Milan Lj. Zlatanović

On geodesic mappings of equidistant generalized Riemannian spaces

Abstract In this paper geodesic mappings of equidistant generalized Riemannian spaces are discussed. It is proved that each equidistant generalized Riemannian space of basic type admits non-trivial geodesic mapping with preserved equidistant congruence. Especially, there exists non-trivial geodesic mapping of equidistant generalized Riemannian space onto equidistant Riemannian space. An example of geodesic mapping of an equidistant generalized Riemannian spaces is presented.

Connections on a non-symmetric (generalized) Riemannian manifold and gravity

Connections with (skew-symmetric) torsion on a non-symmetric Riemannian manifold satisfying the Einstein metricity condition (non-symmetric gravitation theory (NGT) with torsion) are considered. It is shown that an almost Hermitian manifold is NGT with torsion if and only if it is a nearly Kahler manifold. In the case of an almost contact metric manifold the NGT with torsion spaces are characterized and a possibly new class of almost contact metric manifolds is extracted. Similar considerations lead to a definition of a particular class of almost para-Hermitian and almost paracontact metric manifolds. Conditions are given in terms of the corresponding Nijenhuis tensors and the exterior derivative of the skew-symmetric part of the non-symmetric Riemannian metric.

On Equitorsion Geodesic Mappings of General Affine Connection Spaces

In the papers [19], [20] several Ricci type identities are obtained by using non-symmetric affine connection. In these identities appear 12 curvature tensors, 5 of which being independent [21], while the rest can be expressed as linear combinations of the others. In the general case of a geodesic mapping f of two non-symmetric affine connection spaces GAN and GAN it is impossible to obtain a generalization of the Weyl projective curvature tensor. In the present paper we study the case when GAN and GAN have the same torsion in corresponding points. Such a mapping we name ”equitorsion mapping”. With respect to each of mentioned above curvature tensors we have obtained quantities E θ i jmn (θ = 1, · · · , 5), that are generalizations of the Weyl tensor, i.e. they are invariants based on f . Among E θ only E 5 is a tensor. All these quantities are interesting in constructions of new mathematical and physical structures. AMS Subj. Class.: 53B05.

Geodesic mapping onto Kählerian spaces of the first kind

In the present paper a generalized Kählerian space of the first kind is considered as a generalized Riemannian space

Visualization of Infinitesimal Bending of Curves

\mathbb{G}\mathbb{R}_N

Determinantal Representation of Outer Inverses in Riemannian Space

with almost complex structure Fih that is covariantly constant with respect to the first kind of covariant derivative.Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings f: with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives with respect to unknown components of the metric tensor and the complex structure of the Kählerian space .

Equitorsion holomorphically projective mappings of generalized Kählerian space of the first kind

Infinitesimal bending of curves in E 3 is considered and an infinitesimal bending field is determined and discussed. A special case of deformation of a plane curve staying plane under infinitesimal bending is analyzed. Also, the variations of the curvature and the torsion are obtained. Infinitesimal bending of an ellipse and Cassini curve are discussed and presented graphically. We present our tool InfBend aimed at infinitesimal deformation and visualization of curves and surfaces. It is written in C++ and uses OpenGL for modelling three-dimensional curves and surfaces.

New projective tensors for equitorsion geodesic mappings

Starting from a known determinantal representation of outer inverses, we derive their determinantal representation in terms of the inner product in the Euclidean space. We define the double inner product of two miscellaneous tensors of rank 2 in a Riemannian space. The corresponding determinantal representation as well as the general representation of outer inverses in the Riemannian space are derived. A non-zero {2}-inverse X of a given tensor A obeying ρ(X) = s with 1 ≤ s ≤ r = ρ(A) is expressed in terms of the double inner product involving compound tensors with minors of order s, extracted from A and appropriate tensors.