The Nonlinear Connections and Harmonicity (Q2733977)

Let \(\pi : TM \rightarrow M\) be the tangent bundle of a Riemannian manifold (\(M,g\)). The author considers a Riemannian metric of Sasaki type \(G^s\) on \(TM\) defined by means of a nonlinear connection on \(TM\). In the third section he studies the harmonicity of the Riemannian submersion \(\pi : (TM,g^s) \rightarrow (M,g)\) and the geometry of its fibres.NEWLINENEWLINENEWLINEThe last section is devoted to the geometry of the vector fields \(\xi\) on \(M\) regarded as maps from \(M\) to \(TM\). The case where \(\xi\) is an isometric immersion is systematically investigated. The author proves that for any vector field \(\xi\) on \(M\), there is a Riemannian metric of Sasaki type on \(TM\) such that \(\xi\) is a harmonic map.