Rigidity results for complete spacelike submanifolds in plane fronted waves

TL;DR

This paper establishes new rigidity theorems for complete spacelike submanifolds in plane fronted waves, showing they are contained in lightlike hypersurfaces or are totally geodesic wavefronts under certain conditions.

Contribution

It provides novel rigidity results for spacelike submanifolds in plane fronted waves, extending understanding of their geometric structure.

Findings

Complete spacelike submanifolds are contained in characteristic lightlike hypersurfaces.

Certain extremal submanifolds are shown to be totally geodesic wavefronts.

Rigidity results depend on specific geometric assumptions.

Abstract

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.