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In this paper we define a class of torsion-free connections on the totalspace of the (co-)tangent bundle over a base-manifold with a connection and forwhich tangent spaces to the fibers are parallel. Each tangent space to a fiberis flat for these connections and the canonical projection from the(co-)tangent bundle to the base manifold is totally geodesic. In particularcases the connection is metric with signature (n,n) or symplectic and admits asingle parallel totally isotropic tangent n-plane.

Canonical Torsion-Free Connections on the Total Space of the Tangent and the Cotangent Bundle