# Modern Geometric Structures and Fields

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1 Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island

2 Preface to the English Edition Preface Chapter 1. Cartesian Spaces and Euclidean Geometry Coordinates. Space-time Cartesian coordinates Change of coordinates Euclidean geometry and linear algebra Vector spaces and scalar products The length of a curve Affine transformations Matrix formalism. Orientation Affine group Motions of Euclidean spaces Curves in Euclidean space The natural parameter and curvature Curves on the plane Curvature and torsion of curves in R 3 28 Exercises to Chapter 1 32 Chapter 2. Symplectic and Pseudo-Euclidean Spaces Geometric structures in linear spaces Pseudo-Euclidean and symplectic spaces Symplectic transformations The Minkowski space 43 xiii xvii

3 vi Contents The event space of the special relativity theory The Poincare group Lorentz transformations 48 Exercises to Chapter 2 50 Chapter 3. Geometry of Two-Dimensional Manifolds Surfaces in three-dimensional space Regular surfaces Local coordinates Tangent space Surfaces as two-dimensional manifolds Riemannian metric on a surface The length of a curve on a surface Surface area Curvature of a surface On the notion of the surface curvature Curvature of lines on a surface Eigenvalues of a pair of scalar products Principal curvatures and the Gaussian curvature Basic equations of the theory of surfaces Derivational equations as the "zero curvature" condition. Gauge fields The Codazzi and sine-gordon equations The Gauss theorem 80 Exercises to Chapter 3 81 Chapter 4. Complex Analysis in the Theory of Surfaces Complex spaces and analytic functions Complex vector spaces The Hermitian scalar product Unitary and linear-fractional transformations Holomorphic functions and the Cauchy- Riemann equations Complex-analytic coordinate changes Geometry of the sphere The metric of the sphere The group of motions of a sphere Geometry of the pseudosphere Space-like surfaces in pseudo-euclidean spaces The metric and the group of motions of the pseudosphere 102

4 vii Models of hyperbolic geometry Hilbert's theorem on impossibility of imbedding the pseudosphere into R The theory of surfaces in terms of a conformal parameter Existence of a conformal parameter The basic equations in terms of a conformal parameter Hopf differential and its applications Surfaces of constant Gaussian curvature. The Liouville equation Surfaces of constant mean curvature. The sinh-gordon equation Minimal surfaces The Weierstrass-Enneper formulas for minimal surfaces Examples of minimal surfaces 120 Exercises to Chapter Chapter 5. Smooth Manifolds Smooth manifolds Topological and metric spaces On the notion of smooth manifold Smooth mappings and tangent spaces Multidimensional surfaces in R n. Manifolds with boundary Partition of unity. Manifolds as multidimensional surfaces in Euclidean spaces Discrete actions and quotient manifolds Complex manifolds Groups of transformations as manifolds Groups of motions as multidimensional surfaces Complex surfaces and subgroups of GL(n, C) Groups of affine transformations and the Heisenberg group Exponential mapping Quaternions and groups of motions Algebra of quaternions The groups SO(3) and SO(4) Quaternion-linear transformations 173 Exercises to Chapter Chapter 6. Groups of Motions Lie groups and algebras 177

5 viii Contents Lie groups Lie algebras Main matrix groups and Lie algebras Invariant metrics on Lie groups Homogeneous spaces Complex Lie groups Classification of Lie algebras Two-dimensional and three-dimensional Lie algebras Poisson structures Graded algebras and Lie superalgebras Crystallographic groups and their generalizations Crystallographic groups in Euclidean spaces Quasi-crystallographic groups 232 Exercises to Chapter Chapter 7. Tensor Algebra Tensors of rank 1 and Tangent space and tensors of rank Tensors of rank Transformations of tensors of rank at most Tensors of arbitrary rank Transformation of components Algebraic operations on tensors Differential notation for tensors Invariant tensors A mechanical example: strain and stress tensors Exterior forms Symmetrization and alternation Skew-symmetric tensors of type (0, k) Exterior algebra. Symmetric algebra Tensors in the space with scalar product Raising and lowering indices Eigenvalues of scalar products Hodge duality operator Fermions and bosons. Spaces of symmetric and skewsymmetric tensors as Fock spaces Polyvectors and the integral of anticommuting variables Anticommuting variables and superalgebras Integral of anticommuting variables 281 Exercises to Chapter Chapter 8. Tensor Fields in Analysis 285

