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1986 | Buch

Geometry of CR-Submanifolds

verfasst von: Aurel Bejancu

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications (East European Series)

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Über dieses Buch

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non­ trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can us;; Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Inhaltsverzeichnis

Frontmatter
Chapter I. Differential-Geometrical Structures on Manifolds
Abstract
Let N be a real n-dimensional connected differentiable manifold. Throughout the book all manifolds and morphisms are supposed to be differentiable of class C221E;. Denote by {U; xh} a system of coordinate neighborhoods on N, where U is a neighborhood and xh are local coordinates in U, with the indices h, i, j, k, ... taking on values in the range {i, ... , n}. TN and F(N) are respectively the tangent bundle to N and the algebra of differentiable functions on N. Also we denote by Κ (H) the module of differentiable sections of a vector bundle H.
Aurel Bejancu
Chapter II. CR-Submanifolds of Almost Hermitian Manifolds
Abstract
Let N be a n-dimensional almost Hermitian manifold with almost complex structure J and with Hermitian metric g. Let H be a real m-dimensional Riemannian manifold isometrically immersed in N.
Aurel Bejancu
Chapter III. CR-Submanifolds of Kaehlerian Manifolds
Aurel Bejancu
Chapter IV. CR-Submanifolds of Complex Space Forms
Aurel Bejancu
Chapter V. Extensions of CR-Structures to Other Geometrical Structures
Abstract
Let N be a real (2n + l)-dimensional almost contact metric manifold with structure tensors (φ, ξ, n, g), where φ is a tensor field of type (1, 1), ξ is a vector field, n, is a 1-form and g is a Riemannian metric on N. These tensor fields are related by (see §of Chapter I)
$${{\phi }^{2}}X = - X + \eta (X)\xi ; \phi \xi = 0; \eta (\xi ) = 1; \eta (\phi X) = 0$$
(1.1)
$$g(\phi X,\phi Y) = g(X,Y) - \eta (X) \cdot \eta (Y); \eta (X) = g(X,\xi ),$$
(1.2)
for any vector fields X, Y tangent to N.
Aurel Bejancu
Chapter VI. CR-Structures and Pseudo-Conformal, Mappings
Abstract
CR-manifolds were intensively studied from the analytic point of view (see \'lells (1], [2]). As is well known, complex manifolds and normal almost contact manifolds (see Blair [3], p. 62) are examples of ~R-manifolds. Non-trivial CRmanifolds appeared as boundaries of domains in complex spaces, which in fact are real hypersurfaces (i.e., particular CR-submanifolds).
Aurel Bejancu
Chapter VII. CR-Structures and Relativity
Abstract
The main purpose of this chapter is to show that CR-structures on real hypersurfaces of a complex manifold have an interesting application to relativity. It is the merit of Roger Penrose to discover a correspondence between points of a Minkowski space and projective lines of a certain real hypersurface in a complex projective space (see (2.4)).
Aurel Bejancu
Backmatter
Metadaten
Titel
Geometry of CR-Submanifolds
verfasst von
Aurel Bejancu
Copyright-Jahr
1986
Verlag
Springer Netherlands
Electronic ISBN
978-94-009-4604-0
Print ISBN
978-94-010-8545-8
DOI
https://doi.org/10.1007/978-94-009-4604-0