THE HARMONICITY OF THE SPHERICAL GAUSS MAP MARCO RIGOLI 1. Introduction Given an isometric immersion/: M m -* U n , its generalized Gauss map g in the sense of Chern and Osserman is the map g:M ->G m (M n ), the Grassmanian manifold of m-planes in U n , assigning to each point pin M the tangent space at p of M viewed as an w-dimensional plane in U n . Using the fact that G 2 (U n ) has a natural complex structure, Chern [1] proved that if M 2 is a Riemann surface minimally immersed into IR n , then its Gauss map g is anti-holomorphic. Ruh and Vilms [3] gave the beautiful generalization that for arbitrary m, g is harmonic if and only if M has parallel mean curvature. Besides the map g, another Gauss map v, called the spherical Gauss map, plays an important role in the study of immersed submanifolds of U n via the notion of total curvature. Such a map is defined on the unit normal bundle TMf of M and has values in the unit sphere S n ~ l c U n . Its definition is as follows: for each pair (p, £(/>)) e TM^ where £(/?) is a unit normal vector to f{M) at f{p), psM, V (P> C(p)) = C(p) viewed as an element of S"" 1 . The aim of this paper is to show that a result analogous to that of Ruh and Vilms holds, precisely: THEOREM. Let f:M m -*• U n be an isometric immersion. Then its spherical Gauss map v.TM^ -> S"" 1 is harmonic if and only if M has parallel mean curvature. In the next paragraph we begin by describing the Riemannian structure of TM^. 2. The geometry of the normal bundle Let f.M-*N be an isometric immersion of a Riemannian manifold M of dimension w, into a Riemannian manifold N of dimension n. We agree to use the following ranges of indices throughout the paper: 1 ^ i,j, k, ..., ^ m, m + \ ^<x,/3,y, ..., < n, 1 ^ A, B, C, ..., ^ n\ moreover repeated indices imply sum- mation. Let {0 A } be a (local) orthonormal coframe of AT with corresponding Levi-Civita connection forms co$, curvature forms and relative structure equations. Suppose we have chosen {0 A } in such a way that it is a Darboux coframe along/, that is, f*6* = 0. (1) Then any change of Darboux coframe along / i s of the type 9 = KB, (2) where the notation 0 means the column vector of components & A , ~ refers to quantities with respect to the new coframe and AT is a (locally defined) function with values in Received 24 October 1985. 1980 Mathematics Subject Classification 53C42. Bull. London Math. Soc. 18 (1986) 609-612.