A precise determination of the contact CR $$\delta $$-invariant for typical submanifolds within a Lorentz-Sasakian space form

Abstract The concept of a contact CR -submanifold in a Sasakian manifold (Yano and Kon in Geom Dedicata 10:369-391, 1981) relates deeply to CR geometry, which originated from the study of real hypersurfaces in complex manifolds (Bejancu in Geometry of CR submanifolds. D. Reidel Publishing Company, Dordrecht, 1986). Initially introduced for anti-holomorphic submanifolds in complex space forms (Chen in Int J Math 23(3):1250045, 2012), the CR $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -invariant has recently been extended to Sasakian space forms (Mihai and Presura in J Geom 109:34, 2018). This article establishes an optimal inequality concerning the contact CR $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -invariant on contact CR -submanifolds within the Lorentz-Sasakian space form.