Complete classification of parallel Lorentz surfaces in four-dimensional neutral pseudosphere

Abstract

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. 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 four-dimensional (4D) Lorentzian space forms are classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentz space forms,” Tohoku Math. J. 61, 1 (2009)]. Recently, explicit classification of parallel Lorentz surfaces in the pseudo-Euclidean 4-space E24 and in the pseudohyperbolic 4-space H24(−1) are obtained recently by Chen et al. [“Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms,” Int. J. Math. 21, 665 (2010); “Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space,” Cent. Eur. J. Math. 8, 706 (2010)], respectively. In this article, we completely classify the remaining case; namely, parallel Lorentz surfaces in 4D neutral pseudosphere S24(1). Our result states that there are 24 families of such surfaces in S24(1). Conversely, every parallel Lorentz surface in S24(1) is obtained from one of the 24 families. The main result indicates that there are major differences between Lorentz surfaces in the de Sitter 4-space dS4 and in the neutral pseudo 4-sphere S24.