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What are some manifolds other than products of space forms for which the various curvature quantities can be computed easily? I'm interested in odd (real) dimensions just as much even, so I'd like to stay away from complex manifolds.

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up vote 5 down vote accepted

Probably the easiest class of manifolds to compute curvature for are the so called homogeneous spaces, those Riemannian manifolds for which the isometry group acts transitively. Since the isometry group acts transitvely, once one understands curvature at a single point, one understands it everywhere. These examples include (products of) space forms, $\mathbb{C}P^n$ and many other less well known examples.

Explicit computations often end up boiling down to computing commutators of matrices together with projection maps, so are often doable in practice.

For, say, simply connected homogeneous manifolds, up through dimension 5 they're all diffeomorphic to the "usual" things - $S^2, S^3, S^4, S^2\times S^2, \mathbb{C}P^2, S^5,$ and $S^2\times S^3$, plus one exceptional example (the Wu manifold). I would start investigating the curvature of homogeneous spaces with the Wu manifold, or with $S^3, S^5,$ or $S^2\times S^3$ with some nonstandard homogeneous metric (the other examples only have standard metrics, if you require the metrics to be homogeneous.)

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Is that list of simply connected homogeneous 5-manifolds exhaustive? I'm thinking of $\mathbb{R}^5$ and $\mathbb{H}^5$ in particular. –  Jesse Madnick Jun 25 '11 at 5:51
    
@Jesse: These are all the compact simply connected examples. Sorry about that. –  Jason DeVito Jun 25 '11 at 12:07
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