By Kentaro Yano, Masahiro Kon

This quantity offers an advent and survey of

the most modern effects concerning the research of CR sub.

manifolds of Kaehlerian and Sasakian manifolds, a

relatively new box of differential geometry. The

volume additionally comprises all prerequisite details on

Riemannian, Kaehlerian, Sasakian manifolds, f-struc-

tures, and submaitifolds essential to comprehend those

results.

Introduction

Chapter I. buildings on Riemannian manifolds

91. Riemannian manifolds

92. Kaehlerian manifolds

93. Sasakiilll milllifolds

94. f-structure

Chapter II. Submanifolds

91. brought about connection illld moment basic shape

92. Equations of Gauss, Codazzi illld Ricci

93. basic connection

94. Laplacian of the second one primary shape

95. Subnillllifolds of area types

96. Parallel moment basic shape

Chapter III. touch CR submanifolds

91. Submanifolds of Sasakian manifolds

92. f-structure on submanifolds

93. Integrability of distributions

94. absolutely touch umbilical submanifolds

95. EXillnplcs of touch CR submilllifolds

96. Flat basic 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 common connection

99. basic 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 varieties

Chapter VI. Hypersurfaces

Real hypersurfaces of complicated area kinds

Pseudo-Einstein actual hypersurfaces

Generic minimum submanifolds

Semidefinite moment primary shape

Hypersurfaces of S2n+l

(f,g,u,V,A)-structure

Bibliography

Author index

Subject index

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**Extra info for CR Submanifolds of Kaehlerian and Sasakian Manifolds**

**Sample text**

X = Fl (t, x) , or Flt = µα(t) , for all t. α(s) = Proof. (1) =⇒ (4). We have dt d d | µ α(s) = T (µ ). α(t)). α(t) = Fl (t, x). (4) =⇒ (2). This is clear. (2) =⇒ (1). We have d ds α(t)α(s) = d ds (µα(t) α(s)) d = T (µα(t) ) ds α(s) = T (µα(t) )LX (α(s)) = LX (α(t)α(s)) LX X and α(t)α(0) = α(t). So we get α(t)α(s) = FlLX (s, α(t)) = FlL s Flt (e) = FlLX (t + s, e) = α(t + s). (4) ⇐⇒ (5). 14). α(−t). y. t (x (5) =⇒ (3) =⇒ (1) can be shown in a similar way. An immediate consequence of the foregoing lemma is that left invariant and right invariant vector fields on a Lie group are always complete, so they have global flows, because a locally defined one parameter group can always be extended to a globally defined one by multiplying it up.

Tn ). ◦i−1 2 ◦ i1 )(x1 ) So i−1 2 ◦ i1 is a diffeomorphism, as required. (3) Let N be the union of all integral manifolds containing x. Choose the union of all the atlases of these integral manifolds as atlas for N , which is a smooth atlas for N by 2. Note that a connected immersed submanifold of a separable manifold is automatically separable (since it carries a Riemannian metric). Draft from February 21, 2006 Peter W. 24 3. 23. Integrable singular distributions and singular foliations. A smooth (singular) distribution E on a manifold M is called integrable, if each point of M is contained in some integral manifold of E.

24) E(S(W)) is integrable, we can choose the leaf L through x, with the inclusion i. 10) the local vector field [i∗ X, i∗ Y ] ∈ Xloc (L) is i-related to [X, Y ], and [X, Y ](x) ∈ E(S(W))x , as required. 28. Theorem. Let V ⊂ Xloc (M ) be an involutive subset. Then the distribution E(V) spanned by V is integrable under each of the following conditions. (1) M is real analytic and V consists of real analytic vector fields. (2) The dimension of E(V) is constant along all flow lines of vector fields in V.