Peter B Gilkey's Books

Aspects of Differential Geometry V

By: Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

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Aspects of Differential Geometry V

By: Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham complex and the Dolbeault complex, and discuss spinors. In Chapter 19, we discuss complex geometry and establish the Kodaira Embedding Theorem.

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140

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Springer Nature

Published Date

ISBN 10

303102432X

ISBN 13

9783031024320

The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds

By: Peter B. Gilkey

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The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds

By: Peter B. Gilkey

Pseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and StanilovOCoTsankovOCoVidev theory.

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389

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Imperial College Press

Published Date

ISBN 10

1860948588

ISBN 13

9781860948589

Aspects of Differential Geometry I

By: Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

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Aspects of Differential Geometry I

By: Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new. Table of Contents: Preface / Acknowledgments / Basic Notions and Concepts / Manifolds / Riemannian and Pseudo-Riemannian Geometry / Bibliography / Authors' Biographies / Index

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140

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Springer Nature

Published Date

ISBN 10

3031024079

ISBN 13

9783031024078

Asymptotic Formulae in Spectral Geometry

By: Peter B. Gilkey

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Asymptotic Formulae in Spectral Geometry

By: Peter B. Gilkey

A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a leading pioneer in the field, it focuses on the functorial and special cases methods of computi

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312

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CRC Press

Published Date

ISBN 10

0203490460

ISBN 13

9780203490464

Invariance Theory

The Heat Equation and the Atiyah-Singer Index Theorem

By: Peter B. Gilkey

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Invariance Theory

The Heat Equation and the Atiyah-Singer Index Theorem

By: Peter B. Gilkey

This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.

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534

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CRC Press

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ISBN 10

0849378745

ISBN 13

9780849378744

The Irony of Barack Obama

Barack Obama, Reinhold Niebuhr and the Problem of Christian Statecraft

By: R. Ward Holder,Peter B. Josephson

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The Irony of Barack Obama

Barack Obama, Reinhold Niebuhr and the Problem of Christian Statecraft

By: R. Ward Holder,Peter B. Josephson

Drawing on the political theology of Reinhold Niebuhr, described by Barack Obama as 'one of my favourite philosophers', this book assesses the challenges facing the President during his first term. It evaluates his success in adhering to Niebuhr's path of 'Christian realism' when faced with the pragmatic demands of domestic and foreign affairs. In 2008 Candidate Obama used the ideas of 'Hope' and 'Change' to inspire voters and secure the presidency. Obama promised change not only regarding America's policies, but even more fundamentally in the nation's political culture. Holder and Josephson describe the foundations of President Obama's Christian faith and the extent to which it has shaped his approach to politics. Their book explores Obama's journey of faith in the context of a broadly Augustinian understanding of faith and politics, examines the tensions between Christian realism and pragmatic progressivism, explains why a Christian realist interpretation is essential to understanding Obama's presidency, and applies this model of understanding to considerations of foreign and domestic policy. By combining this theological and political analysis the book offers a special opportunity to reflect on the relationship between Christian faith and statesmanship, reflections that are missing from current popular discussions of the Obama presidency. Through consideration of Niebuhr's models of the prophet and the statesman, and the more popular alternative of the political evangelist, Holder and Josephson are better able to explain the president's successes and his failures, and to unveil the Augustinian limits of the political life.

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258

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Routledge

Published Date

ISBN 10

1317026977

ISBN 13

9781317026976

Aspects of Differential Geometry II

By: Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

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Aspects of Differential Geometry II

By: Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.

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143

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Springer Nature

Published Date

ISBN 10

3031024087

ISBN 13

9783031024085

Reinhold Niebuhr in Theory and Practice

Christian Realism and Democracy in America in the Twenty-First Century

By: Peter B. Josephson,R. Ward Holder

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Reinhold Niebuhr in Theory and Practice

