\(P\)-geodesic sections of tangent bundle (Q1897909)

Let \(W\) be a smooth vector field on a manifold \(M\) with connection \(\nabla\). In order for the immersion of \(M\) into the tangent bundle \(TM\) to be completely geodesic with respect to the induced connection it is necessary and sufficient that \(W\) generates an infinitesimal transformation of \(M\) (i.e. \(L_W\nabla= 0\); see p. 133 of [\textit{K. Yano} and \textit{S. Ishihara}; Tangent and cotangent bundles. Differential Geometry, Marcel Dekker (1973; Zbl 0262.53024)]). The author obtains similar results for \(p\)-geodesic sections -- a concept defined in terms of the order of flattening of an integral curve on \(M\).