Minimal surfaces in $4$-dimensional Riemannian manifolds of constant curvature

Abstract

For surfaces immersed in a Euclidean 4-space E 4 , Little Let x\ M-^E* be an immersion of a compact orientable surface in E 4 . Suppose that KN, the curvature of the connection in the normal bundle, is everywhere positive (negative), then x is inflection free immersion and furthermore (N)=-2l(M) (*(AO=2*(M)), where 7L(M) and I(N) denote the Euler characteristics of M and the normal bundle over M respectively.