Isometric immersions with congruent Gauss maps

1. IntroductionThe purpose of this article is to present a refinement of some aspects of abeautiful theorem due to Dajczer and Gromoll [DG], which characterizes pairsof isometric immersions from a given Riemannian manifold into Euclideanspace with congruent Gauss maps. We will reformulate part of the theory ofcircular isometric immersions, which is the proper context for the theorem, bysystematically utilizing the notion of complexification.The simplest case of two distinct isometric immersions with congruent Gaussmaps is provided by the helicoid and the simply connected cover of the catenoid.These are conjugate minimal surfaces, i.e. the real and imaginary parts of aholomorphic map from a Riemann surface into C , and they fit into a one-parameter family of associated minimal surfaces, which are pictured in doCarmo's text [dC, pp. 223-224]. All of the associated minimal surfaces havecongruent Gauss maps.More generally, it was discovered by Dajczer, Gromoll, and Rodriguez thata minimal isometric immersion / from a simply connected Kahler manifoldM into a real Euclidean space R is the real part of a holomorphic map hfrom M into C . The maps fg — re(e' h), for 0gK, are minimal isometricimmersions with congruent Gauss maps, and the collection {fg : 8 e R} iscalled an associated family of minimal isometric immersions.Theorem. Let M be a complete simply connected Riemannian manifold, andsuppose that