Smooth Manifolds and Fibre Bundles with Applications to Theoretical Physics

Book excerpt: Covariant derivative of forms on principal bundles -- The curvature form -- Horizontal lifts of vector fields -- Local sections and trivializations -- Horizontal lifts of curves -- Parallel transport -- Forms in associated bundles -- Covariant derivative of sections in associated vector bundles -- Covariant derivative of tensor fields -- Covariant derivative of sections along smooth maps -- Linear connections -- Koszul connections -- Structure equations -- Geodesics -- Metrical connections -- The Schwarzschild - de Sitter spacetime -- Affine transformations and Killing vector fields -- Conformal transformations -- 11 ISOMETRIC IMMERSIONS AND THE SECOND FUNDAMENTAL FORM -- Connections in reduced subbundles -- The normal bundle and the bundle of adapted orthonormal frames -- The second fundamental form -- The shape tensor -- The shape operator -- The formulae of Gauss and Weingarten -- Strain and vorticity -- The equations of Gauss, Ricci and Codazzi -- Pseudo-Riemannian hypersurfaces -- The Robertson-Walker spacetime -- The Friedmann cosmological models -- 12 JET BUNDLES -- Bundles -- Affine bundles -- Derivations and the Frölicher-Nijenhuis bracket -- First order jet bundles -- Holonomic tangent vectors -- Contact cotangent vectors -- Jet fields and connections -- Equivariant jet fields -- Second order jet bundles -- Prolongation of vector fields -- Calculus of variations -- A: PRELIMINARIES -- Maps -- The permutation group -- Group actions -- Categories and functors -- Connectivity -- Homotopy theory -- Coverings -- Topological groups -- Topological vector spaces -- Bibliography -- Index