## Union of tangent spaces

Suppose a smooth manifold $M$ is embedded in $R^k$ . I wonder what can be said in general about union of tangent spaces $\cup T_{x}(M)$.Like in case of spheres with natural embedding,it is just the complement of "inside". For a torus embedded in $R^3$,it is the whole space.I would also like to know if there is any relation of this space with the tangent bundle of $M$ .

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You get the complement exactly in the case of convex bodies. In general, you may be interested in the notion of evolute: en.wikipedia.org/wiki/Evolute – Pietro Majer May 30 at 11:12