Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 195-202 (1999)

Symmetric Models of the Real Projective Plane

H. R. Farran, Maria do Rosario Pinto, S. A. Robertson

Department of Mathematics, Kuwait University, P. O. Box 5969, 13060 Safat, Kuwait, e-mail:;

Centro de Matematica da Universidade do Porto, Faculdade de Ciencias, 4050 Porto, Portugal, e-mail:;

Faculty of Mathematical Studies, University of Southampton, Southampton SO17 3BJ, U. K., e-mail:

Abstract: We show that the symmetry group of a stable immersion of the real projective plane $P$ in $E^3$ is either trivial or is cyclic of order $3$, and that of a stable map of $P$ in $E^3$ is conjugate to a subgroup of the full tetrahedral group. Thus Boy's surface, in its `standard' form, is the most symmetrical stable immersion of $P$ in $E^3$, and Steiner's surface is given by the most symmetrical stable map of $P$ in $E^3$. We also construct a smooth embedding of $P$ in $E^4$ with symmetry group $SO(2)$ by orthogonal projection of the Veronese surface.

Classification (MSC91): 57R42, 57R40

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