Complete explicit classification of parallel Lorentz surfaces in arbitrary pseudo-Euclidean spaces

Abstract

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in physics since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel spacelike surfaces in an arbitrary indefinite space form are classified in Chen (2010) [20]. Moreover, parallel Lorentz surfaces in 4D indefinite space forms are completely classified in a series of recent articles Chen (submitted for publication) [16], Chen (submitted for publication) [17], Chen (in press) [18], Chen (2010) [19], Chen and Van der Veken (2009) [15] (see also Graves (1979) [12], Graves (1979) [13] and Magid (1984) [14] for some partial results). In this paper, we achieve the complete classification of parallel Lorentz surfaces in a pseudo-Euclidean space with arbitrary codimension and arbitrary index.