Stiefel manifolds and polar decompositions

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?

• 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

First, the identity matrix $I \in O(n)$ represents the standard orthonormal basis $e_1, \dots, e_n$ of $\mathbb{R}^n$, and its coset $I\cdot O(n-k)$ represents the $k$-plane spanned by the first $k$ basis vectors $e_1, \dots, e_k$.
The local splitting you're looking for is equivalent to extending the standard orthonormal basis to a local map from $V_{n,i}$ to $O(n)$, viewed as the space of orthonormal frames in $\mathbb{R}^n$, such that the span of the first $k$ vectors in each frame is equal to the corresponding $k$-plane in $V_{n,k}$. Helgason is basically pointing out that one way to do this is to use the connection naturally induced by the bi-invariant Riemannian metric on $O(n)$ to parallel transport the initial frame along geodesic rays starting at the identity matrix.