| 000 | 09596nam a2201105 i 4500 | ||
|---|---|---|---|
| 001 | 6813736 | ||
| 003 | IEEE | ||
| 005 | 20200413152854.0 | ||
| 006 | m eo d | ||
| 007 | cr cn |||m|||a | ||
| 008 | 090604s2009 cau foab 001 0 eng d | ||
| 020 | _a9781598298208 (electronic bk.) | ||
| 020 | _z9781598298192 (pbk.) | ||
| 024 | 7 |
_a10.2200/S00197ED1V01Y200906MAS005 _2doi |
|
| 035 | _a(CaBNVSL)gtp00534712 | ||
| 035 | _a(OCoLC)426825841 | ||
| 040 |
_aCaBNVSL _cCaBNVSL _dCaBNVSL |
||
| 050 | 4 |
_aQA613 _b.G468 2009 |
|
| 082 | 0 | 4 |
_a516.07 _222 |
| 245 | 0 | 4 |
_aThe geometry of Walker manifolds _h[electronic resource] / _cMiguel Brozos-Vázquez ... [et al]. |
| 260 |
_aSan Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : _bMorgan & Claypool Publishers, _cc2009. |
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| 300 |
_a1 electronic text (xvii, 159 p.) : _bdigital file. |
||
| 490 | 1 |
_aSynthesis lectures on mathematics and statistics, _x1930-1751 ; _v# 5 |
|
| 538 | _aMode of access: World Wide Web. | ||
| 538 | _aSystem requirements: Adobe Acrobat reader. | ||
| 500 | _aPart of: Synthesis digital library of engineering and computer science. | ||
| 500 | _aSeries from website. | ||
| 504 | _aIncludes bibliographical references (p. 129-147) and index. | ||
| 505 | 0 | _aBasic algebraic notions -- Introduction -- A historical perspective in the algebraic context -- Algebraic preliminaries -- Jordan normal form -- Indefinite geometry -- Algebraic curvature tensors -- Hermitian and para-Hermitian geometry -- The Jacobi and skew symmetric curvature operators -- Sectional, Ricci, scalar, and Weyl curvature -- Curvature decompositions -- Self-duality and anti-self-duality conditions -- Spectral geometry of the curvature operator -- Osserman and conformally Osserman models -- Osserman curvature models in signature (2, 2) -- Ivanov-Petrova curvature models -- Osserman Ivanov-Petrova curvature models -- Commuting curvature models -- Basic geometrical notions -- Introduction -- History -- Basic manifold theory -- The tangent bundle, lie bracket, and lie groups -- The cotangent bundle and symplectic geometry -- Connections, curvature, geodesics, and holonomy -- Pseudo-Riemannian geometry -- The Levi-Civita connection -- Associated natural operators -- Weyl scalar invariants -- Null distributions -- Pseudo-Riemannian holonomy -- Other geometric structures -- Pseudo-Hermitian and para-Hermitian structures -- Hyper-para-Hermitian structures -- Geometric realizations -- Homogeneous spaces, and curvature homogeneity -- Technical results in differential equations -- Walker structures -- Introduction -- Historical development -- Walker coordinates -- Examples of Walker manifolds -- Hypersurfaces with nilpotent shape operators -- Locally conformally flat metrics with nilpotent Ricci operator -- Degenerate pseudo-Riemannian homogeneous structures -- Para-Kaehler geometry -- Two-step nilpotent lie groups with degenerate center -- Conformally symmetric pseudo-Riemannian metrics -- Riemannian extensions -- The affine category -- Twisted Riemannian extensions defined by flat connections -- Modified Riemannian extensions defined by flat connections -- Nilpotent Walker manifolds -- Osserman Riemannian extensions -- Ivanov-Petrova Riemannian extensions -- Three-dimensional Lorentzian Walker manifolds -- Introduction -- History -- Three dimensional Walker geometry -- Adapted coordinates -- The Jordan normal form of the Ricci operator -- Christoffel symbols, curvature, and the Ricci tensor -- Locally symmetric Walker manifolds -- Einstein-like manifolds -- The spectral geometry of the curvature tensor -- Curvature commutativity properties -- Local geometry of Walker manifolds with -- Foliated Walker manifolds -- Contact Walker manifolds -- Strict Walker manifolds -- Three dimensional homogeneous Lorentzian manifolds -- Three dimensional lie groups and lie algebras -- Curvature homogeneous Lorentzian manifolds -- Diagonalizable Ricci operator -- Type II Ricci operator -- Four-dimensional Walker manifolds -- Introduction -- History -- Four-dimensional Walker manifolds -- Almost para-Hermitian geometry -- Isotropic almost para-Hermitian structures -- Characteristic classes -- Self-dual Walker manifolds -- The spectral geometry of the curvature tensor -- Introduction -- History -- Four-dimensional Osserman metrics -- Osserman metrics with diagonalizable Jacobi operator -- Osserman Walker type II metrics -- Osserman and Ivanov-Petrova metrics -- Riemannian extensions of affine surfaces -- Affine surfaces with skew symmetric Ricci tensor -- Affine surfaces with symmetric and degenerate Ricci tensor -- Riemannian extensions with commuting curvature operators -- Other examples with commuting curvature operators -- Hermitian geometry -- Introduction -- History -- Almost Hermitian geometry of Walker manifolds -- The proper almost Hermitian structure of a Walker manifold -- Proper almost hyper-para-Hermitian structures -- Hermitian Walker manifolds of dimension four -- Proper Hermitian Walker structures -- Locally conformally Kaehler structures -- Almost Kaehler Walker four-dimensional manifolds -- Special Walker manifolds -- Introduction -- History -- Curvature commuting conditions -- Curvature homogeneous strict Walker manifolds -- Bibliography. | |
