In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature For such submanifold, we investigate curvature properties We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h∗ of the submanifold (associated with the dual connections) satisfy h=-h∗, ie, the submanifold is totally geodesic with respect to the Levi-Civita connection

Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature.

In this paper I will describe two distinct ways by which it obtain a sequence of SOPDEs and a sequences of nonlinear connections on the tangent bundle of k 1 -velocties T 1 k M, starting from a given SOPDE, follows the ideas of papers (25), (26), (27), (28). Some properties about this sequences is also presented. Interesting cases appear in the presence of a regular Lagrangian on T 1 kM.

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Second order partial dierential equations (SOPDEs) and nonlinear connections on the tangent bundle of k 1 -velocities of a manifold

Spectral geometry of eta-Einstein Sasakian manifolds

Nonlinear connections and supersprays in supermanifolds

We study the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the tangent sphere bundle of the n(≥3)-dimensional locally symmetric space, equipped with the standard contact metric structure, is an η-Einstein manifold if and only if the base manifold is a space of constant sectional curvature or .

η-EINSTEIN TANGENT SPHERE BUNDLES OF CONSTANT RADII

To the contact distribution of a contact manifold we associate Hamiltonian type vector fields, called contact Hamiltonian fields. Their properties are investigated and the existence of such vector fields nowhere tangent to a given submanifold is proved. Time-depending contact Hamiltonian vector fields allow us to define the contact energy whose properties are studied. A class of submanifolds in relation to the study of contact Hamiltonian fields is also analyzed.

Hamiltonian Fields and Energy in Contact Manifolds

On the nonlinear connection of a spray

The goal of this lecture is to give an introduction to existence problems of contact structures. So, in the first Section we define the notion of contact structure, as well as some specialized contact structures. We also study the rigidity and the local behavior of such a structure. Some basic problems concerning the geometry of contact manifolds are presented in Sect. 2. The existence of contact forms is studied in the next Section. Specially in the 3–dimensional case, some classical results and the new Geiges–Gonzalo theory of contact circles and contact spheres and the classification manifolds carrying such structures are presented. Some historical considerations pointing important steps in the development of contact geometry are finally presented.

Contact Forms in Geometry and Topology

A contact version of a Laudenbach's engulfing theorem is proved. Some properties of the notions of contact displacement energy and contact Hofer–Zehnder capacities are presented and, under the condition of existence of a modified action selector on a contact manifold, we can prove some inequalities involving these invariants. These inequalities are similar to the ones obtained by Frauenfelder, Ginzburg and Schlenk, in the symplectic case.

Gheorghe Pitiş

Papers

Hamiltonian Fields and Energy in Contact Manifolds

On the nonlinear connection of a spray

Contact Forms in Geometry and Topology

Capacities and displacement energy in contact hofer geometry