Differential Geometric Structure Research Articles

Information-Geometric Approach for a One-Sided Truncated Exponential Family.

In information geometry, there has been extensive research on the deep connections between differential geometric structures, such as the Fisher metric and the α-connection, and the statistical theory for statistical models satisfying regularity conditions. However, the study of information geometry for non-regular statistical models is insufficient, and a one-sided truncated exponential family (oTEF) is one example of these models. In this paper, based on the asymptotic properties of maximum likelihood estimators, we provide a Riemannian metric for the oTEF. Furthermore, we demonstrate that the oTEF has an α = 1 parallel prior distribution and that the scalar curvature of a certain submodel, including the Pareto family, is a negative constant.

Curves Lying on Non-lightlike Surface: Differential Equation for Position Vector

The main purpose of this study is to examine curves lying on a given non-lightlike surface with the help of its position vectors. For this purpose, the darboux frame is used and the position vector of the curve is expressed as a linear combination of the darboux frame with differentiable functions. Then, nonhomogeneous systems of differential equations revealed by the position vector of the curve are obtained for timelike and spacelike surfaces, respectively. For both timelike and spacelike surfaces, the solutions of nonhomogeneous systems of differential equations are obtained depending on the character of the curves and the values kg, kn and tr. The general solutions of the systems of differential equations are obtained separately for each case. Moreover, by considering only the particular solution of the systems of differential equations, new results regarding the differential geometric structure of the curves on the surface are presented with the help of the position vector

A Life in Mathematical Physics: Conference in Honour of Tekin Dereli

1. IntroductionThe friends and colleagues of Prof. Dr. Tekin Dereli came together to celebrate his 72nd birthday and the 50th year of his mathematical physics career. The three-days event took place online over the Zoom on 26-28 November 2021. His scientific and scholarly achievements, his kindness as a colleague, his thoughtful guidance of students and his service to the physics community in Turkey as a whole and last but not the least his open and easy accessibility to anyone needing his expert counseling have been appreciated by all.¨ We are grateful to the Turkish Academy of Sciences (TUBA) for sponsoring the online, open-access publication of the proceedings as a special issue in The Journal of Physics: Conference Series.2. Biography of Tekin DereliTekin Dereli was born in Ankara in 1949 where he completed his early education at Ankara Science High School in 1967. He earned his Bachelor’s degree in physics with high honors in 1971 at the Middle East Technical University (METU), where he also completed his PhD in 1976, with a thesis entitled “Fermions in Yang-Mills Potentials”. He gave his habilitation in 1981 with a thesis on “Dynamics of Neutrinos in Gravitational Fields” (in Turkish). His academic career began at METU as an instructor in 1973, where he later became an associate professor. He was promoted to professorship in 1987 at Ankara University and then returned to METU in 1993. He was invited to Ko¸c University as Professor of Physics in 2001 where he worked till 2021.Tekin Dereli held various research positions abroad, being always on-leave from METU. These include a visiting scientist position at Yale University during 1974-1975 under the hosting of Feza Gürsey and a research associate position at Brandeis University during 1976-1977 under the hosting of Stanley Deser. He collaborated with Peter C. Aichelburg as Einstein-Memorial Foundation Fellow at the Institute for Theoretical Physics, University of Vienna during 1977-1978. He was SRC Research Associate at Lancaster University between 1979-1981 where a long and fruitful collaboration with Robin W. Tucker has begun. He had been an Associate Member of ICTP between 1986-1991 and Alexander von Humboldt Foundation Fellow at Karlsruhe University during 1988-1989 under the hosting of Julius Wess.Tekin Dereli’s research interests cover a broad range of subjects. He published papers on magnetic monopoles in Yang-Mills gauge field theories, supersymmetry and supergravity, the role of quaternions and octonions in physics. He studied Clifford algebras and spin structures associated with the space-time. He worked on generalised theories of gravity, both in higher and lower dimensions than four, emphasizing their differential geometric gauge structure. In particular he investigated various aspects of black hole and gravitational wave solutions.Furthermore he constructed higher dimensional cosmological models. Tekin Dereli also worked on dualities in physics and mathematics in general and deformation quantisation in the context of the Landau problem.List of Scientific Bibliography of Tekin Dereli (1975-2021), Theses Supervised by Tekin Dereli, Alphabetical List of Contributors, Organising Committee are available in this pdf.

