Pseudo-totally umbilical lightlike submanifolds

Authors : Samuel SSEKAJJA

Pages : 1304-1320

Doi:10.15672/hujms.932487

View : 24 | Download : 12

Publication Date : 2022-10-01

Article Type : Research Paper

Abstract :We introduce the geometry of pseudo-totally umbilical lightlike submanifold $M$ of a semi-Riemannian manifold $\bar{M}$. In line with the above, we give a complete classification of pseudo-totally umbilical $1$-lightlike submanifolds, such as the lightlike hypersurfaces and half-lightlike submanifolds. Furthermore, pseudo-totally umbilical screen distributions are also investigated, with a complete classification for any lightlike hypersurfaces whose screen distributions are pseudo-totally umbilical. Closely linked to the above we also show, under some geometric conditions, that some pseudo-totally umbilical leaves $M^{*}$, of the screen distribution over $M$, as non-degenerate submanifolds of $\bar{M}$, are either contained in semi-Euclidean spheres or the hyperbolic spaces. Moreover, tangible examples are constructed in this case. Finally, we introduce the notion of mean lightlike sectional curvatures and relate them to the well-known tensors used in the characterisation of lightlike hypersurfaces in space-times.
Keywords : lightlike submanifolds, pseudo totally umbilical submanifolds, mean lightlike sectional curvatures