A formula for the normal part of the Laplace-Beltrami operator on the foliated manifold
Haruo Kitahara, Shinsuke Yorozu · Pacific Journal of Mathematics · 1977
In this paper, we give a formula for the normal part of the Laplace-Beltrami operator with respect to the second connection on a foliated manifold with a bundle-like metric.This formula is analogous to the formula obtained by S. Helgason.lItroduction* We shall be in C°°-category and manifolds are supposed' to be paracompact, connected Hausdorff spaces.Let M be a complete (p + <7)-dimensional Riemannian manifold and H a compact subgroup of the Lie group of all isometries of M. We suppose that all orbits of H have the same dimension p.Then H defines a ^-dimensional foliation F whose leaves are orbits of H, and the Riemannian metric is a bundle-like metric with respect to the foliation F. A quotient space B = M/F is a Riemannian F-manifold [5].Let L D be the Laplace-Beltrami operator on M with respect to the second connection D[8], and let Δ{L D ) denote the operator defined by (*) in § 4. Our goal in this paper is the following theorem: THEOREM.Let L D be the Laplace-Beltrami operator on M with respect to the second connection D and L B the Laplace-Beltrami operator on B with respect to the Levi-Civita connection associated with the Riemannian metric defined by the normal component of the metric on M. Thenwhere δ is the function given by (**) below.This theorem is analogous to the following result obtained by S. Helgason [2]: Suppose V is a Riemannian manifold, H a closed unimodular subgroup of the Lie group of all isometries of V (with the compact open topology).Let WaV be a submanifold satisfying the condition: For each w eW, (H-w) nW={w}, V w = (H w) w 0 W w , where 0 denotes orthogonal direct sum.Let L v and L w denote the Laplace-Beltrami operators on V and W, respectively.Then Δ(Ly) = δ~1 /2 L w oδ ι/2 -δ~1 /2 L w (δ 1/2 ) 425