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In this paper we study the second covariant derivative of Riemannian curvature tensor. Some new identities for the second covariant derivative are given. Namely, identities obtained by cyclic sum with respect to three indices are given. In the first case, two curvature tensor indices and one covariant derivative index participate in the cyclic sum, while in the second case one curvature tensor index and two covariant derivative indices participate in the cyclic sum.

In this paper, we study warped product bi-slant submanifolds of cosymplectic manifolds. It is shown that there is no proper warped product bi-slant submanifold other than pseudo-slant warped product. Finally, we give an example of warped product pseudo-slant submanifolds.

In this paper, we consider equitorsion third type almost geodesic mappings of non-symmetric affine connection spaces. Applying different computational methods, we find invariants of these mappings obtained from curvature tensors of a corresponding non-symmetric affine connection space. Weyl projective tensor of a symmetric affine connection space is generalized in this way for the case of a third type almost geodesic mapping.

In the present paper we study equitorsion geodesic mappings between two generalized Riemannian spaces f : GR N → G R ‾ N . In this case these spaces have the same torsions at corresponding points. We prove that a generalized Riemannian space GR N admits an equitorsion geodesic mapping onto a generalized Riemannian space G R ‾ N if and only if in GR N certain differential equations of Cauchy type in covariant derivatives of the θ = 1 , … , 4 kinds have a solution with respect to the unknown tensor a i j , the gradient vector λ i ≠ 0 , and the differentiable function μ θ , θ = 1 , … , 4 . In fact we find four systems of PDE all equivalent to the existence of an equitorsion geodesic mapping and discuss the number of linearly independent solutions of this system of PDE. We establish an upper bound for the number of solutions for the geometrical problem under consideration.

Invariants of second-type almost geodesic mappings are obtained in this paper. These invariants are generalizations of Thomas projective parameter and Weyl projective tensor.

Uddin, Siraj/0000-0002-3564-6405 WOS: 000461185300012 In this study, we introduce a new class of pseudo f-structure, called hyperbolic f-structure. We give some classifications of this new structure. Further, we extend the notion of (kappa, mu, nu)-nullity distribution to hyperbolic almost Kenmotsu f-manifolds. Finally, we construct some non-trivial examples of such manifolds.

We study G-almost geodesic mappings of the second type $$\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2$$ , θ = 1,2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e-structures that generate mappings of type $$\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2$$ , θ = 1,2. For a mapping $$\mathop {{\pi _2}}\limits_\theta (e,F),\theta = 1,2$$ , θ = 1,2, we determine the basic equations which generate them.

We introduce a special kind of almost geodesic mappings of the first type \(\pi _1^*\) of spaces with non-symmetric affine connections. Also, we investigate a special class of equitorsion almost geodesic mappings of type \(\pi _1^*\) and find some invariant geometric objects of these mappings.

We investigate two kinds of special almost geodesic mappings of the third type in this paper. We also find some relations for curvature tensors of almost geodesic mappings of the third type.

Vanishing of linearly independent curvature tensors of a non-symmetric affine connection space as functions of vanished curvature tensor of the associated space of this one are analyzed in the first part of this paper. Projective curvature tensors of a non-symmetric affine connection space are expressed as functions of the affine connection coefficients and Weyl projective tensor of the corresponding associated affine connection space in the second part of this paper.

In this paper we investigate a special kind of almost geodesic mapping of the third type of spaces with non-symmetric affine connection. Also we find some relations for curvature tensors of associated affine connection spaces of almost geodesic mappings of the third type. Finally, we investigate equitorsion almost geodesic mapping having the property of reciprocity and find an invariant geometric object of this mapping.

The basic information on generalized Riemannian space GR"N and its submanifold X"M and normal bundle X"N"-"M^N are given. As introduced connections are generally asymmetric, it is possible to define four kinds of covariant derivative. In the present work one considers integrability conditions of derivational equations obtained by using the 1st and the 2nd kind of covariant derivative. We obtain also the corresponding Gauss-Codazzi equations. Similar results using simultaneously the both kinds of covariant derivative are obtained.

