A nondegenerate equi-centroaffne surface in R4 is called homoge-neous if for any two points p and q on the surface there exists an equi-centroaffne transformation in R4 which takes the surface to itself and takes p to q. In this paper we classify the equi-centroaffnely homogeneous surfaces with flat indefinite metric in R4 up to centroa

A nondegenerate centroa

We give the classiffcation of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E3 and 3-dimensional Minkowski space E31.

We introduce the notion of .(0,m)-Codazzi tensors relative to an affine connection which extends the well known concept for m D 2. On compact Riemannian surfaces of genus we determine the dimension of the R-vector space of traceless Codazzi tensors, which depends on m and only, additionally we extend these result for genus zero. We give some exemplary applications to submanifolds in affine and in Riemannian geometry and to Weyl geometries

A hypersurface x: M → Sn+1 without umbilic point is called a Mobius isoparametric hypersurface if its Mobius form Φ=−ρ−2 Σi(ei(H) + Σj (hij−Hδij)ej(log ρ))θi vanishes and its Mobius shape operator S=ρ−1(S−Hid) has constant eigenvalues. Here {ei} is a local orthonormal basis for I=dx·dx with dual basis {θi}, II =Σ ij hijθi ⊗ θj is the second fundamental form, H=1 n Σi hii, ρ2=n n−1 (||II||2−nH2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in Sn+1 is a Mobius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mobius isoparametric hypersurfaces in Sn+1 with two distinct principal curvatures up to Mobius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Mobius isoparametric hypersurface embedded in Sn+1 can take only the values 2, 3, 4, 6.