Vol. 84, No. 2, 1979

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Symmetric planes

Rainer Löwen

Vol. 84 (1979), No. 2, 367–390

Symmetric planes are defined as stable planes carrying an additional structure of a symmetric space whose symmetries are automorphisms of the plane. An example of a stable plane is the geometry induced by a topological projective plane on any of its open subsets. We consider several examples of this type which are, in fact, symmetric planes.

Working with the Lie triple system, we construct a linear local approximation to both the geometric and the differential geometric structure of a symmetric plane M. We show that under some reasonably mild restrictions, this so-called tangent translation plane determines the global structure of M as a symmetric plane. Later, this result will be used in order to determine all symmetric planes in low dimensions. The two-dimensional case of this classification is given in the present paper. Symmetric planes often turn up inside stable planes of sufficient homogeneity, and their classification may then be applied.

Mathematical Subject Classification 2000
Primary: 51H25
Secondary: 51H20
Received: 29 November 1978
Published: 1 October 1979
Rainer Löwen