The Stiefel manifold of orthonormal 
-frames in 
is the collection of vectors (
, ..., 
) where 
is in 
for all 
, and the 
-tuple (
, ..., 
) is orthonormal.
This is a submanifold of 
, having dimension 
.
Sometimes the "orthonormal" condition is dropped in favor of the mildly weaker condition that the 
-tuple (
, ..., 
) is linearly independent. Usually, this does not
affect the applications since Stiefel manifolds are usually considered only during
homotopy theoretic considerations. With respect
to homotopy theory, the two definitions are more
or less equivalent since Gram-Schmidt
orthonormalization gives rise to a smooth deformation retraction of the second
type of Stiefel manifold onto the first.
For the matrix model 
,
the canonical Riemannian metric is
 |
where 
are tangent matrices and 
denotes the matrix trace.
Wang and Lim (2026) proved that, for 
and 
, there are no general
Liouvillian functions expressing the minimizing
geodesic from its endpoints, the geodesic distance,
or the Riemannian logarithm. The logarithm is the initial tangent vector of the minimizing
geodesic on a normal neighborhood. The obstruction concerns
endpoint data. Given an initial point and tangent vector, the geodesic
still has a matrix exponential formula. Wang
and Lim (2026) report AI assistance in finding the 
counterexample used in their proof.