Christian Realism and Democracy in America in the Twenty-First Century

By: Peter B. Josephson,R. Ward Holder

American public life is gripped by a tumult that it has not experienced in at least half a century. Resentment, distrust, despair, fear, envy, and outrage are the passions of the day. Yet it was not long ago that political scientists and theologians could speak of a “Niebuhr renaissance” marked by an appreciation of moral paradox, ethical nuance, and a recognition of the irony of American history. American political leaders from Barack Obama and Hillary Clinton to George Bush and John McCain referenced Reinhold Niebuhr as an important influence on their political understandings. Columnists like David Brooks commented on the political condition of contemporary America, and scholars from Gary Dorrien and Daniel Rice to Richard Crouter developed academic accounts of Niebuhr’s political realism. From an insistence on political purity, to a wariness of international institutions and the claims of expertise, to a rejection of whole categories of public goods – it would be difficult to find a more significant shift from the principles that shaped statecraft and public policy during Niebuhr’s prime to those that are foundational in the age of Trump. Reinhold Niebuhr in Theory and Practice: Christian Realism and Democracy in America in the Twenty-First Century explains the collapse of the Niebuhrian renaissance in public life and the ascendance of the “children of light and the children of darkness” in the 2016 election. Our focus is Niebuhr himself and what the encounter between his own theology and his practical political experience might reveal in our contemporary situation. Niebuhr tells us that he does not offer precise policy prescriptions. But Niebuhr was a prolific author, and his works offer insights both into what realistic and Christian public policies would look like, and perhaps more importantly into how citizens should think for themselves about the political challenges of our times. Our aim, then, is to reassert the possibility of a distinctly Niebuhrian public intellectualism and a distinctly Niebuhrian political practice in the wake of the 2016 election.

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245

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Lexington Books

Published Date

ISBN 10

1498576702

ISBN 13

9781498576703

The Geometry Of Spherical Space Form Groups

By: Peter B Gilkey

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The Geometry Of Spherical Space Form Groups

By: Peter B Gilkey

New Edition: The Geometry of Spherical Space Form Groups (2nd Edition)In this volume, the geometry of spherical space form groups is studied using the eta invariant. The author reviews the analytical properties of the eta invariant of Atiyah-Patodi-Singer and describes how the eta invariant gives rise to torsion invariants in both K-theory and equivariant bordism. The eta invariant is used to compute the K-theory of spherical space forms, and to study the equivariant unitary bordism of spherical space forms and the Pinc and Spinc equivariant bordism groups for spherical space form groups. This leads to a complete structure theorem for these bordism and K-theory groups.There is a deep relationship between topology and analysis with differential geometry serving as the bridge. This book is intended to serve as an introduction to this subject for people from different research backgrounds.This book is intended as a research monograph for people who are not experts in all the areas discussed. It is written for topologists wishing to understand some of the analytic details and for analysists wishing to understand some of the topological ideas. It is also intended as an introduction to the field for graduate students.

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373

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World Scientific

Published Date

ISBN 10

9814522120

ISBN 13

9789814522120

The Cambridge Companion to Science and Religion

By: Peter Harrison

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The Cambridge Companion to Science and Religion

By: Peter Harrison

This book explores the historical relations between science and religion and discusses contemporary issues with perspectives from cosmology, evolutionary biology and bioethics.

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323

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Cambridge University Press

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ISBN 10

0521712513

ISBN 13

9780521712514

Serving Country and Community

Who Benefits from National Service?

By: Peter Frumkin,JoAnn Jastrzab

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Serving Country and Community

Who Benefits from National Service?

By: Peter Frumkin,JoAnn Jastrzab

Who benefits from AmeriCorps, VISTA, and National Civilian Community Corps? Frumkin and Jastrzab make important recommendations on how to improve the programs and resolve some of the political and administrative issues which have plagued these initiatives in the past two decades."ùJames Youniss, Catholic University of America --

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328

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Harvard University Press

Published Date

ISBN 10

0674046781

ISBN 13

9780674046788

The Ethics of Neoliberalism

The Business of Making Capitalism Moral

By: Peter Bloom

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The Ethics of Neoliberalism

The Business of Making Capitalism Moral

By: Peter Bloom

The 21st century is the age of "neo-liberalism" – a time when the free market is spreading to all areas of economic, political and social life. Yet how is this changing our individual and collective ethics? Is capitalism also becoming our new morality? From the growing popular demand for corporate social responsibility to personal desire for "work-life balance" it would appear that non-market ideals are not only surviving but also thriving. Why then does it seem that capitalism remains as strong as ever? The Ethics of Neoliberalism boldly proposes that neoliberalism strategically co-opts traditional ethics to ideologically and structurally strengthen capitalism. It produces "the ethical capitalist subject" who is personally responsible for making their society, workplace and even their lives "more ethical" in the face of an immoral but seemingly permanent free market. Rather than altering our morality, neoliberalism "individualizes" ethics, making us personally responsible for dealing with and resolving its moral failings. In doing so, individuals end up perpetuating the very market system that they morally oppose and feel powerless to ultimately change. This analysis reveals the complex and paradoxical way capitalism is currently shaping us as "ethical subjects". People are increasingly asked to ethically "save" capitalism both collectively and personally. This can range from the "moral responsibility" to politically accept austerity following the financial crisis to the willingness of employees to sacrifice their time and energy to make their neoliberal organizations more "humane" to the efforts by individuals to contribute to their family and communities despite the pressures of a franetic global business environment. Neoliberalism, thus, uses our ethics against us, relying on our "good nature" and sense of personal responsibility to reduce its human cost in practice. Ironically

