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.
