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 XnJ and ildeXnJ. result Explicit calculations of inverse metric matrices for n=2. 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.
Extended metric defined on Siegel-Jacobi space using invariant forms.
problem Defining a metric on the extended Siegel-Jacobi upper half space.
method Matrix embedding, pre-Iwasawa decomposition, invariant forms, sum of squares of forms.
result Invariant metric on the extended Siegel-Jacobi upper half space is derived.
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
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.
Geodesics on extended Siegel-Jacobi upper half-plane determined.
problem Determining geodesics on a complex geometric space.
method Equating parameters in geodesic equations on the extended Siegel-Jacobi upper half-plane.
result Geodesic equations on Siegel-Jacobi, Siegel, and Heisenberg spaces.
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.
Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.
problem Characterizing the geometry of the extended Siegel-Jacobi upper half-plane.
method Parameterized using S-coordinates and expressed in terms of invariant metrics.
result Extended Siegel-Jacobi upper half-plane is a reductive, non-symmetric manifold.
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 …
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.
Let N be the space of Gaussian distribution functions over R, regarded as a 2-dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper we show that the tangent bundle of N, endowed with its natural Kähler structure, is the Siegel-Jacobi space…
This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.
problem Evolution of spectral curves in Siegel Jacobi space through Schrodinger equation.
method Kahler geometry induced on lognormal statistical manifold, Dombrowski's construction.
result Time-dependent Schrodinger equation with varying energy.
In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…
We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.
We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in Cn. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk D1J, which is a parti…
The coherent state representation of the Jacobi group G1J is indexed with two parameters, μ(=ℏ1), describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of th…
We find the homogenous Kähler isomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi domain D1J=C×D1 as the sum of the Kähler two-form on C and the one on the Siegel ball D1. The classical motion and quantum evolution on D1J…
Researchers explore geometric dualities in statistical manifolds.
problem Understanding geometric dualities in statistical manifolds.
method Exploring the dualistic geometry of statistical manifolds, focusing on Hessian manifolds.
result Moduli space of univariate normal distributions corresponds to Siegel half-space and Siegel-Jacobi space.
We show that the Wei-Norman method applied to describe the evolution on the Siegel-Jacobi disk D1J=D1×C1, where D1 denotes the Siegel disk, determined by a hermitian Hamiltonian linear in the generators of the Jacobi group G1J and Berezin's scheme using coherent …
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.
We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…
The real Jacobi group G1J(R), defined as the semi-direct product of the group SL(2,R) with the Heisenberg group H1, is embedded in a 4×4 matrix realisation of the group Sp(2,R). The left-invariant one-forms on G1J(R) and their dual orthogonal left-i…
Study on extended weakly symmetric spaces, classifying and providing an example.
problem Understanding geometric properties of extended weakly symmetric spaces.
method Classification and presentation of a non-trivial example.
result Existence of extended weakly symmetric spaces established.
To give a Cartan calculus on the extended quantum 3d space, the noncommutative differential calculus on the extended quantum 3d space is extended by introducing inner derivations and Lie derivatives.
Study geodesic extendibility on metric spaces and map them to a half-space.
problem Geodesic extendibility on metric spaces.
method Explicit isometry between (Σ(X),dH) and XimesR≥0. result Established group isometry between Iso(X,d) and Iso(Σ(X),d_H) for Hadamard spaces.
Extended Rank-One Theorem to special metric spaces.
problem Extending a theorem to new types of spaces.
method Applied to a new class of metric measure spaces.
result Rank-One Theorem proven for RCD(K,N) spaces. Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
Classification extended for reducible symmetric spaces.
problem Classifying homogeneous foliations on symmetric spaces.
method Extended classification from irreducible to reducible symmetric spaces.
result Classification completed for all noncompact symmetric spaces.
Schwartz functions smoothly extend to real projective spaces.
problem Identifying functions that extend smoothly to compactifications.
method Showed a similar result for real projective spaces.
result Schwartz functions extend smoothly to real projective spaces.
Riemann extension for the anti Mach metric is derived, the solution of geodesic equations for the extended space are given, some properties for the extended space was studied and compared with the basic space and the constructions of a translation surface for the anti Mach metric in four dimension is established.
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
The differential geometry of 3-dimensional Bianchi, Cartan and Vranceanu (BCV) spaces is well known. We introduce the extended Bianchi, Cartan and Vranceanu (EBCV) spaces as a natural seven dimensional generalization of BCV spaces and study some of their main geometric properties, such as the Levi-Civita connec…
Symplectic embedding extended to stratified spaces.
problem Symplectic embedding theorem for stratified spaces.
method Defined symplectic structure on stratified spaces, demonstrated embedding in complex projective space.
result Symplectic embedding theorem extended to stratified spaces.
Study extended Bogomolny equations on curved space with special boundary conditions.
problem Classify solutions to extended Bogomolny equations with gauge group SU(2).
method Relate solutions to holomorphic data via Kobayashi-Hitchin correspondence.
result Completely classify solutions to the extended Bogomolny equations.
This paper extends the C*-signature to non-Witt spaces using noncommutative geometric methods.
problem Extending the signature to non-Witt spaces with noncommutative geometric methods.
method Noncommutative geometric methods, combinatorial framework, and comparison with analytical signature.
result Constructing the C*-signature on non-Witt spaces.
Classifies and constructs all extendable automorphisms of closed surfaces over the 3-sphere.
problem Identifying extendable automorphisms of closed surfaces over the 3-sphere.
method Classification and construction of extendable automorphisms through embeddings and Heegaard surfaces.
result All extendable automorphisms of closed surfaces can be induced by automorphisms of the 3-sphere on Heegaard surfaces.
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
Extends functions on symmetric spaces to analytic functions.
problem Extending functions on symmetric spaces to analytic functions.
method Harmonic analysis on symmetric spaces and representation theory of groups.
result Proves Whitney type extension theorems for symmetric spaces.
Extends Lipschitz functions while preserving local constants.
problem Extending Lipschitz functions on metric spaces while maintaining local constants.
method Extends Lipschitz functions on metric spaces while locally preserving the asymptotic Lipschitz constant.
result Sobolev spaces on metric measure spaces are invariant under isomorphism of mm-structures.
The paper extends Thompson Sampling to infinite action spaces using information theory.
problem Addressing the limitation of finite action spaces in Thompson Sampling.
method Information-theoretic analysis, extending rate-distortion theory to infinite action spaces.
result Derives a near-optimal regret bound for bandits with infinite and continuous action spaces.
The study examines spaces of non-extendable quasimorphisms for group pairs.
problem Analyzing the space of non-extendable quasimorphisms for group pairs.
method Established a five-term exact sequence of cohomology relative to bounded subcomplex.
result Proved stable commutator length equivalent to stable mixed commutator length for certain pairs.
New findings on Frobenius structures on Kodaira manifolds.
problem Understanding Frobenius structures on Kodaira manifolds.
method Extended deformation theory and Frobenius structures.
result Frobenius structure on Kodaira manifolds is trivial on degree-2 component.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.