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This work was conceived as an introduction to global differ ential geometry. It assumes known only the elements of classical differential geometry and Lie groupst. Some theorems are admit ted without proof, but in the majority of cases detailed proofs are given. If this book allows researchers to initiate them selves in contemporary works on the global theory of connections, it will have achieved its goal. The Consiglio Nazionale delle Ricerche has done me the great honour of including my book in its fine collection. I would wish it to find here an expression of my profound gratitude. Monsieur Dalla Volta has graciously provided a skilful and invaluable cooperation with the material cares of publication, which has been a great help to me. Without a doubt this book would never have seen the light of day without the illuminating advice of Monsieur Enrico Bompiani; it was conceived during the course of some weeks spent in 1955 at the University of Rome in the unforgettable atmosphere of the Istituto di Matematica. A. LICHNEROWICZ t The notations used for linear groups are those of Chevalley (Lie Groups) .
List of contents
I General Notions about Differentiable Manifolds.- 1-1: Differentiable Manifold.- 1-2: Exterior Differential Forms.- 1-3: Vector Valued Forms.- II Infinitesimal Connections: Linear Connections.- 2-1: Homotopy Notions.- 2-2: Infinitesimal Connections on a Principal Fibre Bundle.- 2-3: Linear Connections.- 2-4: Riemannian Connections.- III Holonomy Groups and Curvature.- 3-1: General Case and Manifolds with a Linear Connection.- 3-2: Riemannian Manifolds: Reducibility.- IV Harmonic Forms and Forms with Zero Covariant Derivative.- 4·1: Elements of Homology.- 4-2: Harmonic Forms.- 4-3: The Operators Defined by a Form on a Riemannian Manifold.- V Almost Complex Manifolds and Subordinate Structures.- 5-1: Complex Structure on a Real Vector Space.- 5-2: Hermitian Vector Spaces.- 5-3: Almost Complex Structure and Subordinate Structures on a Differentiable Manifold.- 5-4: Almost Complex Connections.- 5-5: Forms on Pseudohermitian and Pseudokählerian Manifolds.- List of Symbols.
