On a structure defined by a tensor field of type (1,1) satisfying \(P^3-P=0\) (Q5928335)

Let \(M\) be a differentiable manifold of dimension \(n\) and \(P\) a tensor field of type (1,1) satisfying \(P^3-P=0\) on \(M\), i.e. a \(P\)-structure on \(M\). The author proves that such a \(P\)-structure of rank \(n-1\) is equivalent to an almost Lorentzian paracontact structure on \(M\). The technique of proof is a modification of the one used by \textit{K. Yano} [Tensor, New Ser. 14, 99-109 (1963; Zbl 0122.40705)] for the case of structures satisfying \(f^3+f=0\). Also the author obtains that a \(CR\)-submanifold of a Lorentzian para-Sasakian manifold carries a \(P\)-structure.