finiteness of fundamental groups in extended complete einstein-type manifolds

Here, an extension of the complete non-compact einstein-type manifolds is studied and shows that it has a finite fundamental group. this paper generalizes wylie's results to the setting of extended einstein-type manifolds for certain parameters. some direct corollaries of this result are also pointed out. for instance, the sphere bundle $sm$ has a finite fundamental group. these findings not only generalize previous results but also offer new insights into the applications of einstein-type manifolds across mathematics and physics, particularly in the structure of associated bundles and the behavior of geometric flows.