The Gauss Map for Surfaces: Part 1. the Affine Case

Let M be a connected oriented surface and let G'2 be the Grassmannian of oriented 2-planes in Euclidean (2 + c)-space. E2 + l. Smooth maps t: M -» (7f are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If í is a Gauss map of an immersion X: M -» E2 + 1, we study the extent to which / uniquely determines X under certain circumstances. Let X: M -* E" + ' be an immersion of an oriented «-dimensional manifold into Euclidean (n + c)-space. Associated to X is the tangent plane map t: M -* G¡¡, where G¡¡ is the Grassmannian of oriented «-planes through the origin of E"4 '. The map t assigns lo p e X the oriented «-plane dX(TpM), where TpM is the tangent space to M at p. This generalizes the classical Gauss map for surfaces in E3, so we call t the Gauss map of X. On the other hand, suppose /: M -* Gcn is a smooth map. Is t the Gauss map of an immersion, or even locally a Gauss map, and to what extent does t determine XI In [9] we consider this question for « = 2 and c = 2 under the assumption that / is an immersion. Y. A. Aminov [2] studied the same existence question under essentially the same assumptions as imposed in [9]. More recently, Hoffman and Osserman [4, 5] studied a closely related question for « = 2 and c arbitrary under the additional assumption that M is a Riemann surface, i.e., M possesses a conformai structure. In this paper, we again consider the given question, but only under the assumption that M is a surface, i.e., « = 2. In Part 2 of this paper, we take a second look at the results of Hoffman and Osserman [4, 5] from the point of view established in this paper. Some of the main results in this paper deal with maps / which are not immersions. Theorem 1 states a necessary and sufficient condition for a rank 1 map v. M —> G2 —i.e., rank(r*| ) = 1 for all p e M—defined on a simply connected plane domain M to be a Gauss map. A corollary to Theorem 1 is that any rank 1 map t: M -» G\ = S2{\) defined on a simply connected plane domain is a Gauss map. In Theorem 2 we establish a sufficient condition on / in order for t to be a Gauss map on a neighborhood of a point p given that rank(f „| ) = 1. We also show that the theory developed in [9] for immersions t: M -» G2 holds exactly in the same form for immersions t: M -* G2, where c > 2, if what we call Received by the editors December 5, 1984. 1980 Mathematics Subject Classification. Primary 53A05; Secondary 53A15. 53C42. Kev words and phrases. Orientable surface. Gauss map of an immersion, Grassmannian. 1 1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 431 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use