Screen conformal invariant light-like hypersurfaces of indefinite Sasakian space forms (Q380753)

The paper begins with a review of the geometry of indefinite Sasakian manifolds including the structure vector field \(\xi\), the fundamental collineation \(\bar\phi\), the notion of \(\bar \phi\)-sectional curvature and indefinite Sasakian space forms \(\bar M(c)\) of constant \(\bar \phi\)-sectional curvature \(c\). NEWLINENEWLINENEWLINEA hypersurface \(M\) of an indefinite Sasakian manifold \(\bar M\) of dimension \(2n+1\) is said to be light-like if the induced metric is of constant rank \(2n-1\) and the normal bundle \(TM^\perp\) is a subbundle of the tangent bundle \(TM\) of rank 1. There exists a (non-unique) non-degenerate complementary vector bundle \(S(TM)\) of \(TM^\perp\) in \(TM\) called a screen distribution. Locally for any null section \(E\) of \(TM^\perp\) there exists a particular null section \(N\) in the orthogonal complement of \(S(TM)^\perp\) in \(T\bar M\). Associated to \(N\) and \(E\) one has shape operators \(A_N\) and \(A^*_E\).NEWLINENEWLINENEWLINEThe hypersurface \(M\) is invariant if \(\xi\) is tangent and \(\bar\phi X\) is tangent for \(X\) tangent. An invariant light-like hypersurface is said to be screen locally conformal if \(A_N=\varphi A^*_E\) for some local non-vanishing smooth function \(\varphi\) on \(M\).NEWLINENEWLINEThe author first proves that for a screen conformal invariant light-like hypersurface of an indefinite Sasakian space form \(\bar M(c)\) one has that \(c=-3\). Moreover for positive type number \(M\) is not totally geodesic and \(\varphi\) can be controlled by a differential equation. NEWLINENEWLINENEWLINEThe author then studies the leaves \(M'\) of the screen distribution and relates the geometric quality (totally geodesic, minimal, etc.) of the leaf to the same quality of the screen conformal invariant light-like hypersurface in \(\bar M(c)\).