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.
