arXiv subjects
Publications and source records attributed to Samuel Ssekajja.
In this note, we show that a lightlike hypersurface of an indefinite Sasakian manifold, which is tangent to structure vector field is not locally symmetric, semi-symmetric or semi-parallel.
We study lightlike hypersurfaces of an indefinite almost contact metric-manifold $\bar{M}$. We prove that there are only two types of such hypersurfaces, known as ascreen and inascreen, with respect to the position of the structure vector field of $\bar{M}$. We also show that the second class of hypersurfaces naturally admits an almost Hermitian structure.
Here, we consider a lightlike hypersurface, tangent to the structure vector field, of an indefinite Sasakian manifold. We prove that no such a hypersurface can either have parallel or recurrent second fundamental forms. In addition to the above, we also prove that no such a hypersurface may have parallel or recurrent induced structural tensors.
In the present paper, we study the geometry of certain classes of null submanifolds of indefinite complex contact manifolds. In particular, we show that quaternion null submanifolds are always totally geodesic. We also present the geometry of distributions on screen real and screen transversal anti-invariant submanifolds.
We study the geometry of null hypersurfaces in indefinite complex contact manifolds. We prove several classification results for a variety of well-known null hypersurfaces, including the totally umbilic, totally screen umbilic, and the screen conformal ones. Furthermore, a characterization of the ambient space is given in case the underlying null hypersurface is totally contact umbilic, totally contact screen umbilic or contact screen conformal, i.e. we have proved that the ambient complex contact manifold must be a space of constant $GH$-sectional curvature of $-3$.
Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expanding horizons in black hole theory. In the present paper, we introduce a degenerate Ricci-type flow in $M'$ via the intrinsic Ricci tensor of $M$. Several new gradient estimates regarding the flow are proved.
First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characterization results, on the above class of null submanifolds, are proved. In particular, we prove that such null submanifolds exist in indefinite Sasakian space forms of constant holomorphic sectional curvatures of $-3$.
We introduce two classes of null hypersurfaces of an indefinite Sasakian manifold, $(\overline{M}, \overlineϕ,ζ, η)$, tangent to the characteristic vector field $ζ$, called; {\it contact screen conformal} and {\it contact screen umbilic} null hypersurfaces. These hypersurfaces come in to fill the existing gap in screen conformal and screen totally umbilic null hypersurfaces. We prove that such hypersurfaces are contained in indefinite Sasakian space forms of constant $\overlineϕ$-sectional curvature of $-3$.
In the present paper, we show that the geometry of a screen integrable null hypersurface can be generated from an isometric immersion of a leaf of its screen distribution into the ambient space. We prove, under certain geometric conditions, that such immersions are contained in semi-Euclidean spheres or hyperbolic spaces, and the underlying null hypersurfaces are necessarily umbilic and screen totally umbilic. Where necessary, examples have been given to illustrate the main ideas.
We introduce higher order mean curvatures of screen almost conformal (SAC) half-lightlike submanifolds of indefinite contact manifolds, admitting a semi-symmetric non-metric connection, and use them to generalize some known results of [6]. Also, we derive a new set of integration formulae via the divergence of some special vector fields tangent to this submanifold. Several examples are also included to illustrate the main concepts.
We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [4].
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.
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 $μ$-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.
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.