On the automorphism group of a $G$ -structure

A linear Lie group is called elliptic if its Lie algebra contains no matrix of rank one.).)$ The purpose of this paper is to prove that the glo- bally defined infinitesimal automorphisms of a G-structure (called G-vector field) are given by a system of linear elliptic differential equations if and only if this G-structure is elliptic.(See Lemma for a precise statement.)It follows easily THEOREM A. The group of diffeomorphisms of $M$ which leave a given ellipticTheorem A is a generalization of the results of Boothby-Kobayashi-Wang [1] and Ruh [8].(In fact, Ruh's sufficient condition clearly implies that the G-structure in question is elliptic.)Both Lemma and Theorem A are contained implicitly in Guillemin-Sternberg [3].Still we feel their explicit statements with proofs would be worth publishing because of their impor- tance.Also we shall provide two examples to show that Theorem A is best possible in a sense, following suggestions of Professor S. Kobayashi and Pro- fessor S. Sternberg.