Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension

A spatial surface of a pseudo-Riemannian space form is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. It is well known that a surface in a pseudo-Riemannian space form is parallel if and only if it is locally invariant under the reflection with respect to the normal space at each point. Such surfaces are important in geometry as well as in general relativity since the extrinsic invariants of the surfaces do not change from point to point. Recently, parallel spatial surfaces in 4-dimensional Lorentzian space forms were classified by Chen and Van der Veken (2009) [6]. In this article, we completely classify parallel spatial surfaces in pseudo-Riemannian space forms with an arbitrary index and dimensions. As an immediate by-product, we achieve the classification of all spatial surfaces in Lorentzian space forms with arbitrary dimensions.