The Gauss Map for Surfaces : Part 2 . the Euclidean Case

We study smooth maps t: M -> Ci of a Riemann surface M into the Grassmannian Gi of oriented 2-planes in E2 ' ' and determine necessary and sufficient conditons on t in order that it be the Gauss map of a conformai immersion X: M -» E2 + '. We sometimes view / as an oriented riemannian vector bundle; it is a subbundle of Ej/'. the trivial bundle over M with fibre E2 + l. The necessary and sufficient conditions obtained for simply connected M involve the curvatures of t and tx , the orthogonal complement of t in "É\f ', as well as certain components of the tension of ; viewed as a map t: M -» Sc (1), where Sc(l) is a unit sphere of dimension C that contains Ci as a submanifold in a natural fashion. If t satisfies a particular necessary condition, then the results take two different forms depending on whether or not t is the Gauss map of a conformai minimal immersion. The case t: M -» Gl is also studied in some additional detail. In [5, 6], Hoffman and Osserman study the following question: Let M be a Riemann surface and /: M -* G2 be a smooth map into the Grassmannian of 2-planes in (2 + c)-space; when is t the Gauss map of a conformai immersion X: M -» E2+c? In [5, 6] necessary and sufficient conditions are established when M is simply connected, in order for / to be a Gauss map. The purpose of this paper is primarily to redo the work of Hoffman and Osserman from the point of view established in [12, 13]. One reason for doing this is to free their results of its dependence on the use of complex variables in order to allow and suggest generalizations from the case of surfaces to higher dimensional manifolds. Also, to some extent, the necessary and sufficient conditions established in [5, 6] in order for t to be a Gauss map are somewhat mysterious (to me at least) and could use some illumination. In particular, we obtain corresponding conditions which are stated directly in terms of traditional geometric invariants. We will briefly describe these invariants and also describe how they appear in the theorems of this paper. Let g0 be the standard metric on G2. We say t: M -* G2 is conformai if g0 = t*g0 induces the given conformai structure on M, where t* does not vanish. One may view / not only as a map but quite naturally as an oriented riemannian rank 2 vector bundle over M, a subbundle of the trivial bundle of E2/' over M with fibre E2 + <. As such we can define k: M -» R to be the curvature of t with respect to the Received by the editors December 5, 1984. 1980 Mathematics Subject Classification. Primary 53A05; Secondary 53C42.