Riemannian submersion and the Laplace-Beltrami operator

Abstract

In the present paper we consider only Riemannian submersions π : (M, g)-» (B, B g) such that fibers F are complete and connected and imbedded in (M, g) regularly as totally geodesic submanifolds.It is well-known that, if φ is an eigenfunction of the Laplacian in (B, B g), the lift φ=φ L is also an eigenfunction of the Laplacian in (M, g) with the same eigenvalue [1].The purpose of the present paper is to find corresponding relations in the case of p-forms.For p-forms we get a little more complicated result.If a p-form ω is an eigenelement of the Laplace-Beltrami operator Δ in (B, B g) f the horizontal lift ω=ω L is not always an eigenelement of the Laplace-Beltrami operator Δ in (M, g).In order that ω be an eigenelement with the same eigenvalue as ω, ω must satisfy a necessary and sufficient condition which is obtained in § 4 of the present paper.In § 1 we recall some properties of Riemannian submersions with totally geodesic fibers.There we use local coordinates adapted to the Riemannian submersion.In § 2 fundamental formulas in covariant differentiation are given.In § 3 a relation between Δω and Δω is obtained when ω-ω L .In § 4 a necessary and sufficient condition to be satisfied by ω such that Δω-λω is obtained in order that ω=ω L satisfy Δω-λω.A simple sufficient condition is also obtained.As an application harmonic forms are studied in some special case.Remark.In the present paper lift always means horizontal lift.§ 1. Riemannian submersions with totally geodesic fibers. Riemanniansubmersions were studied extensively by the authors R. H. Escobales [2], S. Ishihara [3], S. Ishihara and M. Konishi [4], Y. Mutδ [5], T. Nagano [7], B. O'Neill [8], K. Yano and S. Ishihara [10], [11] and others.Riemannian submersions considered in the present paper are limited to those with totally geodesic fibers only, and this means that the tensor T of B. O'Neill vanishes [8].Tensors in the total manifold M, in the base manifold B or in a