Following the basic facts of [Formula: see text]-planar mappings introduced by Mikeš and Sinyukov and further developed by Hinterleitner and Mikeš, we consider the notions of an [Formula: see text]-planar curve and an [Formula: see text]-planar mapping, but in case of manifolds endowed with a non-symmetric linear connection and an affinor structure. Consequently, we investigate infinitesimal [Formula: see text]-planar and particularly geodesic transformations of manifolds with non-symmetric linear connection and find necessary and sufficient conditions for the existence of these transformations.

In the works by Mincic et al. (1998, 1999, 2007) [8,9,11] we have studied the problem (started by M. Prvanovic): Can a generalized Riemannian space GR"N (with non-symmetric basic tensor) contain a Riemannian subspace (possessing a symmetric induced basic tensor)? In the present work the analogous problem is studied for a space L"N with non-symmetric affine connection and its subspace L"M. In the Section 2 it is assumed that the equations defining submanifold are given and that it is valid (2.4), i.e., induced torsion T~=0, and the connection on the manifold M"N is determined so that T 0. At the third section we start from connection on the manifold M"NT 0. From the mentioned condition (2.4) we determine the Eq. (1.1) defining the submanifold M"M@?M"N, so that induced torsion T~=0, i.e., we get L"M^O@?L"N. It is proved that the answer to the question cited above is affirmative and several examples are constructed. The examinations have local character.

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.

In this paper we consider equitorsion geodesic mappings of equiaffine spaces of the ?-kind, ? ? {0, ..., 5}. In the case when ? ? {0, 5}, vector ?i , which determines that mapping, is gradient, which doesn?t hold in general case. We found the condition when the vector ?i is gradient in the case of ? ? {1, ..., 4}. Some invariant geometric objects of such mappings are found. Anti-equiaffine spaces of ?-kind, ? ? {0, ..., 5} are introduced and discussed. Equitorsion geodesic mappings of such spaces are described and some invariants are found.

In this paper we define generalized Kahlerian spaces of the first kind ( $$ G_1^K N $$ ) given by (2.1)–(2.3). For them we consider hollomorphically projective mappings with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces ( $$ G_1^K N $$ and $$ G_1^{\bar K} N $$ ) and for them we find invariant geometric objects.

Using recent work as a foundation, we have considered the infinitesimal deformations f:x^i->x^i+@ez^i(x^j) of a space L"N with non-symmetric affine connection L"j"k^i. On the basis of the non-symmetry of the connection, we use four kinds of covariant derivatives to express the Lie derivative and the deformations. The rigidities of the geometric objects (connection, tensors, curvature) are defined and analyzed using the Lie derivative.

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.

Starting from the definition of generalized Riemannian space (GRN) [1], in which a non-symmetric basic tensor Gij is introduced, in the present paper a generalized K?hlerian space GK2 N of the second kind is defined, as a GRN with almost complex structure Fhi, that is covariantly constant with respect to the second kind of covariant derivative (equation (2.3)). Several theorems are proved. These theorems are generalizations of the corresponding theorems relating to KN. The relations between Fhi and four curvature tensors from GRN are obtained.

Minimal surfaces are the surfaces of the smallest area spanned by a given boundary. The equivalent is the definition that it is the surface of vanishing mean curvature. Minimal surface theory is rapidly developed at recent time. Many new examples are constructed and old altered. Minimal area property makes this surface suitable for application in architecture. The main reasons for application are: weight and amount of material are reduced on minimum. Famous architects like Otto Frei created this new trend in architecture. In recent years it becomes possible to enlarge the family of minimal surfaces by constructing new surfaces.

In this paper we investigate holomorphically projective mappings of generalized Kahlerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kahlerian spaces we obtain five invariant geometric objects for these mappings.

In tins paper we consider conoid surfaces as frequently used surfaces in building techniques, mainly as daring roof structures. Different types of conoids are presented using the programme package Mathematica. We describe the generation of conoids and by means of parametric representation we get their graphics. The geometric approach offers a wide range of possibilities in the research of complicated spatial surface systems.