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)