# multidimensional rotation terminology

Given an element $g$ of the orthogonal group $O(n)$, is there a name for the subspace of $R^n$ that's fixed by $g$, and a name for the orthogonal complement of this space? (The latter is what I really want to know. I'm guessing that the former is typically called the fixed subspace, but the latter subspace is more arcane. I'm inclined to call it the equatorial subspace, but I'd rather not give it a name if it already has one, or use the term "equatorial subspace" if it already means something else.)

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If you do call the latter the "equatorial subspace", clearly you should call the former the "polar subspace". –  Tom Goodwillie Nov 7 '12 at 4:42