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*Akira Mizuhara and Hirohiko Shima*

Invariant projectively flat connections and its applications

(Lobachevskii Journal of Mathematics,
Vol.4, pp.99-107)

In this paper we study invariant projectively flat affine connections
and invariant dualistic structures of constant curvature. We first relate
the existence of invariant projectively flat affine connections to that
of certain affine representation of Lie algebras (Theorem 1). Using such
affine representations we give a correspondence between semi-simple symmetric
spaces with invariant projectively flat affine connections and central-simple
Jordan algebras (Theorem 2). As an application we prove that invariant
dualistic structures of constant curvature come from certain invariant
Hessian structures (Theorem 3).

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