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?