Scalar and φ-Sectional Curvature of a Certain Type of Metric f-Structures

We consider a Riemannian manifold (M,g) equipped with an f-structure of constant rank with parallelizable kernel. We assume certain integrability conditions on such a manifold. We prove some inequalities involving the scalar and *-scalar curvature of g. We prove that the corresponding equalities characterize an -manifold, which is a generalization of a Sasakian manifold. We also give a method of constructing such structures on toroidal bundles. Dedicated to the memory of Professor Aldo Cossu Research supported by the Italian MIUR 60% and GNSAGA.     

Curvature properties of anti-Kähler–Codazzi manifolds

In this paper we shall consider a new class of integrable almost anti-Hermitian manifolds, which will be called anti-Kähler–Codazzi manifolds, and we will investigate their curvature properties.     

Curvature properties of twistor spaces of quaternionic Kähler manifolds

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.Dedicated to Professor Helmut Karzel on the occasion of his 70th birthdayResearch partially supported by Contracts MM 413/1994 and MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.     

Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

Asymptotic Curvature of Θ-Metrics

We give an explicit and reasonably simple expression for the curvature tensor of a -metric at boundary points, in terms of the metric tensor and invariants of the -structure. We examine the behavior of the induced metric on level sets of a defining function near the boundary and describe the asymptotic behavior of its curvature tensor. Some applications of these results are given.     

L-restricted φ-representations of algebras

In the present paper we give a common generalization of subdirect product, direct product and weak direct product of given algebras. Mathematics subject classification numbers, 1980/85 Primary 08A30; Secondary 061310.     

Geometry and Topology of Spaces of Quasi-constant Curvature

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The φ-harmonic approximation and the regularity of φ-harmonic maps

We extend the p-harmonic approximation lemma proved by Duzaar and Mingione for p-harmonic functions to φ-harmonic functions, where φ is a convex function. The proof is direct and is based on the Lipschitz truncation technique. We apply the approximation lemma to prove Hölder continuity for the gradient of a solution of a φ-harmonic system with critical growth.     

D-Homothetic Transforms of φ-Symmetric Spaces

We present a step towards the solution of an open problem in contact Riemannian geometry: whether there exists an example of non-Sasakian (strongly) locally φ-symmetric spaces other than the so-called (κ, μ)-spaces. The main theorem in the present paper says that if such examples exist, they are not D-homothetic to other locally φ-symmetric contact metric spaces.     

On φ-Families of Probability Distributions

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.     

Classification of Flat Lagrangian Surfaces in Complex Lorentzian Plane

One of the most fundamental problems in the study of Lagrangian submanifolds from Riemannian geometric point of view is to classify Lagrangian immersions of real space forms into complex space forms. The main purpose of this paper is thus to classify flat Lagrangian surfaces in the Lorentzian complex plane C1^2. Our main result states that there are thirty-eight families of flat Lagrangian surfaces in C1^2. Conversely, every flat Lagrangian surface in C1^2 is locally congruent to one of the thirty-eight families.     

Some Examples of Metrics with Positive Curvature and Negative Curvature on R￣n

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Some characterizations of φ-Lagrangian dual problems

We give some characterizations of the Lagrangian dual objective function λ_φdefined with respect to a coupling function φ, namely, conditions for the existence of a φ, and characterizations when q> is given. In particular, when φ is the so-called “natural coupling function”, one of these characterizations reduces to the main result of [6].     

A characterization of nonlinear φ-accretive operators

Let E be a Banach space and E(–,] a proper lower semi-continuous convex function. The main purpose of this paper is to characterize those m-accretive operators AE x E that are also -accretive. This is done by using the semigroup S generated by -A, and by first establishing a necessary and sufficient condition for to be a Lyapunov function for S. We also obtain similar results for accretive operators that are not necessarily m-accretive, and deduce invariance and order-preserving criteria for nonlinear semigroups.This research was partially supported by the National Science Foundation under Grant MCS-8102086.     

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kähler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce \(Z(M, [[\omega ]])\) depending on symplectic deformation equivalence class \([[\omega ]]\). We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of \(Z(M, [[\omega ]] )\). Using Z invariants, we set up a Kazdan–Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold \((M, \omega )\) of dimension \(\ge \!\!4\), any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to \(\omega \).