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What does it mean for a vector space/linear subspace to be tangent to a manifold (e.g. in $\mathbb{R}^n$ or $\mathbb{C}^n$)? Does it mean that it is contained in the tangent plane to some point? If not, does it say something about the dimension of the intersection with the tangent plane? It looks like low-dimensional examples don't seem particularly helpful here (e.g. lines vs. tangent planes to a surface may be too simple to say something meaningful). There may also be some confusion involving terminology.

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    $\begingroup$ A linear subspace $V \subset \mathbb{R^n}$ is tangent to a submanifold $M\subset \mathbb{R}^n$ at $p\in M$ if $V \subset T_pM$. Visually, it says that it is tangent to $M$ at $p$ with the usual geometric notion of tangency in euclidean geometry. $\endgroup$ – DIdier_ Nov 9 '20 at 10:11

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