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The author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.

The author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.

The author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).

To the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form).

In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant) of warped product bi-slant submanifolds of Kenmotsu manifolds in terms of the warping function (an intrinsic invariant) and bi-slant angles. The equality case is also considered. Some applications of derived inequality are given.

The 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.

The 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.

The 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.

All results are novel and contribute to further study on lightlike submanifolds of metallic semi-Riemannian manifolds.

In this paper some characterizations for the existence of warped product pointwise semi-slant submanifolds of cosymplectic space forms are obtained. Moreover, a sharp estimate for the squared norm of the second fundamental form is investigated, the equality case is also discussed. By the application of derived inequality, we compute an expression for Dirichlet energy of the involved warping function. Finally, we also proved some classifications for these warped product submanifolds in terms of Ricci solitons and Ricci curvature. A non-trivial example of these warped product submanifolds is provided.