Non-compact conformally flat manifolds with constant scalar curvature and noncompact Kaehler manifolds with vanishing Bochner curvature are studied and classified.
1. Introduction
The following theorems are well known: THEOREM A ([6]). Let M be a compact conformally fiat Riemannian manifold with constant scalar curvature. If the Ricci tensor is positive semidefinite, then the universal covering manifold of M is one of Sn(c), E Γ S"-1(c) or E", the real space form of curvature c being denoted by S"(c) or E" depending on whether c is positive or zero.
THEOREM B ([10]
). Let M be a Kaehler manifold of real dimension n with constant scalar curvature whose Bochner curvature tensor vanishes and whose Ricci tensor is positive and semidefinite. If M is compact, then the universal covering manifold is a complex projective space CP n/2 or a complex space C n/2.
The main purpose of the present paper is to prove analogous theorems when the manifold is complete and non-compact. Manifolds and tensor fields are assumed to be of class CΒ°L The indices h, i,j .... run over the range {1, 2 .... } and we use the Einstein convention.
The authors wish to thank Professor H. Kitahara for many valuable suggestions.
2. KEY LEMMAS
Let M be an n-dimensional Riemannian manifold with metric gij. Let A be the Laplacian operator acting on functions on M, i.e. Af = ViV~f where V~ is the operator of covariant differentiation with respect to g. LEMMA 1 (I-4]). Let B r be a metric ball of radius r about a fixed point Xo ~ M. Then there exists a family of functions ~, such that ~k, is one on B~ and zero outside B2~, ~ takes on values between zero and one, is Lipschitz continuous, *Partially supported by TGRC- KOSEF, 1990.