Affine Mean Curvature Research Articles

Affine umbilical surfaces inR 4

A surface in ℝ4 is called affine umbilical if for each vector belonging to the affine normal plane the corresponding shape operator is a multiple of the identity. We will classify affine umbilical definite surfaces which either have constant curvature or which satisfy ∇⊥ g ⊥. Furthermore, it will be shown that for an affine umbilical definite surface, the affine mean curvature vector can not have constant non-zero length.

The Affine Mean Curvature Vector for Surfaces in R4

The Affine Mean Curvature Vector for Surfaces in R4

W 3 gravity from affine maximal hypersurfaces

Affine differential geometry is used on affine maximal hypersurfaces — surfaces with vanishing affine mean curvature — to describe W 3 gravity. The generators of the W 3 algebra, which are related to the affine line curvature and torsion, are constructed from the Berwald-Blaschke metric h ij and Fubini-Pick cubic form A ijk . The maximality of the surface automatically insures the analyticity of the W 3 generators.

Affine isoperimetric problems and surfaces with constant affine mean curvature

Affine isoperimetric problems and surfaces with constant affine mean curvature

Translationsfl�chen in der �quiaffinen Differentialgeometrie

We discuss with equiaffine methods the surfaces of translation with plane generating curves in the three-dimensional affine space. Using (pseudo-) isothermic parameters we determine in this class all the affine minimal surfaces (which include the affine spheres, the quadrics and the ruled surfaces), all the surfaces with vanishing affine Gauss curvature (which include the surfaces with constant non vanishing affine mean curvature), and all the surfaces with only one family of affine lines of curvature.

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ℝ3. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.