We define natural first order Lagrangians for immersions of Riemannian manifolds and we prove a bijective correspondence between such Lagrangians and the symmetric functions on an open subset of m-dimensional Euclidean space.
A tangent bundle to a Riemannian manifold carries various metrics induced by a Riemannian tensor. We consider harmonic vector fields with respect to some of these metrics. We give a simple proof that a vector field on a compact manifold is harmonic with respect to the Sasaki metric on TM if and only if it is parallel. We also consider the metrics and on a tangent bundle (cf. [YI]) and harmonic vector fields generated by them.
Page 1