id: 00051358 dt: b an: 00051358 au: Postnikov, M. ti: Leçons de géométrie. Variétés différentiables. Traduit du russe par Djilali Embarek. (Lectures in geometry. Differentiable manifolds. Transl. from the Russian by Djilali Embarek.) so: Traduit du Russe: Mathématiques. Moscow: Éditions Mir. 431 p. (1990). py: 1990 pu: Moscow: Éditions Mir la: FR cc: ut: differential geometry of curves and surfaces; differentiable manifolds; Frenet formulas; first fundamental form; second fundamental form; principal curvatures; Gaussian curvature; mean curvature; equations of Weingarten; Theorema egregium; filters; compact spaces; Lie groups; tangent vectors; submanifolds; regular value; imbedding; Sard’s theorem; tensors; vector fields; local flow of vector fields; differential forms; exterior differentiation; de Rham complex; cohomology groups; Poincaré lemma; de Rham-Leray theorem; spectral sequence; integration of densities; Stokes’ theorem; singular homology and cohomology; degree of a map ci: Zbl 0326.53001 li: ab: This book is divided into 29 parts, called lessons. Regarding the subject matter, it should be divided into two large parts. The first five lessons refer to the classical differential geometry of curves and surfaces in ${\bbfR}\sp 3$ (and sometimes in ${\bbfR}\sp n$). The remaining ones concern the theory of differentiable manifolds, directed to the de Rham, de Rham- Leray and Stokes theorems. The individual lessons concern: 1. regular curves parametrized by arc-length in ${\bbfR}\sp n$, 2. Frenet formulas, 3. surfaces (given by one parametrization), the first fundamental form, isometries, 4. the second fundamental form, principal curvatures, Gaussian curvature, mean curvature, 5. equations of Weingarten, Theorema egregium (Gauss), 6. differentiable manifolds (real and ${\bbfC}$-analytical) as sets equipped with atlases, 7. the topology of a differentiable manifold, 8. the dimension of a differentiable manifold as a topological invariant, 9. Lebesgue’s theorem on a covering of a cube, 10. filters, compact spaces, 11. Lie groups of matrices, real: $GL(n,{\bbfR})$, $O(n)$, $O(p,q)$, $Sp(n)$, and complex, 12. tangent vectors to a real (and ${\bbfC}$-analytical) manifold, 13. submanifolds, the theorem on the preimage of a regular value, 14. the Whitney imbedding theorem, 15. Sard’s theorem, 16. tensors of type $(p,q)$, differentiations of the algebra of smooth functions, the Lie algebra of differentiable vector fields, 17. integral curves and the local flow of vector fields, the Lie derivative of tensor fields, 18. the algebra of differential forms, the pullback of forms, 19. the exterior differentiation and the Lie derivative of differential forms, 20. the de Rham complex, cohomology groups, the Poincaré lemma $H\sp m(S\sp n)$, 21. the double complex $C\sp{*,*}({\germ U})$, 22. the de Rham-Leray theorem, the groups $E\sb 2\sp{p,q}$, 23. the spectral sequence of a double complex, the groups $E\sp{p,q}\sb \infty$, 24. paracompact manifolds, integration of densities, 25. integration of differential forms, orientable manifolds, 26. the degree of a proper map and its homotopic invariance, 27. manifolds with a boundary, Stokes’ theorem, 28. ${\bbfR}\sp 3$: grad, div, cases of Stokes’ theorem, 29. singular homology and cohomology. The favourable value of the book (peculiar to the author) is his care for details and choice of proper examples. However, this book (mainly its first part) is written in the time-worn language of differential geometry (compared with the more modern lectures by {\it M. do Carmo} [Differential geometry of curves and surfaces (1976; Zbl 0326.53001)]. Next, the author uses non-modern definitions of the tangent vector to a manifold and of tensors. This is caused by the non-using of multilinear algebra (in particular, tensor algebra). The author also omits the apparatus of vector bundles. Strong points are clear style, figures and suitable examples. rv: J.Kubarski (Łódź)