  We consider point distributions in compact connected two-point homogeneous
spaces (Riemannian symmetric spaces of rank one). All such spaces are known,
they are the spheres in the Euclidean spaces, the real, complex and
quaternionic projective spaces and the octonionic projective plane. Our concern
is with discrepancies of distributions in metric balls and sums of pairwise
distances between points of distributions in such spaces.
  Using the geometric features of two-point spaces, we show that Stolarsky's
invariance principle, well-known for the Euclidean spheres, can be extended to
all projective spaces and the octonionic projective plane (Theorem 2.1 and
Corollary 2.1). We obtain the spherical function expansions for discrepancies
and sums of distances (Theorem 9.1). Relying on these expansions, we prove in
all such spaces the best possible bounds for quadratic discrepancies and sums
of pairwise distances (Theorem 2.2). Applications to $t$-designs on such
two-point homogeneous spaces are also considered. It is shown that the optimal
$t$-designs meet the best possible bounds for quadratic discrepancies and sums
of pairwise distances. (Corollaries 3.1 and 3.2).
