Abstract
We classify all surfaces in R 4 which are homogeneous in the sense of equi-centroaffine differential geometry. There result 21 group classes, some of them depending on one or two real parameters. The classification is cleared up, i.e. each copy is equivalent to exactly one representative. This applies as well to the corresponding groups as to the orbits (and also to the parameter cases). In particular, we can characterize the Clifford tori in a purely affine manner and determine all homogeneous centroaffine spheres. This answers a former question on the existence of centroaffine spheres which are not contained in a hyperplane. The classification and, in particular, the uniqueness is based on geometric insight and is essentially not computer dependent. The leading ideas are of a general nature and may also be applied to homogeneity for higher-dimensional cases and for related geometries.
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Walter, R. Homogeneity for Surfaces in Four-Dimensional Vector Space Geometry. Geometriae Dedicata 71, 129–178 (1998). https://doi.org/10.1023/A:1005042926934
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DOI: https://doi.org/10.1023/A:1005042926934