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Let $G_{n,k}$ be the grassmanian the manifold of the $k$-dimensional vector space contain in $\mathbb{R}^n$.

Prove that there exists a differentiable aplication $W:(a,b)\rightarrow G_{n,k}$ iff there exist $Y_1,\dots Y_k:(a,b)\rightarrow \mathbb{R}^n$ such that for every $t$ we have that $\{Y_1(t),\dots,Y_k(t) \}$ is a base for $W(t)$.

I have an idea for the if part: we clearly should take $W(t)=\langle Y_1(t),\cdots Y_k(t) \rangle$ but I don't know how to prove that $W(t)$ as define is differentiable.

For the only if part we have that locally we can construct $\{Y_k\}$ but I dont know how to extend this for $G_{n,k}$.

Any help?

EQJ
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1 Answers1

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Hint/Suggestion (but not a complete answer):

To prove that $W(t)$ is differentiable, you need to know what it means for a map into a quotient space to be differentiable, since $G_{n,k}$ is just $$SO(n) / SO(n-k),$$ right?

So what's the definition of differentiability for a map into that space?

Hint for making progress on the rest of that first proof: The set $\{Y_1(t), \ldots, Y_k(t)\}$ can be orthogonalized by Gram-Schmidt, and then extended to an orthonormal basis for $\Bbb R^n$, yielding a map into $SO(n)$.

John Hughes
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