We give elements of a general theory of local two-point functions on Riemannian manifolds. Some classical and also recent results in Riemannian geometry are reproved in a unified form.
Let (M,g) be a smooth Riemannian manifold of dimension n > 2.
By a two-point function on M we shall mean a smooth function F(x,y)
defined on an open neighborhood U C M x M of the diagonal A(M x M) such that, whenever (x,y) EC U, there is a shortest geodesic joining
x to y. The most usual case is that a pair of points, (x,y), belongs to t if, and only if, each of the points belongs to some normal neighborhood of the other point. Many "geometric" two-point functions occur in a natural way in Riemannian geometry (see [2], [12] and the first section below). The aim of this paper is to formulate (and prove) some simple general principles for operating with two-point functions. As an application, we reprove in a clear form a number of results on harmonic spaces, spaces with volume-preserving local geodesic symmetries, and special homogeneous Riemannian spaces. Some of these results are classical, some others have been proved recently by the present authors, or may be new.
KOWALSKI -VANHECKE 1. EXAMPLES OF TWO-POINT FUNCTIONS We start with some examples of natural two-point functions on a Riemannian manifold (M,g). A. The simplest function of two points x,y on (M,g) is the Riemannian distance d(x,y). This function is not a two-point function in our sense, because it is not smooth at the diagonal A(M x M). Therefore, one usually works rather with the "distance function" l 2 Q(x,y) = d(x,y) , which is smooth at the diagonal. If (M,g) is connected and complete, then (x,y) is a two-point function on = M x M. In the general case (also for M disconnected), is a local two-point function. B. Let (x ,...,x ) be a system of normal coordinates centered at m M and defined in a normal neighborhood URm. Let (gij), or (g ij) respectively, denote the matrix of the covariant components, or contravariant components respectively, of the metric tensor g with respect to the given coordinates. Consider the characteristic polynomials det(gij -X6ij), det(g i j -X6 i j ) as functions on . It is easy to see that these polynomials are independent of the special choice of the normal coordinates in Cl because the transition matrix m is always a (constant) orthogonal matrix. For x E U , denote by Ik(m,x) (or k(m,x)) the coefficient of Xn-k in the first (or the second) polynomial for k =1,...,n. (In fact Pk and k are elementary symmetric functions of degree k of the eigenvalues.) Now, if the point m M varies and x E M remains in a normal neighborhood of m, then Pk(m,x), k(m,x) are two-point functions in our sense. In particular, for each fixed m E M, the function (m,x) is the square of the "normal volume function" (m,x) = ... , where w ( m ) is the positively oriented volume-element defined in the neighborhood gJm C M. C. For m M and a small positive r, let Gm(r) denote the geodesic sphere with center m and radius r. Let (m,x) denote some intrinsic, or extrinsic scalar invariant of the sphere at the point x C G (r);
for instance, the mean curvature h(m,x), the length of the second