Invariant Submanifolds of Generalized Sasakian-Space-Forms

The new characterizations of invariant submanifolds of generalized Sasakian-space forms (SSF)   in terms of their behavior with respect to the various curvature tensors are obtained in this work. By examining the connections between these submanifolds' second fundamental form σ and certain curvature tensors Wi with i = 2, 3, 4, 6, 7, we derive necessary and sufficient conditions for their geodesicity. We show that total geodesicity corresponds to the claim that tensor products vanish, Q(σ, Wi) = 0, subject to different non-degeneracy conditions for each curvature tensor. The essential characterization comes from the W6 curvature tensor, which suffices to fulfil 2n(f1−f3)≠0, and all other tensors lead to complementary constraints concerning the structural functions f1, f2, and f3. These findings offer various avenues for studying the geometric nature of invariant submanifolds and enhance our insights into the behaviour of generalized Sasakian-space-forms.