关于调和Gauss映照与相对仿射Gauss映照

On Harmonic Gauss Maps and Relatively Affine Gauss Maps

本文用对应的Gauss映照来讨论欧氏球面上的等距浸入的调和性,得到了相应的结果.本文又讨论了相对仿射映照,考虑伪球面的情况,得到与欧氏球面上相类似的结果.

Let Sm+p(c) be a Euclidean (m + p)-sphere of constant curvature c and (?):Mm→Sm+p(c) the isometric immersion of an m-dimensional Riemannian manifold Mm into Sm+p(c). By the idea of A.Ros, there is an isometric immersion f:Sm+p(c)→SM(m+p+1), the space of symmetric matrices of order (m + p + 1), defined by (x) = x'x for x$Sm+p(c). So, we have the composition %= x ψ:Mm→SM(m+p+1). The main results of this paper are as follows. Theorem 2.1. Let ψ, f and φ= fψ be as above. Suppose that g:Mm→Gm,n is the Gauss map of Mm to the Grassmann manifold Gm,n where n =1/2(m + p + 1)(m + p + 2) - m. Then, gφ is harmonic if and only if ψ istotally geodesic. Let Hm+1 be a unit pseudo-sphere of dimension m + 1 and f:Mm→Hm+1 the isometric immersion of an m-dimensional complete Riemannian manifold Mm into Hm+1. Denote by ψ:Mm→Q the generalized Gauss map with respect to f. Theorem 3.1. Let f,ψ be as above. If ψ is non-trivial relatively affine, then Mm is either (i)Mm = Sk(λ2-1)xHm-k(λ-2-1) for some real number or