6 ix 8.1. Tensors of rank 2 in pseudo-euclidean space Electromagnetic field Reduction of skew-symmetric tensors to canonical form Symmetric tensors Behavior of tensors under mappings Action of mappings on tensors with superscripts Restriction of tensors with subscripts The Gauss map Vector fields Integral curves Lie algebras of vector fields Linear vector fields Exponential function of a vector field Invariant fields on Lie groups The Lie derivative Central extensions of Lie algebras 309 Exercises to Chapter Chapter 9. Analysis of Differential Forms Differential forms Skew-symmetric tensors and their differentiation Exterior differential Maxwell equations Integration of differential forms Definition of the integral Integral of a form over a manifold Integrals of differential forms in R Stokes theorem The proof of the Stokes theorem for a cube Integration over a superspace Cohomology De Rham cohomology Homotopy invariance of cohomology Examples of cohomology groups 343 Exercises to Chapter Chapter 10. Connections and Curvature Covariant differentiation Covariant differentiation of vector fields Covariant differentiation of tensors Gauge fields 359

7 Cartan connections Parallel translation Connections compatible with a metric Curvature tensor Definition of the curvature tensor Symmetries of the curvature tensor The Riemann tensors in small dimensions, the Ricci tensor, scalar and sectional curvatures Tensor of conformal curvature Tetrad formalism The curvature of invariant metrics of Lie groups Geodesic lines Geodesic flow Geodesic lines as shortest paths The Gauss-Bonnet formula 389 Exercises to Chapter Chapter 11. Conformal and Complex Geometries Conformal geometry Conformal transformations Liouville's theorem on conformal mappings Lie algebra of a conformal group Complex structures on manifolds Complex differential forms Kahler metrics Topology of Kahler manifolds Almost complex structures Abelian tori 417 Exercises to Chapter Chapter 12. Morse Theory and Hamiltonian Formalism Elements of Morse theory Critical points of smooth functions Morse lemma and transversality theorems Degree of a mapping Gradient systems and Morse surgeries Topology of two-dimensional manifolds One-dimensional problems: Principle of least action Examples of functionals (geometry and mechanics). Variational derivative Equations of motion (examples) 457

8 xi Groups of symmetries and conservation laws Conservation laws of energy and momentum Fields of symmetries Conservation laws in relativistic mechanics Conservation laws in classical mechanics Systems of relativistic particles and scattering Hamilton's variational principle Hamilton's theorem Lagrangians and time-dependent changes of coordinates Variational principles of Fermat type 477 Exercises to Chapter Chapter 13. Poisson and Lagrange Manifolds Symplectic and Poisson manifolds g-gradient systems and symplectic manifolds Examples of phase spaces Extended phase space Poisson manifolds and Poisson algebras Reduction of Poisson algebras Examples of Poisson algebras Canonical transformations Lagrangian submanifolds and their applications The Hamilton-Jacobi equation and bundles of trajectories Representation of canonical transformations Conical Lagrangian surfaces The "action-angle" variables Local minimality condition The second-variation formula and the Jacobi operator Conjugate points 527 Exercises to Chapter Chapter 14. Multidimensional Variational Problems Calculus of variations Introduction. Variational derivatives Energy-momentum tensor and conservation laws Examples of multidimensional variational problems Minimal surfaces Electromagnetic field equations Einstein equations. Hilbert functional 548

9 xii Contents Harmonic functions and the Hodge expansion The Dirichlet functional and harmonic mappings Massive scalar and vector fields 563 Exercises to Chapter Chapter 15. Geometric Fields in Physics Elements of Einstein's relativity theory Principles of special relativity Gravitation field as a metric The action functional of a gravitational field The Schwarzschild and Kerr metrics Interaction of matter with gravitational field On the concept of mass in general relativity theory Spinors and the Dirac equation Automorphisms of matrix algebras Spinor representation of the group SO(3) Spinor representation of the group 0(1,3) Dirac equation Clifford algebras Yang-Mills fields Gauge-invariant Lagrangians Covariant differentiation of spinors Curvature of a connection The Yang-Mills equations Characteristic classes Instantons 612 Exercises to Chapter Bibliography 621 Index 625

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