The Stiefel Manifold is defined as

$$ \mathrm{St}(p,n):= \{ X\in \mathbb{R}^{n\times p} :\ X^T X = I_p \}. $$

Recall that the tangent space at a point $X\in \mathrm{St}(p,n)$ is given by $$ T_X{\mathrm{St}(p,n)} = \{\xi\in \mathbb{R}^{n\times p}:\ X^T\xi + \xi^T X = 0 \}. $$

Given a point $X\in \mathrm{St}(p,n)$ and a tangent vector $\xi \in T_X{\mathrm{St}(p,n)}$, it is possible to express the exponential map $\exp_X(\xi)$ using the matrix exponential function. A formula is given in the paper www.mit.edu/~wingated/introductions/stiefel-mfld.pdf .

My question is whether the inverse $\log_X(\cdot): \mathrm{St}(p,n) \to T_X\mathrm{St}(p,n)$ can also be expressed using the matrix logarithm.

A related question is the following: Instead of the exponential function one may define other retractions such as $$ R_X(\xi) = (X+\xi)(I_p + \xi^T\xi)^{-1/2} $$ or the closest point projection $$ R_X(\xi) = \pi(X+\xi), $$ where $\pi$ maps a matrix $A$ to the closest element in the Stiefel manifold. This projection can be easily computed using SVD.

Again, the question is wheter one can find simple formulas for the inverses of these retractions.

Any helpful comments would be greatly appreciated.

edit: at this point I do not really care which metric is used (the Euclidean metric or the one inherited from the orthogonal group).