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

Ideal boundaries of open manifolds with nonnegative sectional curvature are considered. Unlike the case of nonpositive curvature, such known definitions of ideal boundary as the space of horofunctions, the space of Busemann functions, and the class of equivalent rays can lead to nonhomeomorphic spaces. A corresponding example is given. Moreover, we study rays that are gradient lines of horofunctions and prove inequalities which connect horofunctions with Busemann functions. We introduce a metric (the angle at infinity) in the space and find sufficient conditions for a Busemann function to be exhausting. We also introduce the $dl$-functions that are generalization of horofunctions and prove topological triviality of the corresponding ideal boundary.