 The Levi-Civita connection of "M" gives rise to a splitting of the tangent bundle
into a vertical space "V" = kerπ* and horizontal space "H" on which π* is a linear isomorphism at each point of UT"M".This splitting induces a metric on UT"M" by declaring that this splitting be an orthogonal direct sum, and defining the metric on "H" by the pullback