Geometry of CR-submanifolds

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.

I: Differential-Geometrical Structures on Manifolds.- 1. Linear connections on a manifold.- 2. The Levi-Civita connection.- 3. Submanifolds of a Riemannian manifold.- 4. Distributions on a manifold.- 5. Kaehlerian manifolds.- 6. Sasakian manifolds.- 7. Quaternion Kaehlerian manifolds.- II: CR-Submanifolds of Almost Hermitian Manifolds.- 1. CR-submanifolds and CR-structures.- 2. Integrability of distributions on a CR-submanifold.- 3. ?-connections on a CR-submanifold and CR-products of almost Hermitian manifolds.- 4. The non-existence of CR-products in S6.- III: CR-Submanifolds of Kaehlerian Manifolds.- 1. Integrability of distributions and geometry of leaves.- 2. Umbilical CR-submanifolds of Kaehlerian manifolds.- 3. Normal CR-submanifolds of Kaehlerian manifolds.- 4. Normal anti-holomorphic submanif olds of Kaehlerian manifolds.- 5. CR-products in Kaehlerian manifolds.- 6. Sasakian anti-holomorphic submanifolds of Kaehlerian manifolds.- 7. Cohomology of CR-submanifolds.- IV: CR-Submanifolds of Complex Space Forms.- 1. Characterization of CR-submanifolds in complex space forms.- 2. Riemannian fibre bundles and anti-holomorphic submanifolds of CPn.- 3. CR-products of complex space forms.- 4. Mixed foliate CR-submanifolds of complex space forms.- 5. CR-submanifolds with semi-flat normal connection.- 6. Pinching theorems for sectional curvatures of CR-submanifolds.- V: Extensions of CR-Structures to Other Geometrical Structures.- 1. Semi-invariant submanifolds of Sasakian manifolds.- 2. Semi-invariant products of Sasakian manifolds.- 3. Semi-invariant submanifolds with flat normal connection.- 4. Generic submanifolds of Kaehlerian manifolds.- 5. QR-submanifolds of quaternion Kaehlerian manifolds.- 6. Totally umbilical and toally geodesic QR-submanifolds of quaternion Kaehlerian manifolds.- VI: CR-Structures and Pseudo-Conformal Mappings.- 1. CR-manifolds and f-structures with complemented frames.- 2. Generic submanifolds of complex manifolds.- 3. Anti-holomorphic submanifolds of complex manifolds.- 4. Pseudo-conformal mappings.- VII: CR-Structures and Relativity.- 1. Geometrical Structures of space-time.- 2. The twistor space and Penrose correspondence.- 3. Physical interpretations of CR-structures.- References.- Author Index.