Date of Award
Mathematics and Statistics
Pure sciences, G-structures
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Differential geometry is the study of a differentiable manifold on which we are given some "geometric structure." We shall define a G r-structure on a differentiable manifold and also a field of geometric objects of order r. The purpose of chapter I is to show the relation between these two concepts as well as to summarize some of the results bearing on these topics. In chapter II we define a GJ-structure on a differentiable manifold by a field of linear operators J and consider a riemannian metric G defined on the same manifold. Introducing the compatibility condition JG = 0 we obtain a degenerate riemannian structure which is investigated with the help of special bases adapted to the structure. We also define special linear connections on the structure and obtain a characterization of these connections in terms of J and G. Finally we obtain a characterization of these degenerate riemannian structures in terms of the holonomy groups of the linear connections. Chapter III begins by defining special cases of the GJ-structures which we call r-tangent structures. We introduce the operators C and M of A. Lichnerowicz. Next we construct a tensor determined by the r-tangent structure which we call the torsion tensor and derive the relation [special characters omitted]where f is any 1-form and T is the 2-form corresponding to the torsion tensor. Using this relation we are able to obtain an expression of the torsion tensor in local coordinates. A major result of this chapter is the establishment of the fundamental relation [special characters omitted]where f is any 1-form and S is a 2-form depending only on the constants of structure. In the remaining chapters we consider the almost tangent structures of H. A. Eliopoulos, which are a special case of the r-tangent structures. We restrict ourselves in chapters IV and V to "real" almost tangent structures. Given a riemannian metric G we say that G is hermitian with respect to J if [special characters omitted]and call the resulting structure an almost hermitian structure subordinate to the almost tangent structure. We define special bases adapted to the almost hermitian structure and special connections which we are able to characterize by conditions on J and G. We prove among other results, two theorems characterizing the almost hermitian structures with relation to connections and their holonomy groups. The chapter V consists of a brief study of particular almost hermitian structures. Here we define pseudohermitian and almost kahlerian structures, giving a number of ways in which such structures can be characterized. In the chapter VI we extend the notion of almost tangent structures to homogeneous almost tangent structures defined on homogeneous Lie spaces and derive necessary and sufficient conditions that such structures exist. We then investigate structures integrable in the sense of Mme Lehmann-Lejeune and obtain similar results for such integrable structures.
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