  It is known, that every function on the unit sphere in $\bbr^n$, which is
invariant under rotations about some coordinate axis, is completely determined
by a function of one variable. Similar results, when invariance of a function
reduces dimension of its actual argument, hold for every compact symmetric
space and can be obtained in the framework of Lie-theoretic consideration. In
the present article, this phenomenon is given precise meaning for functions on
the Grassmann manifold $G_{n,i}$ of $i$-dimensional subspaces of $\bbr^n$,
which are invariant under orthogonal transformations preserving complementary
coordinate subspaces of arbitrary fixed dimension. The corresponding integral
formulas are obtained. Our method relies on bi-Stiefel decomposition and does
not invoke Lie theory.
