Harmonic Sections of Tangent Bundles

Let M be an m dimensional smooth Riemannian manifold with metric g. The tangent bundle T(M) over M is endowed with the Riemannian metric g^D, the diagonal lift of g [3], [5]. Let X be a vector field on M. Then it is regarded as a mapping φx of M to T(M). The purpose of this paper is to study under what conditions the mapping φx of Riemannian manifolds is harmonic. § 1 is devoted to describe some basic facts on geometry of tangent bundles. We will see in §2 that the natural projection, π: T(M)→M is a totally geodesic submersion. In the last section, it is proved that when M is compact and orientable, φx: M→T(M) is harmonic iff the first covariant derivative of X vanishes.

公開日:2010年1月24日で登録したコンテンツは、国立情報学研究所において電子化したものです。