Author: Armand BorelPublisher:
ISBN:
Category :
Languages : en
Pages : 14
Book Description
On the Curvature Tensor of the Hermitian Symmetric Manifolds
Author: Armand BorelMetric Rigidity Theorems on Hermitian Locally Symmetric Manifolds
Author: Ngaiming MokPublisher: World Scientific
ISBN: 9789971508005
Category : Mathematics
Languages : en
Pages : 296
Book Description
This monograph studies the problem of characterizing canonical metrics on Hermitian locally symmetric manifolds X of non-compact/compact types in terms of curvature conditions. The proofs of these metric rigidity theorems are applied to the study of holomorphic mappings between manifolds X of the same type. Moreover, a dual version of the generalized Frankel Conjecture on characterizing compact Khler manifolds are also formulated.
Real Hypersurfaces in Hermitian Symmetric Spaces
Author: Jürgen BerndtPublisher: Walter de Gruyter GmbH & Co KG
ISBN: 311068991X
Category : Mathematics
Languages : en
Pages : 249
Book Description
Hermitian symmetric spaces are an important class of manifolds that can be studied with methods from Kähler geometry and Lie theory. This work gives an introduction to Hermitian symmetric spaces and their submanifolds, and presents classifi cation results for real hypersurfaces in these spaces, focusing on results obtained by Jürgen Berndt and Young Jin Suh in the last 20 years.
The Geometry of Walker Manifolds
Author: Miguel Brozos-VázquezPublisher: Morgan & Claypool Publishers
ISBN: 1598298194
Category : Mathematics
Languages : en
Pages : 178
Book Description
Basic 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.
Almost Complex And Complex Structures
Author: Chuan-chih HsiungPublisher: World Scientific
ISBN: 9814518107
Category : Mathematics
Languages : en
Pages : 328
Book Description
This book gives a self-contained fundamental study of the subject. Besides the following special features it contains the author's detailed solution to the long-standing unsolved problem in the theory of complex manifolds: Does there exist a complex structure on the six-sphere? The special features of the book are: a classification of almost complex (and similarly, almost Hermitian) structures together with inclusion relations; discussions about various known almost Hermitian structures; a necessary and sufficient condition for a general almost Hermitian manifold to have constant holomorphic sectional (or bisectional) curvature and similar conditions for various special almost Hermitian manifolds; some complex Laplacians together with some of their relationships with the real Laplacian; the spectral geometry of Riemannian manifolds and some general almost Hermitian manifolds including Kählerian manifolds as a special case; conditions for an almost complex structure to be a complex structure; some vanishing theorems for Riemannian and almost Hermitian manifolds.
Geometric Realizations Of Curvature
Author: Miguel Brozos-vazquezPublisher: World Scientific
ISBN: 1908977744
Category : Mathematics
Languages : en
Pages : 263
Book Description
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.
Riemannian Manifolds of Conullity Two
Author: Eric BoeckxPublisher: World Scientific
ISBN: 981022768X
Category : Mathematics
Languages : en
Pages : 319
Book Description
This book deals with Riemannian manifolds for which the nullity space of the curvature tensor has codimension two. These manifolds are ?semi-symmetric spaces foliated by Euclidean leaves of codimension two? in the sense of Z I Szab¢. The authors concentrate on the rich geometrical structure and explicit descriptions of these remarkable spaces. Also parallel theories are developed for manifolds of ?relative conullity two?. This makes a bridge to a survey on curvature homogeneous spaces introduced by I M Singer. As an application of the main topic, interesting hypersurfaces with type number two in Euclidean space are discovered, namely those which are locally rigid or ?almost rigid?. The unifying method is solving explicitly particular systems of nonlinear PDE.
Almost Hermitian Manifolds with J-invariant Riemann Curvature Tensor
Author: L. VanheckeOsserman Manifolds in Semi-Riemannian Geometry
Author: Eduardo Garcia-RioPublisher: Springer
ISBN: 3540456295
Category : Mathematics
Languages : en
Pages : 178
Book Description
The subject of this book is Osserman semi-Riemannian manifolds, and in particular, the Osserman conjecture in semi-Riemannian geometry. The treatment is pitched at the intermediate graduate level and requires some intermediate knowledge of differential geometry. The notation is mostly coordinate-free and the terminology is that of modern differential geometry. Known results toward the complete proof of Riemannian Osserman conjecture are given and the Osserman conjecture in Lorentzian geometry is proved completely. Counterexamples to the Osserman conjuncture in generic semi-Riemannian signature are provided and properties of semi-Riemannian Osserman manifolds are investigated.
Geometry of Manifolds
Author: K. ShiohamaPublisher: Elsevier
ISBN: 0080925782
Category : Mathematics
Languages : en
Pages : 536
Book Description
This volume contains the papers presented at a symposium on differential geometry at Shinshu University in July of 1988. Carefully reviewed by a panel of experts, the papers pertain to the following areas of research: dynamical systems, geometry of submanifolds and tensor geometry, lie sphere geometry, Riemannian geometry, Yang-Mills Connections, and geometry of the Laplace operator.