Minimal immersion of pseudo-Riemannian manifolds

Abstract

j=3+l where (xu x2,･■■, xn)is a rectangular coordinate system of E%. (E%, g)isa flat Pseudo-Riemannian manifold of signature (q,n―q). Let c be a point in E%+1 (or Effi) and r>0. We put S%(c,r)= (iE￡,a+1: g(x-c, x-c)=r2} H≫(ctr)={x^E≫#'.g{x-c, x-c)=-r2). Itis known that S%(c,r) and H%(c, r)arecomplete Pseudo-Riemannian manifolds ofsignature(q,n―q)and respective constant sectionalcurvatures r~2and ―r~2. Sq(c,r) and H%(c, r) are calledthe Pseudo-Riemannian sphere and the Pseudohyperbolic space,respectively. The point c is calledthe center of S%(c,r) and H*{c, r). In the following, SJ(O,r) and Hj(0, r) are simply denoted by S%(r) and Hq(r), respectively. AT^ denotes the Pseudo-Riemannian manifold with metric tensor of signature (p, n―p). The Pseudo-Riemannian manifold, the Pseudo-Euclidean space,thePseudo-Riemannian sphere and the Pseudo-hyperbolic space are simply denoted by the P―R manifold, the P―E space, the P―R sphere and theP―h space. The P―R manifold A/?is calledthe Lorentz manifoldand the P―E space E＼is calledthe Minkowski space. Let /: M ^>Ni be an isometric immersion of a P― R manifold M in another P―R manifold A^. That is f*g=g, where g and g are theindefinite metric tensors of M$ and A7￡,respectively. T(M%) and TL{Mf) denote the tangent bundle and the normal bundle of M|＼ 7, 7 and 1L denote theRiemannian connections and thenormal connection on Mf, N% and TX(M^), respectively. Then for any vectorfieldsX, Y<bT(M$), v(=Tl{M ),we have theGauss formula VYY=VrY + B(X.Y).