Are all latitudes of $S^2$ closed submanifolds of $S^2$(closed subspace rather than compact)?
I think it should be true, because latitudes can be viewed as $S^1$.
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1I have no idea what you mean "closed subspace rather than compact" but yes, the (non-degenerate) latitudes of $S^2$ are one-dimensional closed submanifolds diffeomorphic to $S^1$. – levap Apr 18 '17 at 17:58
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1And the degenerate latitudes of $S^2$ are zero-dimensional closed submanifolds, diffeomorphic to $\mathbb{R}^0$. But it's, perhaps, worth pointing out that "can be viewed as $S^1$" does not necessarily give a submanifold. For example, a square in $\mathbb{R}^2$ "can be viewed as $S^1$" in the sense that it is homeomorphic to $S^1$, but it is not a submanifold of $\mathbb{R}^2$. – Jason DeVito - on hiatus Apr 18 '17 at 17:59