Connection Towers and Sasaki Metrics on Higher-Order Tangent Bundles

Abstract: Higher-order tangent bundles possess a rich tower of fibrations, suggesting the existence of geometric structures compatible with their iterated bundle structure. In this paper, we introduce the notion of a connection tower on a higher-order tangent bundle and study the geometric structures induced by such towers. In particular, we show that connection towers determine natural multiconnections, adapted splittings of the tangent bundle, and canonical vector bundle structures on higher-order tangent bundles. We then construct a specific connection tower induced by the Levi-Civita connection of a Riemannian manifold. This construction extends the classical Dombrowski connection map on the tangent bundle and leads naturally to a family of higher-order Sasaki metrics. We study the associated lifts of vector fields and derive explicit Lie bracket formulas for these lifts, together with structural identities for the induced multiconnection. Finally, we determine the Levi-Civita connection of the higher-order Sasaki metrics and derive explicit geodesic equations on the second- and third-order tangent bundles. We also obtain characterization results relating geodesics of the higher-order Sasaki metrics to geodesics on the base manifold.