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CRC Press Inc
27 July 1999
Differential & Riemannian geometry
This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and p: Z . Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change. Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;

R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area. Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture. Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.
By:   Peter B. Gilkey (University of Oregon Eugene USA), John V Leahy (University of Oregon, Eugene, Oregon, USA), JeongHyeong Park (Sungkyunkwan University)
Imprint:   CRC Press Inc
Country of Publication:   United States
Volume:   30
Dimensions:   Height: 235mm,  Width: 156mm,  Spine: 21mm
Weight:   594g
ISBN:   9780849382772
ISBN 10:   0849382777
Series:   Studies in Advanced Mathematics
Pages:   296
Publication Date:   27 July 1999
Audience:   College/higher education ,  Professional and scholarly ,  Professional & Vocational ,  A / AS level ,  Further / Higher Education
Format:   Hardback
Publisher's Status:   Active
ELLIPTIC OPERATORS Introduction The Real and Complex Laplace Operators Spinors Spectral Resolutions Manifolds with Boundary Spectral Invariants The Eta Invariant Computing the Eta Invariant DIFFERENTIAL GEOMETRY Introduction Riemannian Submersions Characteristic Classes The Geometry of Sphere and Principal Bundles The Geometry of Circle Bundles The Hopf Fibration The Scalar Curvature Levi-Civita and Spin Connections POSITIVE CURVATURE Introduction Manifolds with Positive Ricci Curvature Bordism and Connective K Theory Calculations Involving the Eta Invariant The Eta Invariant and Connective K Theory Computing Connective K Theory Groups SPECTRAL GEOMETRY OF RIEMANNIAN SUBMERSIONS Introduction Intertwining the Coderivitives The Real Laplacian The Complex Laplacian The Spin Laplacian Riemannian Submersions with Boundary Heat Trace and Heat Content Unresolved Questions REFERENCES Introduction Main Bibliography Bibliography of Harmonic Morphisms Parabolic PDE Bibliography NOTATION INDEX

Gilkey\, Peter B.; Leahy\, John V; Park\, JeongHyeong

Reviews for Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture

Altogether at ease with the mores of the ancien regime, Fraser (Marie Antoinette, 2001, etc.) eschews a detailed biography of Louis XIV to focus instead on the women who shaped and were shaped by France's most glorious ruler.He was king for 72 years, time enough to build Versailles, wage numerous, mostly unsuccessful, wars and accumulate a rich history with the opposite sex. Louis's mother, Queen Anne, gave birth at 36-a then-astoundingly late stage in life for pregnancy-lending an immediate air of the miraculous to the future monarch. Anne established an unusually close relationship with her son, who never entirely erased from his mind her mixture of beauty and piety as a template of female perfection. Louis abandoned his teenage liaison with the unsuitable Marie Mancini to marry Spanish Infanta Maria Teresa. As Queen Marie-Therese, she accommodated two important mistresses, Louise de La Valliere and the Marquise de Montespan, and drew from Louis the final tribute that she gave him no trouble except by dying. After Marie-Therese, he secretly married Madame de Maintenon, whose demeanor was remarkably like that of his mother. Fraser paints each of these women in full and offers sketches of a succession of minor mistresses, one night stands, sisters-in-law (including the hilariously vulgar and bitchy wife of Louis's homosexual brother), princesses and even an exiled foreign queen, all of whom engaged the king's genuine interest. To help keep track of this large cast, the profusion of changing titles and the dizzying succession of bastards, the author provides a useful guide to the principal characters. Courtiers meticulously charted the king's amorous adventures, and Fraser excels at reproducing the hothouse atmosphere in which the monarch's raised eyebrow might portend a serious change in someone's fortune. Uncomfortably aware of the Church's opposition to his notorious love life, Louis fully indulged himself during his heyday before turning in old age to a greater concern for his salvation.An acutely sensitive group portrait of the women who basked in the Sun King's reflected glory. (Kirkus Reviews)


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