# Differential of a canonical map

While studying about curvatures I came up with the following but was unable to work it out fully. Let $$M \subset \mathbb{R^n}$$ be an embedded submanifold of dimension $$k$$. Then there is a natural (smooth?) map $$\alpha: M \rightarrow \text{Gr}(k,\mathbb{R}^n),$$

given by $$p \mapsto T_p M$$. I know the grassmanian admits a natural description of its tangent spaces, namely: $$T_V\text{Gr}(k,\mathbb{R}^n) \cong \text{Hom}(V , V^+)$$

where by $$V^+$$ I mean the orthogonal complement. Is there a nice description of its differential at an arbitrary point $$p$$? I couldn’t work it out. Also, a closesly related question: suppose we look instead of at $$p \mapsto T_pM$$ at $$p \mapsto (T_p M)^+$$, in the codimension 1 case, I think the differential + a choice of normal vector field gives you a bilinear form on $$T_pM$$ automatically. Intuitively I would expect this is something like the second fundamental form of the manifold , but again, when trying to work it out it becomes a mess. Any ideas? Thanks!

P.S. If someone knows a universal property the grassmanians satisfy, I’m also very interested!

• "Is there a nice description of this map?" Which map are you referring to? You already described the map $\alpha$. – Michael Albanese Jul 16 at 14:18
• sorry I edited it – M. Van Jul 16 at 14:20
• A duplicate – Arctic Char Jul 16 at 14:51
• @ArcticChar Yes, but the other post does not have an answer ... – Paul Frost Jul 17 at 17:04