| 506 | 1 | _aAbstract freely available; full-text restricted to subscribers or individual document purchasers. | |
| 510 | 0 | _aCompendex | |
| 510 | 0 | _aINSPEC | |
| 510 | 0 | _aGoogle scholar | |
| 510 | 0 | _aGoogle book search | |
| 520 | 3 | _aThis book, which focuses on the study of curvature, is an introduction to various aspects of pseudo- Riemannian geometry. We shall use Walker manifolds (pseudo-Riemannian manifolds which admit a non-trivial parallel null plane field) to exemplify some of the main differences between the geometry of Riemannian manifolds and the geometry of pseudo-Riemannian manifolds and thereby illustrate phenomena in pseudo-Riemannian geometry that are quite different from those which occur in Riemannian geometry, i.e. for indefinite as opposed to positive definite metrics. Indefinite metrics are important in many diverse physical contexts: classical cosmological models (general relativity) and string theory to name but two. Walker manifolds appear naturally in numerous physical settings and provide examples of extremal mathematical situations as will be discussed presently. To describe the geometry of a pseudo-Riemannian manifold, one must first understand the curvature of the manifold. We shall analyze a wide variety of curvature properties and we shall derive both geometrical and topological results. Special attention will be paid to manifolds of dimension 3 as these are quite tractable. We then pass to the 4 dimensional setting as a gateway to higher dimensions. Since the book is aimed at a very general audience (and in particular to an advanced undergraduate or to a beginning graduate student), no more than a basic course in differential geometry is required in the way of background. To keep our treatment as self-contained as possible,we shall begin with two elementary chapters that provide an introduction to basic aspects of pseudo-Riemannian geometry before beginning on our study of Walker geometry. An extensive bibliography is provided for further reading. | |
| 530 | _aAlso available in print. | ||
| 588 | _aTitle from PDF t.p. (viewed on June 4, 2009). | ||
| 650 | 0 | _aManifolds (Mathematics) | |
| 650 | 0 | _aRiemannian manifolds. | |
| 650 | 0 | _aCurvature. | |
| 690 | _aAffine connection | ||
| 690 | _aAffine surface | ||
| 690 | _aAlmost Hermitian | ||
| 690 | _aAlmost Kaehler | ||
| 690 | _aChristoffel symbols | ||
| 690 | _aCodazzi Ricci tensor | ||
| 690 | _aCommuting curvature model | ||
| 690 | _aConformally flat | ||
| 690 | _aConformally Kaehler | ||
| 690 | _aConformally Osserman | ||
| 690 | _aContact Walker manifold | ||
| 690 | _aCurvature commuting | ||
| 690 | _aCyclic parallel Ricci tensor | ||
| 690 | _aEinstein | ||
| 690 | _aFlat connection | ||
| 690 | _aFoliated Walker manifold | ||
| 690 | _aGray identity | ||
| 690 | _aGeometry of the curvature operator | ||
| 690 | _aHomogeneous space | ||
| 690 | _aHyper Hermitian | ||
| 690 | _aHyper-Kaehler | ||
| 690 | _aIvanov-Petrova | ||
| 690 | _aJacobi operator | ||
| 690 | _aLevi-Civita connection | ||
| 690 | _aLocally symmetric | ||
| 690 | _aLorentzian | ||
| 690 | _aNijenhuis tensor | ||
| 690 | _aNilpotent Walker manifold | ||
| 690 | _aNull distribution | ||
| 690 | _aOsserman curvature model | ||
| 690 | _aPara-Hermitian | ||
| 690 | _aPara-Kaehler | ||
| 690 | _aParallel null distribution | ||
| 690 | _aProjectively flat | ||
| 690 | _aRicci anti-symmetric | ||
| 690 | _aRicci curvature | ||
| 690 | _aRicci flat | ||
| 690 | _aScalar curvature | ||
| 690 | _aRiemannian extension | ||
| 690 | _aTorsion free connection | ||
| 690 | _aSchouten tensor | ||
| 690 | _aSectional curvature | ||
| 690 | _aSkew-symetric curvature operator | ||
| 690 | _aTricerri-Vanhecke decomposition | ||
| 690 | _aVaisman manifold | ||
| 690 | _aVanishing scalar invariants | ||
| 690 | _aWalker coordinates | ||
| 690 | _aWalker manifold | ||
| 690 | _aWeyl curvature | ||
| 690 | _aWeyl scalar invariants | ||
| 700 | 1 | _aBrozos-Vázquez, Miguel. | |
| 730 | 0 | _aSynthesis digital library of engineering and computer science. | |
| 830 | 0 |
_aSynthesis lectures on mathematics and statistics, _x1930-1751 ; _v# 5. |
|
| 856 | 4 | 2 |
_3Abstract with links to resource _uhttp://ieeexplore.ieee.org/servlet/opac?bknumber=6813736 |
| 999 |
_c561686 _d561686 |
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