🚀 go-pugleaf

I wish to compute the formula for parallel translation along geodesics
in the (compact) Stiefel manifold V(n,k) = O(n)/O(n-k), relative to
its natural invariant connection.

As discussed below, this problem can be solved by computing a formula
for either (i) the geodesic symmetry of V(n,k), or (ii) an involutive
automorphism of O(n) that fixes O(n-k).


The compact Stiefel manifold V(n,k) is the set of orthogonal k-frames
in R^n.  V(n,k) is a Riemannian symmetric space; its group of
isometries G := I(V(n,k)) is the group O(n) of n-by-n orthogonal
matrices, and the isotropy group K at p in V(n,k) is the group

        {   ( I 0 )  -1               }                     ( I )
        { g (     ) g   : Q in O(n-k) }  where p = g.o, o = (   ).
        {   ( 0 Q )                   }                     ( 0 )

Note that o in an n-by-k matrix whose columns lie in V(n,k).
Therefore, V(n,k) = G/K.

Let X, Y in T_p.  Denote the point exp_p(tX) by p_t.  Let s_t be the
geodesic symmetry at p_t in V(n,k).

The parallel translation tau_t of Y along the geodesic t -> exp_p(tX)
is induced by the isometry T_t = s_(t/2)s_0, i.e.,

        tau_t(Y) = (T_t) (Y).
                        *

Therefore, a formula for the geodesic symmetry s_p (at any point p)
provides a formula for parallel translation.


Alternatively, given an involutive automorphism sigma (sigma: G -> G,
sigma^2 = id) that fixes K, the following diagram commutes,

               sigma
            G -------> G
            |          |
         pi |          | pi
            V          V
           G/K -----> G/K
                s_o

and we may use sigma to determine a formula for s_o, and hence for
parallel translation.


I would appreciate any help computing a formula for the geodesic
symmetry of V(n,k), an involutive automorphism of O(n) that fixes
O(n-k), or a reference to a paper or book that discusses these ideas
in the case of Stiefel manifolds.

Steven Smith