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I'm stuck on the following two problems in Milnor and Stasheff's book Characteristic Classes. Really can't get my head around this material and I'm hoping that more worked examples would help. Even some hints would be much appreciated.

5-B: Show that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\textrm{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of $\gamma^n(\mathbb{R}^{n+k})$ in $\varepsilon^{n+k}$. Now consider a smooth manifold $M \subset \mathbb{R}^{n+k}$. If $\bar{g}: M \to G_n(\mathbb{R}^{n+k})$ denotes the generalized Gauss map, show that $D\bar{g}: DM \to DG_n(\mathbb{R}^{n+k})$ gives rise to a cross-section of the bundle $\textrm{Hom}(r_M,\mathrm{Hom}(r_M,\nu)) \cong \mathrm{Hom}(r_M \oplus r_M, \nu)$.

5_D: Show that $G_n(\mathbb{R}^{n+k})$ has the following symmetry property. Given any two $n$-planes $X, Y \subset \mathbb{R}^{n+k}$ there exists an orthogonal automorphism $\mathbb{R}^{n+k}$ which interchanges $X$ qand $Y$. Define the angle $\alpha(X,Y)$ between $n$-planes as the maximum over all unit vectors $x \in X$ of the angle between $x$ and $Y$. Show that $\alpha$ is a metric for the topological space $G_n(\mathbb{R}^{n+k})$ and show that $\alpha(X,Y)=\alpha(Y^\perp, X^\perp)$.

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    Exercise 5D has already been discussed on this site and is has complete, detailed answers: [MSE/830277](http://math.stackexchange.com/q/830277).2015-12-26

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