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Milnor writes on p. 11

If $M'$ is a manifold which is contained in $M$, it has already been noted that $TM'_x$ is a subspace of $TM_x$ for $x \in M'$. The orthogonal complement of $TM'_x$ in $TM_x$ is then a vector subspace of dimension $m - m'$ called the space of normal vectors to $M'$ in $M$ at $x$.

But what does "orthogonal complement" mean here? $M$ is merely a manifold, not Riemannian.

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  • $\begingroup$ Maybe they are using quotient bundles $ TM_x/TM'_x$? $\endgroup$ – gary Apr 9 '15 at 7:45
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    $\begingroup$ All smooth manifolds in that book are subsets of $\mathbb R^n$, as explained in the definition on page 1, line(-6), so that they inherit the riemannian structure from $\mathbb R^n$. $\endgroup$ – Georges Elencwajg Apr 9 '15 at 7:50
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    $\begingroup$ Dear Ricardo, you are welcome and I'm glad I could help you. Thanks for your kind offer but I would rather not earn reputation in this way. If you like you can answer yourself (this is encouraged on this site) in order to get the question off the "unanswered" queue. $\endgroup$ – Georges Elencwajg Apr 9 '15 at 8:13
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In Milnor's book all manifolds are submanifolds of $\mathbf{R}^n$, so they inherit the Riemannian structure from $\mathbf{R}^n$.

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