A note on surfaces with prescribed oriented Euclidean Gauss map

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

GENERALIZED WEIERSTRASS-ENNEPER INDUCING, CONFORMAL IMMERSIONS, AND GRAVITY

Basic quantities related to 2D gravity. such as Polyakov extrinsic action, Nambu-Goto action, geometrical action and the Euler characteristic, are studied using generalized Weierstrass-Enneper (GWE) inducing of surfaces in R3. Connection of the GWE inducing with conformal immersion is made and various aspects of the theory are shown to be invariant under the modified Veselov-Novikov hierarchy of flows. The geometry of [Formula: see text] surfaces (h ~ mean curvature) is shown to be connected with the dynamics of infinite- and finite-dimensional integrable systems. Connections with Liouville-Beltrami gravity are indicated.

EXTRINSIC GEOMETRY OF STRINGS AND THE GAUSS MAP OF SURFACES IN ℝn

A two-dimensional Euclidean string world sheet realized as a conformal immersion in ℝn is mapped into the Grassmannian G2,n through the generalized Gauss map. In order for the Grassmannian to represent tangent planes to a given surface, n–2 integrability conditions must be satisfied by the G2,n fields. These conditions are explicitly derived for arbitrary n by realizing G2,n as a quadric in [Formula: see text]. Both the intrinsic and the extrinsic geometrical properties of the string world sheet are expressed in terms of the Kähler σ model fields.

A REALIZATION OF W GRAVITIES IN THE CONFORMAL GAUGE THROUGH GENERALIZED GAUSS MAPS

String dynamics in ℝn with extrinsic geometry is studied in order to understand their hidden symmetries. String world sheet, realized as a conformal immersion in ℝn, is mapped into the Grassmannian G2, n through the Gauss map. This enables us to study the role of the extrinsic curvature in determining the WSO (n) gravities in the conformal gauge. It is shown that, classically, in ℝ3 and ℝ4 the geometry of surfaces of constant mean curvature densities is equivalent to WSO (n) (n = 3, 4) gravities, the corresponding W algebras being Virasoro (Vir) and Vir ⊕ Vir, respectively.