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.
