arXiv research

It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…

Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.

After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary results from Riemannian geometry are proven using these generalizations. In particul…

Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible wit…

The abstract discusses how generalized Calabi-Gray manifolds help solve non-Kähler geometry questions.

Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…

The paper extends group constructions to coset geometries, creating new ways to combine geometries.

Study of optical geometries with intrinsic torsion in general relativity.

Develops Weyl structures for path geometries, simplifying their study.

These are the lecture notes from the 26th Winter School "Geometry and Physics", Czech Republic, Srni, January 14 - 21, 2006. These lectures are an introduction into the realm of generalized geometry based on the tangent plus the cotangent bundle. In particular we discuss the relation of this geometry to physics, namely…

As a natural application of the {\it theory of geometric averaging} in Finsler geometry and generalized Finsler geometry, a new approach to investigate {\it generalized Finsler geometry}, based on a convex invariance of the average structures, is introduced.

New geometries derived from symplectic Monge-Ampère structures.

We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the eq…

We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the p…

We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted …

Introduces new deformation classes in generalized Kähler geometry.

The study extends inscription problems to non-Euclidean geometries.

It has been known for a while that the effective geometrical description of compactified strings on $d$-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an $O(d,d)$ pairing $η$ and an $O(2d)$ generalized metric $\m…

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex o…

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…

Defines generalized Sasakian structures in contact geometry.

New geometries defined for string models, filling gaps in the literature.

We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.

We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metr…

Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.

Researchers generalize cosmological models using Finsler geometry.

We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distr…

A closed 3-form $H \in Ω^3_0(M)$ defines an extension of $Γ(TM)$ by $Ω^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian redu…

We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with ass…

We give a physical derivation of generalized Kahler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri regarding the equivalence between generalized Kahler geometry and the bi-hermitean geometry of Gates-Hull-Rocek. When cast in the language of supersymmetri…

New approach uses isotropic geometry to solve Euclidean problems.

This work is based on the talk delivered at Poisson 2008. We review the recent advances in Generalized Kahler geometry while stressing the use of Poisson and symplectic geometry. The derivation of the generalized Kahler potential is sketched and the relevant global issues are discussed.

Survey of global geometry for double field theory.

In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the $dd^c$-lemma of Kähler geometry.

This paper is the first of a 3-part series that classifies the 5-dimensional Thurston geometries. The present paper (part 1 of 3) summarizes the general classification, giving the full list, an outline of the method, and some illustrative examples. This includes phenomena that have not appeared in lower dimensional geo…

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing th…

Develops Palatini formalism in generalized geometry for string theory.

New definition of Born geometry connects to known geometries.

The paper studies hyperbolic geometry of links formed by adding trivial components to knots.

New approach to symmetries in teleparallel geometries with non-trivial isotropy groups.

Unsupervised domain mapping has attracted substantial attention in recent years due to the success of models based on the cycle-consistency assumption. These models map between two domains by fooling a probabilistic discriminator, thereby matching the probability distributions of the real and generated data. Instead of…

This paper reviews Douglas curvature in Finsler geometry.

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometr…

In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker…

We survey the concept of multiplicativity from its initial appearance in the theory of Poisson-Lie groups to the far-reaching generalizations, for multivectors and differential forms in the geometry and the generalized geometry of Lie groupoids, as well as their infinitesimal counterparts in the theory of Lie algebroid…

Metric SYZ conjecture proved using non-archimedean geometry.

This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitean symmetri…

Optimization geometry affects deep learning performance.