Suppose that $S$ is a connected compact embedded submanifold of $O(n)$ with dimension equal to $O(n)$, is it true that $S$ has to be a component of $O(n)$?
My guess is that it should be true, but I do not know where to start.
2026-05-04 19:16:45.1777922205
The maximal connected compact submanifold of $O(n)$
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Submanifolds of the same dimension must be open. Since it's open, closed and connected, it must be a component. There's nothing special about $O(n)$ here.