Vol. 92, No. 1, 1981

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Two theorems on general symmetric spaces

Herbert Busemann and Bhalchandra B. Phadke

Vol. 92 (1981), No. 1, 39–48

An important result in the theory of Riemannian symmetric spaces is the theorem that the universal covering space of a complete locally symmetric space is symmetric. The proof uses the highly nontrivial property enjoyed by Riemann (but by neither Finsler nor G-) spaces that they are automatically analytic when locally symmetric and of class C1. Our first theorem, nevertheless, extends the above result to locally symmetric G-spaces, which need not be smooth and which even when smooth are only Finsler, and not necessarily Riemann, spaces. Our second theorem states that a generic locally symmetric G-space is locally Minkowskian. This theorem has no analogue in Riemannian geometry.

Mathematical Subject Classification 2000
Primary: 53C70
Received: 15 June 1979
Revised: 29 October 1979
Published: 1 January 1981
Herbert Busemann
Bhalchandra B. Phadke