pseudolinearity on difierentiable manifolds

After 1970, the classic convexity theory was generalized from the Euclidean to the Riemannian setting . The convex sets were deflned by the property of containing the geodesic segments between any two points (in the same way that line segments behave, as Euclidean geodesics) and the convex (difierentiable) functions by the positiveness of their Rieman- nian Hessian. In some previous papers, we made a step forward, by extending the Rie- mannian convexity to a-ne difierential convexity: the geodesic segments were replaced by segments of auto-parallel curves of some arbitrary linear connections and the Riemannian Hessian by the a-ne difierential one. In this paper, we generalize in a similar way the classic notion of ·- pseudolinearity, jumping from the classic setting directly to the difierential setting. An unexpected link between the ·-pseudolinearity and submer- sions is also discovered.