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Feb 14, 2018 at 9:05 comment added Robert Bryant @BrianKlatt: There are lots of places to read about basic representation theory for compact groups that don't require you to read Fulton & Harris (which is a great book, by the way, though it's not focused on things you need to know for Riemannian geometry). I like "Representation Theory of Compact Groups" by Bröker and tom Dieck for a quick introduction. Everyone who works in Riemannian geometry should know the representations of $\mathrm{SO}(3)$ and $\mathrm{SU}(2)$ (the first nontrivial cases) and their tensor product decompositions, but that's not hard to pick up.
Feb 13, 2018 at 17:21 comment added Brian Klatt Do you have any good references for where to pick this stuff up? Hoping there's a better answer than "read Fulton & Harris.") I'm increasingly annoyed that there aren't any books along the lines of "Representation Theory for (Riemannian) Geometers".
Feb 3, 2015 at 9:24 history edited Robert Bryant CC BY-SA 3.0
Fixed some careless errors and edited for clarity
Feb 2, 2015 at 18:20 vote accept Paul
Feb 2, 2015 at 15:21 history edited Robert Bryant CC BY-SA 3.0
added a missing case in the traceless Ricci terms
Feb 2, 2015 at 13:51 history answered Robert Bryant CC BY-SA 3.0