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Let $M$ be a manifold (which you can assume compact if it helps) and consider the natural action of the isometry group of $M$, Iso($M$) on $M$. The we can define the symmetry rank of $M$ as the rank of Iso($M$). Here is were I have trouble: I think I'm misunderstanding the definition because apparently the following should be clear or obvious:

Symrank($M$)= $k$ iff there exists a k-torus $T^{k}$ acting isometrically on M.

Please correct me if I'm wrong: Is the rank of a group the minimum number of generators?

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up vote 1 down vote accepted

The rank of a compact Lie group $G$ is the dimension of any maximal torus $T \subset G$. This is well-defined, since by standard results any two maximal tori are conjugate.

In your setup, it seems that $\mathrm{Symrank}(M)$ is equal to $k$ if there is a $k$-torus acting effectively and isometrically on $M$, and there does not exist an effective isometric action of a $k'$-torus on $M$ for any $k' > k$.

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