An investigation on the existence of warped product irrotational screen-real lightlike submanifolds of metallic semi-Riemannian manifolds

PurposeThe purpose of this paper is to study the geometry of screen real lightlike submanifolds of metallic semi-Riemannian manifolds. Also, the authors investigate whether these submanifolds are warped product lightlike submanifolds or not.Design/methodology/approachThe paper is design as follows: In Section 3, the authors introduce screen-real lightlike submanifold of metallic semi Riemannian manifold. In Section 4, the sufficient conditions for the radical and screen distribution of screen-real lightlike submanifolds, to be integrable and to be have totally geodesic foliation, have been established. Furthermore, the authors investigate whether these submanifolds can be written in the form of warped product lightlike submanifolds or not.FindingsThe geometry of the screen-real lightlike submanifolds has been studied. Also various results have been established. It has been proved that there does not exist any class of irrotational screen-real r-lightlike submanifold such that it can be written in the form of warped product lightlike submanifolds.Originality/valueAll results are novel and contribute to further study on lightlike submanifolds of metallic semi-Riemannian manifolds.

Some characterization theorems on GCR-lightlike warped product submanifolds of indefinite nearly Kaehler manifolds

It is shown that for a proper Generalized Cauchy–Riemann ([Formula: see text])-lightlike submanifold of an indefinite nearly Kaehler manifold such that [Formula: see text] defines a totally geodesic foliation in [Formula: see text], there does not exist any warped product [Formula: see text]-lightlike submanifold of the type [Formula: see text]. Then, the existence of [Formula: see text]-lightlike warped product submanifolds of the type [Formula: see text] in indefinite nearly Kaehler manifolds is obtained by establishing a characterization in terms of the shape operator. Further, we prove that for a proper [Formula: see text]-lightlike warped product submanifold of an indefinite nearly Kaehler manifold, the induced connection [Formula: see text] can never be a metric connection. Finally, we derive some characterizations in terms of the canonical structures [Formula: see text] and [Formula: see text] on a [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold enabling it to be a [Formula: see text]-lightlike warped product.

The weighted mixed curvature of a foliated manifold

We introduce the weighted mixed curvature of an almost product (e.g. foliated) Riemannian manifold equipped with a vector field. We define several qth Ricci type curvatures, which interpolate between the weighed sectional and Ricci curvatures. New concepts of the ?mixed-curvature-dimension condition? and ?synthetic dimension of a distribution? allow us to renew the estimate of the diameter of a compact Riemannian foliation and splitting results for almost product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. We also study the Toponogov?s type conjecture on dimension of a totally geodesic foliation with positive weighted mixed sectional curvature.