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321

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Routledge

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ISBN 10

1317212665

ISBN 13

9781317212669

Tornado God

American Religion and Violent Weather

By: Peter J. Thuesen

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Tornado God

American Religion and Violent Weather

By: Peter J. Thuesen

One of the earliest sources of humanity's religious impulse was severe weather, which ancient peoples attributed to the wrath of storm gods. Enlightenment thinkers derided such beliefs as superstition and predicted they would pass away as humans became more scientifically and theologically sophisticated. But in America, scientific and theological hubris came face-to-face with the tornado, nature's most violent windstorm. Striking the United States more than any other nation, tornadoes have consistently defied scientists' efforts to unlock their secrets. Meteorologists now acknowledge that even the most powerful computers will likely never be able to predict a tornado's precise path. Similarly, tornadoes have repeatedly brought Americans to the outer limits of theology, drawing them into the vortex of such mysteries as how to reconcile suffering with a loving God and whether there is underlying purpose or randomness in the universe. In this groundbreaking history, Peter Thuesen captures the harrowing drama of tornadoes, as clergy, theologians, meteorologists, and ordinary citizens struggle to make sense of these death-dealing tempests. He argues that, in the tornado, Americans experience something that is at once culturally peculiar (the indigenous storm of the national imagination) and religiously primal (the sense of awe before an unpredictable and mysterious power). He also shows that, in an era of climate change, the weather raises the issue of society's complicity in natural disasters. In the whirlwind, Americans confront the question of their own destiny-how much is self-determined and how much is beyond human understanding or control.

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313

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Oxford University Press

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ISBN 10

0190680296

ISBN 13

9780190680299

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

By: Peter B. Gilkey,John V Leahy,JeongHyeong Park

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

By: Peter B. Gilkey,John V Leahy,JeongHyeong Park

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 pi: 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.

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294

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CRC Press

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ISBN 10

0849382777

ISBN 13

9780849382772

Geometric Realizations of Curvature

By: Miguel Brozos-Vázquez,Peter B. Gilkey,Stana Nikcevic

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Geometric Realizations of Curvature

By: Miguel Brozos-Vázquez,Peter B. Gilkey,Stana Nikcevic

A central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including the latest developments and proofs. Even though many authors have worked in this area in recent years, many fundamental questions still remain unanswered. Many studies begin by first working purely algebraically and then later progressing onto the geometric setting and it has been found that many questions in differential geometry can be phrased as problems involving the geometric realization of curvature. Curvature decompositions are central to all investigations in this area. The authors present numerous results including the Singer Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri Vanhecke decomposition, the Gray Hervella decomposition and the De Smedt decomposition. They then proceed to draw appropriate geometric conclusions from these decompositions. The book organizes, in one coherent volume, the results of research completed by many different investigators over the past 30 years. Complete proofs are given of results that are often only outlined in the original publications. Whereas the original results are usually in the positive definite (Riemannian setting), here the authors extend the results to the pseudo-Riemannian setting and then further, in a complex framework, to para-Hermitian geometry as well. In addition to that, new results are obtained as well, making this an ideal text for anyone wishing to further their knowledge of the science of curvature.

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263

Publisher

World Scientific

Published Date

ISBN 10

1848167415

ISBN 13

9781848167414

Geometry Of Spherical Space Form Groups, The (Second Edition)

By: Peter B Gilkey

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Geometry Of Spherical Space Form Groups, The (Second Edition)

By: Peter B Gilkey

This volume focuses on discussing the interplay between the analysis, as exemplified by the eta invariant and other spectral invariants, the number theory, as exemplified by the relevant Dedekind sums and Rademacher reciprocity, the algebraic topology, as exemplified by the equivariant bordism groups, K-theory groups, and connective K-theory groups, and the geometry of spherical space forms, as exemplified by the Smith homomorphism. These are used to study the existence of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group.This volume is a completely rewritten revision of the first edition. The underlying organization is modified to provide a better organized and more coherent treatment of the material involved. In addition, approximately 100 pages have been added to study the existence of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group. We have chosen to focus on the geometric aspect of the theory rather than more abstract algebraic constructions (like the assembly map) and to restrict our attention to spherical space forms rather than more general and more complicated geometrical examples to avoid losing contact with the fundamental geometry which is involved.

