Modern Geometry - Methods and Applications: Part I. The Geometry of Surfaces, Transformation Groups, and FieldsSpringer Science & Business Media, 14 mars 2013 - 464 pages manifolds, transformation groups, and Lie algebras, as well as the basic concepts of visual topology. It was also agreed that the course should be given in as simple and concrete a language as possible, and that wherever practic able the terminology should be that used by physicists. Thus it was along these lines that the archetypal course was taught. It was given more permanent form as duplicated lecture notes published under the auspices of Moscow State University as: Differential Geometry, Parts I and II, by S. P. Novikov, Division of Mechanics, Moscow State University, 1972. Subsequently various parts of the course were altered, and new topics added. This supplementary material was published (also in duplicated form) as Differential Geometry, Part III, by S. P. Novikov and A. T. Fomenko, Division of Mechanics, Moscow State University, 1974. The present book is the outcome of a reworking, re-ordering, and ex tensive elaboration of the above-mentioned lecture notes. It is the authors' view that it will serve as a basic text from which the essentials for a course in modern geometry may be easily extracted. To S. P. Novikov are due the original conception and the overall plan of the book. The work of organizing the material contained in the duplicated lecture notes in accordance with this plan was carried out by B. A. Dubrovin. |
Table des matières
1 | |
9 | |
3 Riemannian and pseudoRiemannian spaces | 17 |
CHAPTER 2 | 20 |
4 The simplest groups of transformations of Euclidean space | 23 |
5 The SerretFrenet formulae | 38 |
6 PseudoEuclidean spaces | 50 |
The Theory of Surfaces | 61 |
24 Lie algebras | 212 |
CHAPTER 4 | 234 |
27 Differential forms on complex spaces | 266 |
29 Covariant differentiation and the metric | 284 |
30 The curvature tensor | 295 |
CHAPTER 5 | 313 |
32 Conservation laws | 320 |
33 Hamiltonian formalism | 333 |
10 Spacelike surfaces in pseudoEuclidean space | 90 |
12 Analytic functions | 100 |
13 The conformal form of the metric on a surface | 109 |
CHAPTER 3 | 145 |
17 The general definition of a tensor | 151 |
19 Tensors in Riemannian and pseudoRiemannian spaces | 168 |
21 Rank 2 tensors in pseudoEuclidean space and their eigenvalues | 194 |
22 The behaviour of tensors under mappings | 203 |
36 The second variation for the equation of the geodesics | 367 |
The Calculus of Variations in Several Dimensions Fields | 375 |
38 Examples of Lagrangians | 409 |
40 The spinor representations of the groups SO3 and O3 1 Diracs | 427 |
41 Covariant differentiation of fields with arbitrary symmetry | 439 |
42 Examples of gaugeinvariant functionals Maxwells equations and | 449 |
455 | |
Autres éditions - Tout afficher
Modern Geometry - Methods and Applications: Part I. The Geometry of Surfaces ... B.A. Dubrovin,A.T. Fomenko,S.P. Novikov Aucun aperçu disponible - 2014 |
Modern Geometry — Methods and Applications: Part I: The Geometry of ... B.A. Dubrovin,A.T. Fomenko,S.P. Novikov Aucun aperçu disponible - 2011 |
Expressions et termes fréquents
3-dimensional a₁ arbitrary co-ordinate changes co-ordinate system co-ordinates x¹ commutation complex components connexion const corresponding covariant covector curve defined definition denote differential dl² dx¹ dx² dx³ dz¹ e₁ Euclidean 3-space Euclidean co-ordinates Euler-Lagrange equations əx³ əxi Əza Əzi follows formula function g₁ Gaussian curvature geodesic given gradient Hamiltonian Hence integral invariant isometry isomorphic Killing metric Lagrangian lattice Lemma Lie algebra linear transformation matrix metric g Minkowski space non-singular obtain operation orthogonal parameter phase space plane pseudo-Euclidean quadratic form region Riemannian Riemannian metric rotation scalar product skew-symmetric skew-symmetric tensor SO(n Stokes formula subgroup symmetric T₁ tangent space tangent vector tensor of type Theorem translations underlying space vector field whence zero дн др ду дх дха дхі