Abstract: 
An unresolved question in Riemannian geometry is the extent to which rigidity results for nonpositively curved manifolds extend to the case of no conjugate points. After a survey of previously known results in this area, it will be shown that each compact manifold with no conjugate points admits a family of functions whose integrals vanish exactly when central Busemann functions split linearly. These functions vanish when central Busemann functions are sub- or superharmonic. This may be used to generalize splitting theorems of O’Sullivan for manifolds with no focal points and Eberlein for those with nonpositive curvature.