Partially Totally Real Submanifolds of Sasakian Manifolds

Abstract

Partially totally real (PTR) submanifolds were introduced in Kähler geometry by distinguishing a totally real distribution and leaving its orthogonal complement unrestricted. In this paper we develop the corresponding framework for submanifolds of Sasakian manifolds tangent to the Reeb vector field. After separating the Reeb direction, we define the totally real and ambiguous distributions and show that anti-invariant, contact CR, hemi-slant and pointwise hemi-slant submanifolds occur as special cases of the Sasakian PTR framework. We establish the basic tangential and normal decompositions, study maximality and integrability, and derive the additional restrictions produced by the Reeb field. We then investigate the canonical morphisms \(P\) and \(F\), the geometry of the associated distributions, and PTR-submanifolds in Sasakian space forms. Explicit models are included to illustrate the principal structures and the differences from the Kähler case.