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**Modern geometric structures and fields. Transl. from the Russian by D. Chibisov.**
*(English)*
Zbl 1108.53001

Graduate Studies in Mathematics 71. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3929-2/hbk). xx, 633 p. (2006).

Geometric structures considered in this textbook are mainly Riemannian, complex, and symplectic structures on manifolds. Secondly, some special elements of submanifold theory, are incorporated, and the third component is a number of topics of modern physics based on these structures. The spectrum of themes covers a wide range, including classical analysis, analytic functions, linear algebra as well as ideas of differential geometry and topology, relativity, supersymmetry.

It starts in chapters 1. and 2. with basic geometric, analytic and algebraic material, introducing notions of Euclidian, symplectic and pseudo-Euclidian geometry in linear spaces, and the event space of special relativity as well as Lorentz transformations and the PoincarĂ© group.

Chapter 3 deals with surfaces in 3-dimensional Euclidean space, ending with the sine-Gordon equation that governs the theory of surfaces of constant Gaussian curvature \(k=-1\) as well as a number of physical phenomena. In chapter 4, analytic functions and methods of complex geometry are used to introduce models of spherical and hyperbolic geometry as well as conformal parametrization of surfaces. Hilbert’s result, stating that the pseudosphere can not be imbedded in \({\mathbb R}^3\) is proved. The sinh-Gordon in it’s close relation to surfaces of constant mean curvature, the Hopf differential and the Weierstrass-Enneper representation formula for minimal surfaces are established.

Real and complex differentiable manifolds are introduced in chapter 5, including examples, such as quotient manifolds of discrete group actions, especially real, complex and quaternionic projective spaces. A discussion of manifolds underlying groups of linear transformations appears as entering guide to Lie groups and Lie algebras, introduced in chapter 6. As examples of Lie algebras with richer structure, Poisson algebras and Lie superalgebras are considered. Chapter 6 ends presenting the classification of crystallographic groups. Chapter 7 is concerned with tensor algebra, exterior forms, polyvectors, Hodge operator, fermions and bosons, Fock spaces, anticommuting variables, and superalgebras. The following chapters 8 and 9 offer the analytic features of tensors; we mention action of mappings on tensors, vector fields and their exponentials, Lie derivative, central extension of Lie algebras, exterior differentation and integration of differential forms, Maxwell equations, theorem of Stokes, de Rham cohomology, integration on a superspace.

In chapter 10, an introduction to covariant differentiation in the tangent bundle, curvature and geodesics is given, followed in chapter 11 by ‘Conformal and Complex geometries’. Elements of Morse theory, mapping degree, ideas of transversality are combined in chapter 12 with calculus of variation in one variable including application to Hamiltons variational principle, conservation laws in classical and relativistic mechanics. Symplectic, Poisson and Lagrange manifolds are introduced in chapter 13. Multidimensional variational problems in chapter 14 are discussed with minimal surfaces, electromagnetic field- and Einstein equations. The final chapter 15, entitled ‘Geometric Fields in Physics’ is devoted to elements of Einstein’s gravitational field, spinors and the Dirac equation, and Yang-Mills fields and instantons.

Every chapter is completed by challenging exercises, their total number being 191. All notions are illustrated by interesting examples and applications. The textbook offers an abundance of general theories, concrete examples, and algebraic computations. It is a readable introduction to a wide number of areas of geometrical and algebraic themes in its interrelation with physics, appropriate for students of mathematics and theoretical physics.

It starts in chapters 1. and 2. with basic geometric, analytic and algebraic material, introducing notions of Euclidian, symplectic and pseudo-Euclidian geometry in linear spaces, and the event space of special relativity as well as Lorentz transformations and the PoincarĂ© group.

Chapter 3 deals with surfaces in 3-dimensional Euclidean space, ending with the sine-Gordon equation that governs the theory of surfaces of constant Gaussian curvature \(k=-1\) as well as a number of physical phenomena. In chapter 4, analytic functions and methods of complex geometry are used to introduce models of spherical and hyperbolic geometry as well as conformal parametrization of surfaces. Hilbert’s result, stating that the pseudosphere can not be imbedded in \({\mathbb R}^3\) is proved. The sinh-Gordon in it’s close relation to surfaces of constant mean curvature, the Hopf differential and the Weierstrass-Enneper representation formula for minimal surfaces are established.

Real and complex differentiable manifolds are introduced in chapter 5, including examples, such as quotient manifolds of discrete group actions, especially real, complex and quaternionic projective spaces. A discussion of manifolds underlying groups of linear transformations appears as entering guide to Lie groups and Lie algebras, introduced in chapter 6. As examples of Lie algebras with richer structure, Poisson algebras and Lie superalgebras are considered. Chapter 6 ends presenting the classification of crystallographic groups. Chapter 7 is concerned with tensor algebra, exterior forms, polyvectors, Hodge operator, fermions and bosons, Fock spaces, anticommuting variables, and superalgebras. The following chapters 8 and 9 offer the analytic features of tensors; we mention action of mappings on tensors, vector fields and their exponentials, Lie derivative, central extension of Lie algebras, exterior differentation and integration of differential forms, Maxwell equations, theorem of Stokes, de Rham cohomology, integration on a superspace.

In chapter 10, an introduction to covariant differentiation in the tangent bundle, curvature and geodesics is given, followed in chapter 11 by ‘Conformal and Complex geometries’. Elements of Morse theory, mapping degree, ideas of transversality are combined in chapter 12 with calculus of variation in one variable including application to Hamiltons variational principle, conservation laws in classical and relativistic mechanics. Symplectic, Poisson and Lagrange manifolds are introduced in chapter 13. Multidimensional variational problems in chapter 14 are discussed with minimal surfaces, electromagnetic field- and Einstein equations. The final chapter 15, entitled ‘Geometric Fields in Physics’ is devoted to elements of Einstein’s gravitational field, spinors and the Dirac equation, and Yang-Mills fields and instantons.

Every chapter is completed by challenging exercises, their total number being 191. All notions are illustrated by interesting examples and applications. The textbook offers an abundance of general theories, concrete examples, and algebraic computations. It is a readable introduction to a wide number of areas of geometrical and algebraic themes in its interrelation with physics, appropriate for students of mathematics and theoretical physics.

Reviewer: Hubert Gollek (Berlin)

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

53C20 | Global Riemannian geometry, including pinching |

53C80 | Applications of global differential geometry to the sciences |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

70H25 | Hamilton’s principle |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

53C27 | Spin and Spin\({}^c\) geometry |