Explicit Classification of Parallel Lorentz Surfaces in 4d Indefinite Space Forms with Index

A Lorentz surface in 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 general relativity since extrinsic invariants of such surfaces do not change from point to point. Parallel Lorentz surfaces in 4D Lorentzian space forms are classified in [16] by Chen and Van der Veken. Moreover, explicit classification of parallel Lorentz surfaces in 4D indefinite space forms with index 2 are obtained recently in a series of papers by Chen, Dillen and Van der Veken [12, 13, 14]. In this paper, we obtain the complete classification of parallel Lorentz surfaces in 4D indefinite space forms with index 3. Consequently, the complete classification of parallel Lorentz surfaces in 4D indefinite space forms are achieved.