Deformations of G-structures Part A: General Theory of Deformations

This paper is concerned with deformations of structures on manifolds. It is divided into two parts: Part A treats the question of defining a general theory of deformations which will generalize the theory in, e.g. [1 ], [7], and [10], while at the same time retaining some geometric or analytical significance. The second section, Part B, investigates in more detail the implications of our general theory on the "classical" structures in differential geometry. Let G c= GL(n, R) be a linear Lie group, and let X be an n-dimensional manifold. A G-Structure on X is a reduction of the structure group of the tangent bundle of X from GL(n, R) to G; geometrically, a G-structure gives a principal fibre bundle G-> B~-> X where Ba consists of all G-frames on X. The G-structure is integrable if X has a coordinate covering such that the coordinate frames are G-frames. A deformation theory of integrable G-structures has been given in [11 ], and, in case n == 2 k and G ~ GL (k, C) C GL (2 k, R), some considerable progress has been made towards obtaining general results generalizing the well known variation of complex analytic structure. In [1] a deformation theory of Riemannian manifolds of constant curvature was proposed; a variant of this was used in [13], although the problem in these cases was specifically to prove the "rigidity" of a structure, rather than to discuss the geometric significance of the existence of deformations. After some preliminaries in § I, we shall, in § II, give a general definition of deformations of G-structures generalizing the theories described above. Our definition may be verbally stated as follows: A 1-parameter deformation of a G-structure G ~ BG-~ X is given by a 1-parameter family of G-structures G-> B a (t)-~ X, B a (0) = B G, such that the deformed structures have precisely the same local properties as the original G-structure. In other words, we shall assume the local triviality of our deformations, and then seek the global implications of this hypothesis. Clearly, such a theory generalizes the special cases given above. In § I I I we discuss the relationship of our theory with the theory of sheaves; the possibility of such a relationship was one of the motivating factors in our definition. Paragraph IV is devoted to the higher order theory of deformations; as was indicated, for complex …