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508

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World Scientific

Published Date

ISBN 10

9813220805

ISBN 13

9789813220805

Geometric Properties Of Natural Operators Defined By The Riemann Curvature Tensor

By: Peter B Gilkey

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Geometric Properties Of Natural Operators Defined By The Riemann Curvature Tensor

By: Peter B Gilkey

A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition.The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed.

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316

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World Scientific

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ISBN 10

9814490091

ISBN 13

9789814490092

Kingdom through Covenant (Second Edition)

A Biblical-Theological Understanding of the Covenants

By: Peter J. Gentry,Stephen J. Wellum

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Kingdom through Covenant (Second Edition)

A Biblical-Theological Understanding of the Covenants

By: Peter J. Gentry,Stephen J. Wellum

Kingdom through Covenant is a careful exposition of how the biblical covenants unfold and relate to one another—a widely debated topic, critical for understanding the narrative plot structure of the whole Bible. By incorporating the latest available research from the ancient Near East and examining implications of their work for Christology, ecclesiology, eschatology, and hermeneutics, scholars Peter J. Gentry and Stephen J. Wellum present a thoughtful and viable alternative to both covenant theology and dispensationalism. This second edition features updated and revised content, clarifying key material and integrating the latest findings into the discussion.

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765

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Crossway

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ISBN 10

1433553104

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9781433553103

The Scouting Notebook 2002

By: Peter Gammons,Tracy Ringolsby

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The Scouting Notebook 2002

By: Peter Gammons,Tracy Ringolsby

Once again, STATS has unleashed a team of national experts to give readers the best analysis of every player on every major league baseball team. Includes analysis of every team, top to bottom, plus top minor league prospects.

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724

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STATS Publishing

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188406499X

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9781884064999

Exception to the Rule: The Surprising Science of Character-Based Culture, Engagement, and Performance

By: Peter J. Rea,James K. Stoller,Alan Kolp

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Exception to the Rule: The Surprising Science of Character-Based Culture, Engagement, and Performance

By: Peter J. Rea,James K. Stoller,Alan Kolp

The antidote to navigating turbulent times isn’t more rules. It is timeless virtue that creates sustainable value. Thoughtful leaders are keenly aware of the enormous challenge they face to drive high performance in a world that continues to ratchet up pressure and uncertainty. Some leaders respond by getting tough and establishing strict rules. They get people in line, but they don’t inspire excellence. Wise leaders, on the other hand, help their people practice character to navigate their way through the turbulence—without lowering performance expectations. As a result, their people are more reliable under pressure. Exception to the Rule links ancient wisdom with contemporary science on high performance, teamwork, and engagement. Building an organizational culture based on classical virtues―of trust, compassion, courage, justice, wisdom, temperance and hope―is both strategically smart and a better way to live. Exception to the Rule walks you through the steps of helping everyone in your organization focus on character defined by virtue. The word virtue means excellence, which is why each one is essential to help people perform at a high level despite uncertainty and pressure. Under character-based leadership, teams work better together, creativity flourishes and engagement increases. The most powerful idea of Exception to the Rule is this: character defined by virtue is not based on birthright; it can be learned and practiced. Everyone can develop habits to become better than they were. While character cannot be legislated, character can be cultivated. As virtue proves its value, the culture you have can evolve into the culture you need.

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305

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McGraw Hill Professional

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ISBN 10

1260026841

ISBN 13

9781260026849

Managerial Leadership

By: Peter A. Topping

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Managerial Leadership

By: Peter A. Topping

Annotation Effective management in today's competitive business environment requires solid leadership skillsshy;shy;skills for which companies often don't provide adequate training. Managerial Leadership fills that void, providing a four-tiered approach that helps professionals from all walks of life develop both strong management and leadership skills. Top leadership models and frameworks, tools for assessing leadership strengths, techniques for handling change and growth, and more make Managerial Leadership a veritable learning laboratory.

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244

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McGraw Hill Professional

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ISBN 10

0071450947

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9780071450942

Aspects of Differential Geometry V

By: Esteban Calviño-Louzao,Eduardo García-Río,Peter B Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

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Aspects of Differential Geometry V

By: Esteban Calviño-Louzao,Eduardo García-Río,Peter B Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo

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158

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Morgan & Claypool Publishers

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ISBN 10

1636391117

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9781636391113

Aspects of Differential Geometry II

By: Peter Gilkey,JeongHyeong Park

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Aspects of Differential Geometry II

By: Peter Gilkey,JeongHyeong Park

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.

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159

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Morgan & Claypool Publishers

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ISBN 10

1627057846

ISBN 13

9781627057844

Book's by Peter B Gilkey

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