Two-Point Homogeneous Spaces

Abstract

Most of the classical metric spaces with metric d have the property that given any two pairs of points a, , a2 , bi, b2 of the space with d(al , a2) = d(b1, b2), there is an isometry of the space carrying a1, a2 to b1, b2 respectively. This property was called two-point homogeneity by Birkhoff [2] and called the (*)-property by Busemann. For simplicity, we shall call such a metric space a (*)-space. Concerning these spaces, very few properties are known even under the further assumption that they are Riemannian. Busemann conjectured that if they are S. L. [6] and closed (i.e., each geodesic is congruent to the euclidean circle), then they are elliptic [6, p. 233]. In this paper, we shall determine all the compact and connected (*)-spaces. It is to be noted that whereas the compact and connected metrically homogeneous spaz es (in the sense of van Dantzig) can be quite pathological such as solenoids and infinite-dimensional spaces, the connected and compact (*)-spaces are found to be closed manifolds with very simple homology properties. In these discussions, we make extensive use of the theory of transformation groups developed by Montgomery and Zippin. In fact, let E be a compact and connected (*)-space. The group G of all its isometries is compact. We shall first show that each non-discrete normal subgroup of G is transitive on E so that G has no arbitrarily small normal subgroup. Thus G is a compact Lie group and E a closed manifold. When the dimension of E is odd, a Riemannian metric can be introduced to render E a Clifford-Klein space form. When the dimension of E is even, the situation becomes more complicated. It will be proved that there exists an isometry of E which has only a finite number of fixed points and the index of each fixed point is equal to unity. It follows then that E has nonvanishing Euler characteristic. Using this property, the underlying topological space of E can be determined. We shall show that a connected, compact (*)space is homeomorphic with either a sphere Sn , a real projective space pn' a. complex projective space K, a quaternion projective space Q4f or the Cayley projective plane M. Conversely, the spheres and the various projective spaces are really the underlying topological spaces of certain (*)-spaces. In the case of Sn and pfnl this is well known. In the case of K2n, Q4' and M'6, we shall construct a convex metric for each of them to render them two-point homogeneous. The resulting metric spaces will be called the complex elliptic spaces, the quaternion elliptic spaces and the Cayley elliptic plane respectively. Let E be a (*)-space with metric d. Suppose d* to be an equivalent metric related with d by an equation of the form: d*(x, y) = f(d(x, y)), x, y e E, where f is a function in a single real variable. The space E with the new metric 177