Recurrent tensors on a linearly connected differentiable manifold

Abstract

Introduction. Let M be a connected smooth (i.e., CI) manifold of dimension n, and B the total space of the frame bundle over M. Let a linear connection be given on M, and let B [zO] be the submanifold of B consisting of all the points which can be joined to a given point zo in B by sectionally smooth horizontal curves. The purpose of this paper is to elaborate on a natural correspondence set up by S. S. Chern [2, pp. 78-79] between tensors of type (r, s) on M and sets of nr+, functions of a particular type on B, and use it to prove the theorem that a tensor S on M is recurrent (i.e., S is not a zero tensor and its covariant derivative is equal to the tensor product of a covariant vector and S itself) if the restrictions to B [zo] of its corresponding functions on B have no common zero and are proportional to a set of constants. We give several applications of this theorem, obtaining among others the following results: (i) A recurrent tensor on M has no zero (Theorem 3.8); (ii) A property obtained by K. Nomizu [4, p. 73] characterizing linear connections with covariantly constant curvature tensor or covariantly constant torsion tensor (Theorem 4.3); (iii) A property characterizing linear connections with recurrent curvature or recurrent torsion similar to (ii) above (Theorem 4.2); (iv) The holonomy group of a linear connection with recurrent curvature is at most of dimension n(n1)/2 (Corollary 4.4). The author is deeply indebted to Professor S. S. Chern for his generous help during the first half of the year 1959 at the University of Chicago when the author tried to acquaint himself with the modern theory of linear connections. The author is also grateful to the referee for pointing out a gap in the original proof of our main theorem (3.9) and for his valuable suggestions which have resulted in several improvements in this paper. 1.1. Bundle of frames. In ? ?1.1-1.5, we summarize some of the results on linear connections which are needed for our later work. We assume as known the classical theory of linear connections and the elementary properties of n-dimensional smooth manifolds (class Cand with countable base). All the vector fields, tensor fields, q-forms, etc. defined on a smooth manifold or on part of it are assumed to be of class Co unless stated otherwise. Each of the