## Modern Geometry - Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Google eBook)This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised. |

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### Contents

v | |

1 | |

9 | |

Riemannian and PseudoRiemannian Spaces | 17 |

The Simplest Groups of Transformations of Euclidean Space | 23 |

The SerretFrenet Formulae | 38 |

PseudoEuclidean Spaces | 50 |

The Theory of Surfaces | 61 |

The Behaviour of Tensors Under Mappings | 205 |

Vector Fields | 208 |

Lie Algebras | 216 |

The Differential Calculus of Tensors | 238 |

SkewSymmetric Tensors and the Theory of Integration | 248 |

Differential Forms on Complex Spaces | 270 |

Covariant Differentiation | 275 |

Covariant Differentiation and the Metric | 288 |

The Second Fundamental Form | 76 |

The Metric on the Sphere | 86 |

Spacelike Surfaces in PseudoEuclidean Space | 90 |

The Language of Complex Numbers in Geometry | 95 |

Analytic Functions | 100 |

The Conformal Form of the Metric on a Surface | 109 |

Transformation Groups as Surfaces in NDimensional Space | 120 |

Conformal Transformations of Euclidean and PseudoEuclidean Spaces of Several Dimensions | 136 |

Tensors The Algebraic Theory | 145 |

The General Definition of a Tensor | 151 |

Tensors of Type 0 k | 161 |

Tensors in Riemannian and PseudoRiemannian Spaces | 170 |

The Crystallographic Groups and the Finite Subgroups of the Rotation Group of Euclidean 3Space Examples of Invariant Tensors | 176 |

Rank 2 Tensors in PseudoEuclidean Space and Their Eigenvalues | 197 |

The Curvature Tensor | 299 |

The Elements of the Calculus of Variations | 317 |

Conservation Laws | 324 |

Hamiltonian Formalism | 337 |

The Geometrical Theory of Phase Space | 348 |

Lagrange Surfaces | 362 |

The Second Variation for the Equation of the Geodesies | 371 |

The Calculus of Variations in Several Dimensions Fields and Their Geometric Invariants | 379 |

Examples of Lagrangians | 413 |

The Simplest Concepts of the General Theory of Relativity | 416 |

The Spinor Representations of the Groups SO3 and O3 1 Diracs Equation and Its Properties | 431 |

Covariant Differentiation of Fields with Arbitrary Symmetry | 443 |

Examples of GaugeInvariant Functionals Maxwells Equations and the YangMills Equation Functionals with Identically Zero Variational Derivative C... | 453 |

### Common terms and phrases

3-dimensional analytic arbitrary Cartesian co-ordinate change co-ordinate system co-ordinates x1 co-ordinatized commutator completes the proof complex components connexion consider const constant corresponding covector curvature tensor curve defined definition denote derivative electromagnetic field element Euclidean 3-space Euclidean co-ordinates Euler-Lagrange equations example Exercise follows function Gaussian curvature geodesic geometry given GL(n gradient gravitational Hamiltonian Hence identity integral invariant isometries isomorphic Jacobian Lagrangian lattice Lemma length Lie algebra linear transformation Lorentz matrix metric gu Minkowski Minkowski metric Minkowski space momentum neighbourhood non-zero notation Note obtain operation orthogonal parameter particle particular phase space pseudo-Euclidean quadratic form region relative respect Riemannian Riemannian metric right-hand side rotation satisfying scalar product skew-symmetric skew-symmetric tensor smooth sphere spinor Stokes formula subgroup Suppose symmetric tangent space tangent vector tensor of type Theorem theory translations underlying space vector field whence zero

### Popular passages

Page 462 - Electrodynamics of Continuous Media, Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford, UK, 1982.

Page v - YN (LD Landau Institute for Theoretical Physics, Academy of Sciences of the USSR, Moscow, USSR).

Page vi - Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in...

Page vi - The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University.

Page 2 - The chief aim of analytic (or co-ordinate) geometry is to describe geometrical figures by means of algebraic formulae referred to a Cartesian system of co-ordinates of the plane or 3-dimensional space.