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
Chapter I. buildings on Riemannian manifolds
91. Riemannian manifolds
92. Kaehlerian manifolds
93. Sasakiilll milllifolds
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
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
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Extra info for CR Submanifolds of Kaehlerian and Sasakian Manifolds
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.