Reduction of codimension of surfaces (Q1262553)

Let x: \(M^ 2\to Q^ n\) be a smooth immersion of a two-manifold \(M^ 2\) into a space form Q such that its mean curvature vector lies in a parallel subbundle E of its normal bundle. The authors show that the codimension of x can be reduced to the following values k under suitable assumptions: 1) If x is analytic then \(k=\dim E+1\) or \(x(M^ 2)\) is minimal in an (n-dim E)-dimensional totally umbilic submanifold of \(Q^ n\). 2) If \(M^ 2\) is a 2-sphere then \(k=\dim E\) or \(x(M^ 2)\) is minimal as above. [Due to some undefined notation in the main theorems this has to be concluded from the proofs given in this paper.]