Parallel Radical and Screen Distributions on Lightlike Hypersurfaces of Indefinite Statistical Manifolds

Abstract

We study the radical and screen distributions of a lightlike hypersurface of an indefinite statistical manifold with respect to the induced connections arising from the ambient dual affine connections. Necessary and sufficient conditions are obtained for each distribution to be parallel with respect to both induced connections. These conditions are expressed through the screen shape operators, the local screen fundamental forms, the mean induced connection, and the tangential difference tensor. When both distributions are parallel, we characterize the screen projection and obtain a block decomposition of the tangential difference tensor, whose mixed radical-screen part vanishes. We also obtain equivalent conditions for the induced connections to be metric connections. In the metric case, the induced screen connections coincide with the Levi--Civita connection on each integral manifold of the screen distribution, and the curvature tensors of the three induced connections agree on screen directions. We further determine the action of these curvature tensors on the radical direction. An explicit example shows that both distributions may be parallel although neither induced connection is a metric connection and their curvature tensors need not vanish.