On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group $\mathrm{\Gamma }$ is said to be {\it rigid}, if $\mathrm{\Gamma }$ determines the boundary up to homeomorphisms of a CAT(0) space on which $\mathrm{\Gamma }$ acts geometrically. C.Croke and B.Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are rigid CAT(0) groups then so is ${\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2$.