Horofunctions, busemann functions, and ideal boundaries of open manifolds of nonnegative curvature

Abstract

1. Definitions An open manifold X n is defined to be a complete noncompact manifold. The metric on X ~ will be denoted by d. Open manifolds are obtainable, for example, as follows: Let ~n be a compact submanifold in R N with boundary and let f be a continuous function on ~n whose zero set coincides with the boundary 0X '~. Construct an embedding r of the interior of ~n into the space R x x R by assigning r = (x,f.-l(x)), x 6 Int(X'~). In this case the boundary 0)( '~ "goes to infinity" and the set X n = ~(Int(X'~)) endowed with the metric induced by the embedding is an open manifold diffeomorphic to the interior of the compact manifold ~-,z with boundary. For an open manifold X n there are known several ways of constructing the ideal boundary that corresponds to 0)~ r~ in the example presented above (see the definitions of the sets O(X), B(X), and X(oc) below). If X is an "Hadamard manifold", i.e. a simply connected manifold of nonpositive sectional curvature, then all the definitions of ideal boundary determine pairwise homeomorphic sets It]. In the present article we restrict ourselves to the case of manifolds of nonnegative or almost nonnegative curvature. We will exhibit an example of a surface X for which the above-indicated definitions give pairwise nonhomeomorphic sets, namely: the space of horofunctions O(X) is homeomorphic to a closed interval, the space B(X) of Busemann functions is a two-point set, and the space X(oo) of equivalence classes of rays is a singleton. On the exhibited surface there are exactly two poles. The same two points are the unique souls of X (see the definition of a soul of an open manifold of nonnegative curvature in_ [2]), and the points of the shortest geodesic joining them are the minimum points of the horofunctions. The correspondence indicated is one-to-one. The surface constructed is a space of nonnegative curvature in the sense of the comparison theorem but is not a smooth Cl-surface. However, it admits of approximation in the Hausdorff metric by smooth surfaces and since the convergence of surfaces in the Hausdorff metric implies the convergence of their ideal boundaries, the example constructed ensures existence of smooth surfaces with pairwise nonhomeomorphic ideal boundaries. 1.1. Rays and Busemann functions. Any open manifold is unbounded. Therefore, for every point p, there exists a sequence of points pi such that d(p, Pl) -+ oo as i --+ ~c. Connect the points p and pi by the shortest geodesics li(s) as follows: From the sequence {i} one can choose a subsequence {j} such that the sequence constituted by the directions of the shortest geodesics lj(s) at the point p = lj(O) converges to some vector v. Then every segment of the geodesic l(s) = expp(sv) is a limit of the corresponding segments of the geodesics lj(s). By that reason, the geodesic l(s), 0 <_" s < co, is a shortest geodesic on each of its segments. Such a geodesic is calIed a ray. The set of all rays issuing from the point p will be denoted by 77.p. The geodesic l(s), -oo < s < cc (infinite to both sides), is called a line in case it is a shortest geodesic over each of its segments. DEFINITION 1.1. Let l(s), 0 <_ s < co, be a ray. The function