Locally symmetric half lightlike submanifolds in an indefinite Kenmotsu manifold (Q2904301)

A codimension \(2\) submanifold \(M\) of a pseudo-Riemannian manifold \((\bar M, \bar g)\) is called half light-like if the radical distribution \(\mathrm{Rad}(TM)=TM \cap TM ^{\perp}\) is a vector subbundle of \(TM\) and \(TM^\perp\) has rank one. Because of possible ranks of \(\mathrm{Rad}(TM)\), all codimension \(2\) light-like submanifolds are either half light-like or coisotropic. Given a half light-like submanifold \(M\) of \((\bar M,\bar g)\), one has NEWLINE\[NEWLINE T\bar M = TM \oplus \mathrm{tr}(TM)=\mathrm{Rad}(TM) \oplus \mathrm{tr}(TM)\oplus _{\mathrm{orth}} S(TM),NEWLINE\]NEWLINE where \(\mathrm{tr}(TM)\) and \( S(TM)\) respectively denote the transversal vector bundle and the screen distribution and \(\oplus _{\mathrm{orth}}\) denotes the \(\bar g\)-orthogonal direct sum.NEWLINENEWLINEIn the paper under review, the authors consider half light-like submanifolds of indefinite Kenmotsu manifolds \((\bar M, J,\zeta,\theta,\bar g)\). The following results are proved:NEWLINENEWLINE\(\bullet\) The structure \(1\)-form \(\theta\) is closed on \(M\).NEWLINENEWLINE\(\bullet\) If \(M\) is locally symmetric and \(tr(TM)\) is parallel (with respect to \(\bar \nabla\)), then the induced Ricci type tensor \(R^{(0,2)}\) of \(M\) is symmetric.NEWLINENEWLINE\(\bullet\) If \(M\) is irrotational and locally symmetric and \(tr(TM)\) is parallel, then \(M\) has constant sectional curvature \(0\) and \(\zeta\) is normal to \(M\).