Quasi Generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds

In this paper, we introduce and study a new class of CR-lightlike submanifold of an indefinite nearly Sasakian manifold, called Quasi Generalized Cauchy-Riemann (QGCR) lightlike submanifold. We give some characterization theorems for the existence of QGCR-lightlike submanifolds and finally derive necessary and sufficient conditions for some distributions to be integrable.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1604.05436; text overlap with arXiv:1006.1329 by other author

A note on quasi generalized CR-lightlike geometry in indefinite nearly μ\mu-Sasakian manifold

The concept of quasi generalized CR-lightlike was first introduced by the authors in [18]. In this paper, we focus on ascreen and co-screen quasi generalized CR-lightlike submanifolds of indefinite nearly $\mu$-Sasakian manifold. We prove an existence theorem for minimal ascreen quasi generalized CR-lightlike submanifolds admitting a metric connection. Classification theorems on nearly parallel and auto-parallel distributions on a co-screen quasi generalized CR-lightlike submanifold are also given. Several examples are also constructed, where necessary, to illustrate the main ideas.Comment: 18 page

On three dimensional affine Szab\'o manifolds

In this paper, we consider the cyclic parallel Ricci tensor condition, which is a necessary condition for an affine manifold to be Szab\'o. We show that, in dimension $3$, there are affine manifolds which satisfy the cyclic parallel Ricci tensor but are not Szab\'o. Conversely, it is known that in dimension $2$, the cyclic parallel Ricci tensor forces the affine manifold to be Szab\'o. Examples of $3$-dimensional affine Szabo manifolds are also given. Finally, we give some properties of Riemannian extensions defined on the cotangent bundle over an affine Szab\'o manifold.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1604.05420, arXiv:1604.0542

Some symmetry properties of four-dimensional Walker manifolds

In this paper, we investigate geometric properties of some curvature tensors of a four-dimensional Walker manifold. Some characterization theorems are also obtained.Comment: 10 page

Some remarks on quasi generalized CR-null geometry in indefinite nearly cosymplectic manifolds

In [21], the authors initiated the study of quasi generalized CR (QGCR)-null submanifolds. In this paper, attention is drawn to some distributions on ascreen QGCR-null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifold.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1604.05426, text overlap with arXiv:1006.1329 by other author

On twisted Riemannian extensions associated with Szab\'o metrics

Let $M$ be an $n$-dimensional manifold with a torsion free affine connection $\nabla$ and let $T^*M$ be the cotangent bundle. In this paper, we consider some of the geometrical aspect of a twisted Riemannian extension which provide a link between the affine geometry of $(M,\nabla)$ and the neutral signature pseudo-Riemannian geometry of $T^*M$. We investigate the spectral geometry of the Szab\'o operator on $M$ and on $T^*M$.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1604.0542

Induced and intrinsic Hashiguchi connections on Finsler submanifolds

We study the geometry of Finsler submanifolds using the pulled-back approach. We define the Finsler normal pulled-back bundle and obtain the induced geometric objects, namely, induced pullback Finsler connection, normal pullback Finsler connection, second fundamental form and shape operator. Under a certain condition, we prove that induced and intrinsic Hashiguchi connections coincide on the pulled-back bundle of Finsler submanifold.Comment: 13 page

Sasakian Finsler structures on pulled-back bundle

Under a pulled-back approach given in [1] and firstly presented in [2], we introduce, in this paper, the concepts of almost contact and normal almost contact Finsler structures on the pulled-back bundle. Properties of structures partly Sasakians are studied. Using the hh-curvature tensor of Chern connection given in [2], we obtain some characterizations of horizontally Finslerian K-contact structures via the horizontal Ricci tensor and the flag curvature.Comment: 16 page

Affine Szab\'o connections on smooth manifolds

In this paper, we introduce a new structure, namely, affine Szab\'o connection. We prove that, on $2$-dimensional affine manifolds, the affine Szab\'o structure is equivalent to one of the cyclic parallelism of the Ricci tensor. A characterization for locally homogeneous affine Szab\'o surface is obtained. Examples of two- and three-dimensional affine Szab\'o manifolds are also given.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1604.0542

Warped products with a Tripathi connection

The warped product $M_1 \times_F M_2$ of two Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$ is the product manifold $M_1 \times M_2$ equipped with the warped product metric $g=g_1 + F^2 g_2$, where $F$ is a positive function on $M_1$. The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such a notion plays very important roles in differential geometry as well as in physics, especially in general relativity. In this paper we study warped product manifolds endowed with a Tripathi connection. We establish some relationships between the Tripathi connection of the warped product $M$ to those $M_1$ and $M_2$.Comment: 13 page