Statistical geometry and Hessian structures on pre-Leibniz algebroids

We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids are special cases of local pre-Leibniz algebroids, which are still general enough to include many physically motivated algebroids such as Lie, Courant, metric and higher-Courant algebroids. They create a natural framework for generalizations of differential geometric structures on a smooth manifold. The symmetrization of the bracket on an anti-commutable pre-Leibniz algebroid satisfies a certain property depending on a choice of an equivalence class of connections which are called admissible. These admissible connections are shown to be necessary to generalize aforementioned structures on pre-Leibniz algebroids. Consequently, we prove that, provided certain conditions are met, statistical and conjugate connection structures are equivalent when defined for admissible connections. Moreover, we also show that for ‘projected-torsion-free’ connections, one can generalize Hessian metrics and Hessian structures. We prove that any Hessian structure yields a statistical structure, where these results are completely parallel to the ones in the manifold setting. We also prove a mild generalization of the fundamental theorem of statistical geometry. Moreover, we generalize a-connections, strongly conjugate connections and relative torsion operator, and prove some analogous results.

О структурных формах проективной структуры

A projective structure on a smooth manifold is a maximal atlas such that all its transition maps are the fractional linear transformations. Our aim is to interpret this notion in terms of the higher order frame bundles and their structure forms. It is shown that the projective structure gener­ates the sequence of differential geometric structures. The construction is following: Step 1. For a smooth manifold the so-called quotient frame bundle as­sociated to the 2nd order frame bundle on the manifold is constructed. Step 2. Given projective structure on the manifold, the mappings from the quotient frame bundle to the higher order frame bundles are con­structed. These mappings are the differential geometric structures. Step 3. The pullbacks of the structure forms of the frame bundles via the mappings are considered. These are called structure forms of the pro­jective structure. The expressions of their exterior differentials in terms of the forms themselves are found. These expressions coincide with the structure equations of a projective space.

Geometric characterization of continuously defective elastic crystals

We show how some differential geometric structures associated with a concept of a homogeneous space appear naturally in a kinematic model of continuously distributed defects in an elastic crystal solid and discuss how one can use them to describe the defectiveness of such a continuum.

Characterizing symplectic Grassmannians by varieties of minimal rational tangents

We show that if the variety of minimal rational tangents (VMRT) of a uniruled projective manifold at a general point is projectively equivalent to that of a symplectic or an odd-symplectic Grassmannian, the germ of a general minimal rational curve is biholomorphic to the germ of a general line in a presymplectic Grassmannian. As an application, we characterize symplectic and oddsymplectic Grassmannians, among Fano manifolds of Picard number $1$, by their VMRT at a general point and prove their rigidity under global Kähler deformation. Analogous results for $G/P$ associated with a long root were obtained by Mok and Hong–Hwang a decade ago by using Tanaka theory for parabolic geometries. When $G/P$ is associated with a short root, for which symplectic Grassmannians are most prominent examples, the associated local differential geometric structure is no longer a parabolic geometry and standard machinery of Tanaka theory cannot be applied because of several degenerate features. To overcome the difficulty, we show that Tanaka’s method can be generalized to a setting much broader than parabolic geometries, by assuming a vanishing condition that certain vector bundles arising from Spencer complexes have no nonzero sections. In our application of this method to the characterization of symplectic (or odd-symplectic) Grassmannians, this vanishing condition is checked by exploiting geometry of minimal rational curves.

Suitable Spaces for Shape Optimization

The differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov–Poincaré metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to H^{1/2}-shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of H^{1/2}-shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces.

Rigidity of Transformation Groups in Differential Geometry

In this survey, symmetry provides a framework for classification of manifolds with differential-geometric structures. We highlight pseudo-Riemannian metrics, conformal structures, and projective structures. A range of techniques have been developed and successfully deployed in this subject, some of them based on algebra and dynamics and some based on analysis. We aim to illustrate this variety.

