We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H. Conversely, any nowhere antiholomorphic harmonic map into H is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of compl...

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 map G and formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determin...

The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is t...

The aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra A considered separately by F. Boca and D. Mundici. The center of A is isomorphic to C[0, 1], so we first consider the action of the Gauss map on C[0, 1] and then extend the map to A and show...

We study the biharmonic stress-energy tensor S2 of Gauss map. Adding few assumptions, the Gauss map with vanishing S2 would be harmonic.

We discover a bijective map between the Gauss Map and the left-half of the Stern-Brocot Tree. The domain of the Gauss Map is then extended to cover all reals, and the coverage of the Stern-Brocot Tree is extended to include all positive and negative rationals in a manner that preserves the map between the two constructions.