Suppose I have $M$ and $N$, two $k$-manifolds in $\mathbb{R}^n$. Is it true that $M\cup N$ is also a manifold? What is a sufficient condition for positive answer?
Is the union of two manifolds a manifold?
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differential-geometry
manifolds
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0related: http://mathoverflow.net/questions/78733/when-is-the-union-of-embedded-smooth-manifolds-a-smooth-manifold – 2017-01-28
1 Answers
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No. Take two lines in $\mathbb{R}^n$ which intersect only at the origin. Disjointness is sufficient, although not necessary, for a positive answer.
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0I don't think the OP is asking about disjoint unions since he is considering both manifolds as subsets of $\mathbb{R}^n$ (so the honest union makes sense), but if he is asking about disjoint unions then yes, the answer to his question is that it's always true. – 2012-09-03