Differential Structures of Frölicher Spaces on Tangent Curves

Differential-geometric structures of Frolicher spaces for a singular manifold consisting of two tangent curves are considered. Calculations for two types of structures lead either to the ∞-flatness of all curves passing from one branch to another at a singular point or to the ∞-flatness of functions. In the second case, smooth curves can change their branch of motion, their velocity vector vanishes at the singular point.

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

On the differential geometry of numerical schemes and weak solutions of functional equations

We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe the geometric framework, highlight several examples and describe how two well-known proofs fit with our setting. The first one is a re-interpretation of the classical proof of an implicit functions theorem in an inverse limit of branch setting, for which our setting enables us to state an implicit functions theorem without additional norm estimates, and the second one is the finite element method of a Dirichlet problem where the set of triangulations appears as a smooth set of parameters. In both cases, smooth dependence on the set of parameters is established. Before that, we develop the necessary theoretical tools, namely the notion of Cauchy diffeology on spaces of Cauchy sequences and a new generalization of the notion of tangent space to a diffeological space.

An Elementary Introduction to Information Geometry.

In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential geometry. Proofs are omitted for brevity.

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-geometric structure

There are reviewed modern investigations devoted to studying nonlinear dispersiveless heavenly type integrable evolutions systems on functional spaces within the modern differential-geometric and algebraic tools. Main accent is done on the loop diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, generating dispersionless heavenly type integrable systems.

Differential-Geometric Structure and the Lax–Sato Integrability of a Class of Dispersionless Heavenly-Type Equations

This short communication is devoted to the study of differential-geometric structure and the Lax – Sato integrability of the reduced Shabat-type, Hirota, and Kupershmidt heavenly equations.

The study on the Einstein's connection in $5$-dimensional $ES$-manifold for the second class

The manifold $ {}^*{g} - ESX_n $ is a generalized $ n $-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $ {}^*{g}^{ \lambda \nu } $ through the $ ES $-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to derive a new set of powerful recurrence relations and to prove a necessary and sufficient condition for a unique Einstein's connection to exist in $5$-dimensional ${}^*{g}-ESX_5$ and to display a surveyable tnesorial representation of $5$-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the second class.

On a special class of submanifolds in pseudoeuclidean space En^(2n)

A special class of 2m dimensional submanifolds M→E n 2n with structure of double fiber bundle is studied. Using the Cartan's method of exterior forms on manifold the structure equations of M are discovered and the differential geometric structure on M is described. M has the structure of Rashevsky-Einstein space.

Connecting Information Geometry and Geometric Mechanics

The divergence function in information geometry, and the discrete Lagrangian in discrete geometric mechanics each induce a differential geometric structure on the product manifold Q × Q . We aim to investigate the relationship between these two objects, and the fundamental role that duality, in the form of Legendre transforms, plays in both fields. By establishing an analogy between these two approaches, we will show how a fruitful cross-fertilization of techniques may arise from switching formulations based on the cotangent bundle T * Q (as in geometric mechanics) and the tangent bundle T Q (as in information geometry). In particular, we establish, through variational error analysis, that the divergence function agrees with the exact discrete Lagrangian up to third order if and only if Q is a Hessian manifold.

SU(3) gauge theory of nuclear rotations

The legacy Bohr-Mottelson model of collective rotational modes has a hidden differential geometric structure that enables its natural generalization to a nuclear model that has the mathematical structure of Yang-Mills theory. The essential differential geometry ingredients for Yang-Mills are a base manifold, a gauge group, and a connection or covariant derivative. In this letter, the base manifold is the space of nuclear orientations and quadrupole-monopole deformations, the gauge group is either SO(3) or SU(3), and the covariant derivative determines a new gauge-invariant “magnetic-type” interaction. The high-lying energy states of the legacy irrotational flow model enter, as a direct result of gauge coupling, the domain of low-energy yrast rotational bands, as observed by experiment. Although the relevant SU(3) representation for a deformed nucleus is the same as the Elliott model, the non-Abelian SU(3) gauge group's physical interpretation is very different and concerns the Kelvin circulation.