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I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.

More specifically, I need to use results about the eigenvalues of the operator $R:T_pM \rightarrow T_pM$; $X \mapsto -[[X,c],c]$ for a given $c \in T_pM$. The fact is, using the book of Besse "Manifolds all of whose geodesics are closed", it seems some eigenvalues are $-4$, but the results used in "The free loop space of globally symmetric spaces" imply eigenvalues $\frac{1}{4}$. I think I'm computing something wrong, but I don't know where to find the right computations done explicitly. Any reference?

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I assume that you are interested in globally compact symmetric spaces of compact type. The curvature depends on the choice of a Riemannian metric. The vector $c$ in your formula should be a unit vector with respect to that metric. Depending on the metric you choose, in the case of complex or quaternionic projective space, you get eigenvalues like $1$ and $4$, or $\frac 14$ and $1$. If you get negative eigenvalues, then there probably is a sign mistake (or a strange sign convention, or you are in fact dealing with a symmetric space of noncompact type).

The formulas in Ziller's article that you mention look correct, so you may want to check the references he gives. Another reference I like is [J. Cheeger, D. Ebin: Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam (1975)]. In Chapter 3, he considers homogeneous spaces with normal metrics, of which symmetric spaces are an easy special case.

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  • $\begingroup$ Thanks for helping! Shouldn't the curvature tensor be independent of the metric chosen on a symmetric space? $\endgroup$ – Aloizio Macedo Oct 3 '15 at 3:12
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    $\begingroup$ Yes, the curvature tensor is independent. But to compute eigenvalues, you have to insert some vector c. And there, you want to pick a unit vector with respect to the metric you have chosen. Otherwise, it only makes sense to talk about the signs and ratios of eigenvalues. $\endgroup$ – Sebastian Goette Oct 3 '15 at 8:02

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