Homogeneous Parabolic Surfaces in $ R^4$

Abstract: Parabolic surfaces in $\bf{R}^4 $ bear their name because they satisfy parabolic partial differential equations. In this paper we shall determine and classify all parabolic non-ruled surfaces in $\bf{R}^4 $ which are homogeneous in the sense of equiaffine geometry. In the course of the general discussion it will also come out that there are no compact parabolic surfaces, thus answering a question which has been open for some time.

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