INTRODUCTION

Let M be a 2-dimensional smooth manifold. Q an n-dimensional space form of constant sectional curvature and x: M→ Q an immersion. We say that we can reduce the codimension to k < n-2 if there is a (k+2)-dimensional totally geodesic submanifold Q’ ‹ Q such that x(M) ‹ Q’. We always equip M with the induced metric. Let N be the normal bundle with its induced connection D and H the mean curvature vector of the immersion.