Kentaro Yano, Masahiro Kon's CR Submanifolds of Kaehlerian and Sasakian Manifolds PDF

By Kentaro Yano, Masahiro Kon

This quantity provides an advent and survey of
the newest effects touching on the examine of CR sub.
manifolds of Kaehlerian and Sasakian manifolds, a
relatively new box of differential geometry. The
volume additionally includes all prerequisite details on
Riemannian, Kaehlerian, Sasakian manifolds, f-struc-
tures, and submaitifolds essential to comprehend those


Chapter I. constructions on Riemannian manifolds
91. Riemannian manifolds
92. Kaehlerian manifolds
93. Sasakiilll milllifolds
94. f-structure

Chapter II. Submanifolds
91. precipitated connection illld moment basic shape
92. Equations of Gauss, Codazzi illld Ricci
93. general connection
94. Laplacian of the second one primary shape
95. Subnillllifolds of house kinds
96. Parallel moment primary shape

Chapter III. touch CR submanifolds
91. Submanifolds of Sasakian manifolds
92. f-structure on submanifolds
93. Integrability of distributions
94. completely touch umbilical submanifolds
95. EXillnplcs of touch CR submilllifolds
96. Flat common connection
97. minimum touch CR submanifolds

Chapter IV. CR submanifolds
91. Submanifolds of Kaehlerian manifolds
92. CR submanifolds of IIermitian manifolds
93. Characterization of CR submanifolds
94. Distributions
95. Parallel f-structure
96. completely umbilical submanifolds
97. Examples of CR submanifolds
98. Semi-flat basic connection
99. common connection of invariant submanifolds
910. Parallcl suggest curvature vector
911. vital fonnulas

912. CR submanifolds of em

Chapter V. Submanifolds and Riemannian fibre bundles
91. Curvature tensors
92. suggest curvature vector
93. Lengths of the second one primary types

Chapter VI. Hypersurfaces
Real hypersurfaces of advanced area kinds
Pseudo-Einstein actual hypersurfaces
Generic minimum submanifolds
Semidefinite moment primary shape
Hypersurfaces of S2n+l

Author index
Subject index

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Sample text

19b) to obtain two other relations of the same kind. 16) where the {e;} denote a basis. rh ;1 + . rJikr hii - rJ;ir ik. 26) The Rf,,. are the components of the famous Riemann-Christoffel curvature tensor, a central concept of differential geometry and of general relativity and the name "tensor" will be justified at Level 1. All formulas that we have just obtained keep their validity without any change of notations in a hyperplane defined in EN. Up to now, we considered the plane only, but what can we say about the sphere?

10) where x(u) = x(u1, u 2 ). lox uu+ ·1 . 2 .!.. u 2)2 u+ x vu 1 x vu where ilk = duk /ds. 11) 0 and we deduce . 12) We also remark that is a first integral of the system. The two following points are in need of some attention: first, we did not take into account (as boundary conditions) the condition on the curve C: it is to go through the points A and B and second, we did not care about the nature of the extremum of s - . If the first problem is a classical one (with AB a solution in a limited number of cases), the second one is one of the main problems of variational calculus.

One should point out as well that Gauss' curvature which was introduced through metric considerations will lead us to a simple topological invariant. 15) that we shall write for a curvilinear triangle ~ and as follows: i 3 K(u,v)dS(u,v) = 3 L::O·; - 7r = 27r- L)7r- a;). 19) Now let us apply this formula to every triangle on a triangulated closed surface and let us take the sum. One can then realize that the three terms contribute to the three terms in Eq. 18) and that it finally leads to the remarkable relation i K(x)dS(x) = 27rx(S) .

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