The geometric characteristics of SGL submanifolds in an indefinite Sasakian statistical manifold equipped with a quarter symmetric metric connection

This research paper examines the geometric structure of screen generic lightlike (SGL) submanifolds in an indefinite Sasakian statistical manifold equipped with a quarter-symmetric (QS) metric connection. The study focuses on analyzing the integrability conditions and the parallelism properties of various distributions associated with these submanifolds. It explores the characteristics of totally geodesic foliations and mixed geodesic submanifolds, providing significant insights into their geometric behavior. In addition to the theoretical development, the paper also presents an illustrative example of a contact SGL submanifold within an indefinite Sasakian statistical manifold.

💡 Research Summary

The paper investigates the geometry of screen generic lightlike (SGL) submanifolds situated in an indefinite Sasakian statistical manifold equipped with a quarter‑symmetric (QS) metric connection. The authors begin by recalling the basic notions of statistical manifolds, namely a semi‑Riemannian metric (\tilde\rho) together with a pair of torsion‑free affine connections (\bar\nabla) and its dual (\bar\nabla^*) satisfying the Codazzi‑type condition ((\bar\nabla_X\tilde\rho)(Y,Z)=(\bar\nabla_Y\tilde\rho)(X,Z)). The difference tensor (K=\bar\nabla-\bar\nabla^\circ) (where (\bar\nabla^\circ) is the Levi‑Civita connection) is symmetric and compatible with (\tilde\rho).

A Sasakian structure ((\phi,\nu,\eta,\tilde\rho)) is introduced, characterized by (\bar\nabla^\circ_X\nu=-\phi X) and ((\bar\nabla^\circ_X\phi)Y=\tilde\rho(X,Y)\nu-\eta(Y)X). When a statistical connection (\bar\nabla) satisfies the additional algebraic condition (K(X,\phi Y)+\phi K(X,Y)=0), the quadruple ((\bar\nabla,\tilde\rho,\phi,\nu)) defines an indefinite Sasakian statistical manifold.

The quarter‑symmetric metric connection (\tilde D) is defined by
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