Preface to the second edition
Preface
Contents
Conventions
I. REVIEW OF FUNDAMENTAL NOTIONS
OF ANALYSIS
1.Graded algebras
2.Berezinian
3.Tensor product of algebras
4.Clifford algebras
5.Clifford algebra as a coset of the tensor algebra
6.Fierz identity
7.Pin and Spin groups
8.Weyl spinors, helicity operator; Majorana pinors, charge
conjugation
9.Representations of Spin(n, m), n + m odd
10.Dirac adjoint
11.Lie algebra of Pin(n, m) and Spin(n, m)
12.Compact spaces
13.Compactness in weak star topology
14.Homotopy groups, general properties
15.Homotopy of topological groups
16.Spectrum of closed and self-adjoint linear operators
II. DIFFERENTIAL CALCULUS ON BANACH SPACES
1.Supersmooth mappings
2.Berezin integration; Gaussian integrals
3.Noether's theorems I
4.Noether's theorems II
5.Invariance'of the equations of motion
6.String action
7.Stress--energy tensor; energy with respect to a timelike vector
field
III. DIFFERENTIABLE MANIFOLDS
1.Sheaves
2.Differentiable submanifolds
3.Subgroups of Lie groups. When are they Lie subgroups?
4.Cartan-Killing form on the Lie algebraof a Lie group G
5.Direct and semidirect products of Lie groups and their Lie
algebra
6.Homomorphisms and antihomomorphisms of a Lie algebra into
spaces of vector fields
7.Homogeneous spaces; symmetric spaces
8.Examples of homogeneous spaces, Stiefel and Grassmann
manifolds
9.Abelian representations of nonabelian groups
10.Irreducibility and reducibility
11.Characters
12.Solvable Lie groups
13.Lie algebras of linear groups
14.Graded bundles
IV. INTEGRATION ON MANIFOLDS
1.Cohomology. Definitions and exercises
2.Obstruction to the construction of Spin and Pin bundles;
Stiefel-Whitney classes
3.Inequivalent spin structures
4.Cohomology of groups
5.Lifting a group action
6.Short exact sequence; Weyl Heisenberg group
7.Cohomology of Lie algebras
8.Quasi-linear first-order partial differential equation
9.Exterior differential systems (contributed by B. Kent Harrison)
10.Bicklund transformations for evolution equations (contributed
by N.H. Ibragimov)
11.Poisson manifolds I
12.Poisson manifolds II (contributed by C. Moreno)
13.Completely integrable systems (contributed by C. Moreno)
V. RIEMANNIAN MANIFOLDS. KAHLERIAN
MANIFOLDS
1.Necessary and sufficient conditions for Lorentzian signature
2.First fundamental form (induced metric)
3.Killing vector fields
4.Sphere Sn
5.Curvature of Einstein cylinder
6.Conformal transformation of Yang-Mills, Dirac and Higgs
operators in d dimensions
7.Conformal system for Einstein equations
8.Conformal transformation of nonlinear wave equations
9.Masses of"homothetic" space-time
10.Invariant geometries on the squashed seven spheres
11.Harmonic maps
12.Composition of maps
13.Kaluza-Klein theories
14.Kihler manifolds; Calabi-Yau spaces
V BIS. CONNECTIONS ON A PRINCIPAL FIBRE
BUNDLE
1.An explicit proof of the existence of infinitely many connections
on a principal bundle with paracompact base
2.Gauge transformations
3.Hopf fibering S3S2
4.Subbundles and reducible bundles
5.Broken symmetry and bundle reduction, Higgs mechanism
6.The Euler-Poincare characteristic
7.Equivalent bundles
8.Universal bundles. Bundle classification
9.Generalized Bianchi identity
10.Chern-Simons classes
11.Cocycles on the Lie algebra of a gauge group; Anomalies
12.Virasoro representation of(Diff Sl ). Ghosts. BRST operator
VI. DISTRIBUTIONS
1.Elementary solution of the wave equation in d-dimensional
spacetime
2.Sobolev embedding theorem
3.Multiplication properties of Sobolev spaces
4.The best possible constant for a Sobolev inequality on Rn, n > 3
(contributed by H. Grosse)
5.Hardy-Littlewood-Sobolev inequality (contributed by
H. Grosse)
6.Spaces H,.
7.Spaces Hs(Sn) and H,.,
8.Completeness of a ball on Wp in Ws
9.Distribution with laplacian in L2(In)
10.Nonlinear wave equation in curved spacetime
11.Harmonic coordinates in general relativity
12.Leray theory of hyperbolic systems. Temporal gauge in general
relativity
13.Einstein equations with sources as a hyperbolic system
14.Distributions and analyticity: Wightman distributions and
Schwinger functions (contributed by C. Doering)
15.Bounds on the number of bound states of the Schr6dinger
operator
16.Sobolev spaces on Riemannian manifolds
SUPPLEMENTS AND ADDITIONAL PROBLEMS
1.The isomorphism HHM4. A supplement to
Problem 1.4 (I. 17)
2.Lie derivative of spinor fields (III.15)
3.Poisson-Lie groups, Lie bialgebras, and the generalized classical
Yang-Baxter equation (IV. 14) (contributed by Carlos Moreno
and Luis Valero)
4.Volume of the sphere Sn. A supplement to Problem V.4 (V. 15)
5.Teichmuller spaces (V.16)
6.Yamabe property on compact manifolds (V.17)
7.The Euler class. A supplement to Problem Vbis.6 (Vbis. 13)
8.Formula for laplacians at a point of the frame bundle (Vbis.14)
9.The Berry and Aharanov-Anandan phases (Vbis.15)
10.A density theorem. A supplement to Problem VI.6 "Spaces
11.Tensor distributions on submanifolds, multiple layers, and
shocks (VI. 18)
12.Discrete Boltzrnann equation (VI. 19)
Subject Index
Errata to Part I
