Invariant Distances and Metrics in Complex Analysis, Volume 47, Number 5

C onstructing a distance that is invariant under a given class of mappings is one of the fundamental tools for the geometric approach in mathematics. The idea goes back to Klein and even to Riemann. In this article we will consider distances invariant under biholomorphic mappings of complex manifolds. There will be many such distances. A number of these will come from functions on the tangent spaces, in the way that a Riemannian metric on a manifold yields a distance on the manifold. Following Riemann’s lead, we think of a suitable function on the tangent spaces as a way to measure “lengths” of tangent vectors, and we can use it to define lengths of curves and ultimately a distance on the manifold. We do not insist that this function be related to an inner product. If the complex manifold is M and its tangent spaces are denoted Tp(M), then we shall work with any nonnegative function f(p, v) , for v ∈ Tp(M) , such that f(p, v) suitably respects scalar multiplication (real or complex as appropriate) in v . In most cases the function will be continuous in (p, v), but sometimes we allow the continuity to be slightly relaxed. Motivated by the Riemannian case, we shall refer to this function as a metric if f(p, v) vanishes only for v = 0, or a pseudometric in general. We do not assume that f(p, v) satisfies the triangle inequality in v if p is fixed; if the triangle inequality is in fact satisfied, the function will be called a norm or pseudonorm. If the “length” function comes from an inner product, then the metric is said to be Riemannian as usual. If that inner product is the real part of a Hermitian inner product, then we call the metric Hermitian. If the Hermitian inner product behaves almost like the Euclidean metric (in the sense that they agree to order two), then we call the metric Kählerian. If, on the other hand, there is no inner product present at all and only the notion of vector length is defined, then we call the metric Finslerian. The way that the word “Hermitian” is used here may be unfamiliar to some readers, and we offer a note of clarification by considering the simple example of Cn. If we think of Cn as a complex space, then it is natural to use the inner product, for z = (z1, . . . , zn) and w = (w1, . . . , wn) in Cn, given by 〈z,w〉 = n ∑