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The real Stiefel manifold $V_{n,k}$ of orthogonal $k$-frames in $\mathbb{R}^n$ can be viewed as the reductive homogeneous space $G/H=O(n)/O(n-k)$. If ${\frak{so}}(n)$ is the Lie algebra of $O(n)$, then we have the reductive decomposition $$ {\frak{so}}(n)={\frak{m}}+{\frak{h}} $$ where $$ {\frak{m}}=\left \{ \begin{pmatrix} A & B\\ -B^T & O \end{pmatrix}: A\>\> \text{is a}\>\> k\times k \>\>\text{skew-symmetric matrix} \right \} $$ and $$ {\frak{h}}=\left \{ \begin{pmatrix} O & O\\ O & C \end{pmatrix}: C\>\> \text{is a}\>\> (n-k)\times (n-k) \>\>\text{skew-symmetric matrix} \right \} $$ with ${\frak{so}}(n)$ and ${\frak{h}}$ a reductive pair.

According to Helgason, there is a local diffeomorphism $$ (\text{exp}X,h)\mapsto (\text{exp}X)h,\quad \text{where}\>\> X\in {\frak{m}}, h\in H. $$ However, there doesn't seem to be anything in the literature where this decomposition is calculated explicitly. I did find the following ``polar decomposition": $$ V_{n,k}\times P_k\rightarrow M_{n,k},\quad (v,r)\mapsto vr^{1/2} $$ where $P_k$ is the set of positive semi-definite symmetric matrices and $M_{n,k}$ is the set of all $n\times k$ real matrices.

Does this last decomposition have any connection with the polar decompositions of Lie groups? If not, is there an explicit description of the polar decomposition involving Stiefel manifolds somewhere?

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For a Cartan decomposition, you need $[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$ and that is certainly not the case here. –  Fran Burstall Aug 17 '13 at 13:06
    
@Fran: I'm not sure of the correct terminology here. I've changed Cartan to polar. –  Oliver Jones Aug 17 '13 at 21:22
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