I am looking for a lot of interesting examples of locally symmetric Riemannian Einstein manifold of nonnegative sectional curvature. The only one that I know is the complex projective space.
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$\begingroup$ You certainly also know the sphere and Euclidean space, too. $\endgroup$ – Deane Yang Feb 22 '17 at 15:21

$\begingroup$ sure, they are kind of trivial. $\endgroup$ – abcd Feb 22 '17 at 15:22

4$\begingroup$ You know quaternionic projective space. $\endgroup$ – Ben McKay Feb 22 '17 at 16:07

8$\begingroup$ How about any compact simple Lie group endowed with its biïnvariant metric reduced modulo any finite subgroup? More generally, finite isometric free quotients of irreducible Riemannian symmetric spaces of compact type all satisfy this condition. Most are only locally symmetric. $\endgroup$ – Robert Bryant Feb 22 '17 at 16:42

1$\begingroup$ I agree with Robert, and his list is almost everything you are going to get. You may still take products of examples with the same Einstein constant and some strange quotients of those, but that is more or less it. So look up a table of symmetric spaces to see what you can get. Or tell us what `interesting' means for you. $\endgroup$ – Sebastian Goette Feb 22 '17 at 20:38