the extended Siegel-Jacobi upper half space
result Equations of motion attached to linear Hamiltonians in the generators of the real Jacobi group
Defines gauge transformations for Jacobi structures and their effects on contact groupoids.
problem Understanding transformations of Jacobi structures and their implications.
method Definition and discussion of gauge transformations for Jacobi structures and their impact on contact groupoids.
result Gauge transformations affect the contact structure of contact groupoids.
Holomorphic Jacobi manifolds integrate to complex contact groupoids.
problem Integrating holomorphic Jacobi manifolds.
method Homogenization scheme to identify and integrate holomorphic Jacobi manifolds to complex contact groupoids.
result Holomorphic Jacobi manifolds integrate to complex contact groupoids.