Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
66
result(s) for
"Gilkey, Peter B"
Sort by:
Geometric properties of natural operators defined by the riemann curvature tensor
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.
eBook
The Geometry of Curvature Homogenous Pseudo-riemannian Manifolds
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 Stanilov-Tsankov-Videv theory.Sample Chapter(s)Chapter 1: The Geometry of the Riemann Curvature Tensor (1,291 KB)Contents:The Geometry of the Riemann Curvature TensorCurvature Homogeneous Generalized Plane Wave ManifoldsOther Pseudo-Riemannian ManifoldsThe Curvature TensorComplex Osserman Algebraic Curvature TensorsStanilov-Tsankov TheoryReadership: Researchers in differential geometry and mathematical physics.
eBook
The geometry of curvature homogeneous pseudo-riemannian manifolds
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.
eBook
Geometric realizations of curvature
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.
eBook
The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order
We study the eta invariants of compact flat spin manifolds of dimension
n
with holonomy group
, where
p
is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when
p
=
n
= 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.
Journal Article
The Eta Invariant and the Connective K-Theory of the Classifying Space for Cyclic 2 Groups
We use the eta invariant to study the connective K-theory groups kom(B\\Bbb{Z}[cursive l]) of the classifying space for the cyclic group \\Bbb{Z}[cursive l] where [cursive l] = 2[nu] > or = 2. [PUBLICATION ABSTRACT]
Journal Article
Geometry of differential operators on Weyl manifolds
We define natural operators of Laplace type for a Weyl manifold which transform conformally. We use the asymptotics of the heat equation for these operators to construct global invariants in Weyl geometry.
Journal Article
Eigenvalues of the form valued Laplacian for Riemannian submersions
Let π:Z→Y\\pi :Z\\rightarrow Y be a Riemannian submersion of closed manifolds. Let Φp\\Phi _{p} be an eigen pp-form of the Laplacian on YY with eigenvalue λ\\lambda which pulls back to an eigen pp-form of the Laplacian on ZZ with eigenvalue μ\\mu. We are interested in when the eigenvalue can change. We show that λ≤μ\\lambda \\le \\mu, so the eigenvalue can only increase; and we give some examples where λ>μ\\lambda >\\mu, so the eigenvalue changes. If the horizontal distribution is integrable and if YY is simply connected, then λ=μ\\lambda =\\mu, so the eigenvalue does not change.
Journal Article
Volume density asymptotics of central harmonic spaces
We show the asymptotics of the volume density function in the class of central harmonic manifolds can be specified arbitrarily and do not determine the geometry.
Paper