Isometric immersions of pseudo-Riemannian space forms

Abstract

In this paper we study local isometric immersions f: M s n ( K)→ N s+ q 2 n−1 ( c) of a time-like n-submanifold M s n ( K) with constant sectional curvature K and index s into a pseudo-Riemannian space form N s+ q 2 n−1 ( c) with constant sectional curvature c and index s+ q, where q≥0, 1≤ s≤ n−1 and K≠ c. We first prove the existence of Chebyshev coordinates of a time-like submanifold M s n ( K) in certain conditions. Afterwards, we generalize the classical Bäcklund theorem for space-like (or time-like) submanifolds in N n−1 2 n−1 ( c) and N 1 2 n−1 ( c). Finally as an application, in the Chebyshev coordinates, we use the Bäcklund theorem to give a Bäcklund transformation and a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.