New Results on the Geometry of the Translation Surfaces

Abstract

Abstract. In this paper we study the second mean curvature for different hypersurfaces in space forms. We furnish some examples and we remind some connections between II-minimality and biharmonicity. The main result consists in proving that there are no II-minimal translation surfaces in the Euclidean 3-space.

On Some Weingarten Surfaces in the Special Linear Group SL(2,R)

“…Later on, generalized Weingarten surfaces in the Euclidean space of dimension 3 were studied. See, e.g., [19][20][21]. The study of Weingarten surfaces may be extended to other ambient spaces (e.g., [22,23]).…”

“…(c) The projection π : SL(2, R), ds 2 → H + , g + is also a Riemannian submersion. To see this, recall that {AE, AF, AH}, with E, F and H defined in (21), spanning the tangent space of SL(2, R) at A. It is known by Patragenaru (see [24]) that all left-invariant metrics on SU(1, 1) are isometric to one of the 3-parameter families of metrics g(c 1 , c 2 , c 3 ) with c 1 ≥ c 2 > 0 > c 3 , and its isometry group has dimension 4 if and only if c 1 = c 2 .…”