scholarly journals WHAT IS...a CR Submanifold?

2017 ◽  
Vol 64 (07) ◽  
pp. 722-724
Author(s):  
Phillip S. Harrington ◽  
Andrew Raich
Keyword(s):  
Cr Submanifold

scholarly journals TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A NEARLY KAEHLER MANIFOLD

1996 ◽  
Vol 27 (2) ◽  
pp. 145-149
Author(s):  
S. H. KON ◽  
SIN-LENG TAN
Keyword(s):  
Kaehler Manifold ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Cr Submanifold ◽  
Nearly Kaehler Manifold ◽  
Totally Real ◽  
Cr Submanifolds

The geometry of a CR-submanifold in a Kaehler manifold has been extensively studied. B.Y . Chen has classified the totally umbilical CR-submanifolds of a Kaehler manifold and showed that they are either totally geodesic, or totally real or dim$(D^{\perp}) =1$. In this paper we show that such a result is also true in a nearly Kaehler manifold.

scholarly journals CR-SUBMANIFOLDS OF A QUASl-KAEHLER MANIFOLD

1995 ◽  
Vol 26 (3) ◽  
pp. 261-266
Author(s):  
S. H. KON ◽  
SIN-LENG TAN
Keyword(s):  
Sufficient Conditions ◽  
Kaehler Manifold ◽  
Complex Submanifold ◽  
Cr Submanifold ◽  
Almost Complex ◽  
Cr Submanifolds

Let $M$ be a CR-submanifold of a quasi-Kaehler manifold $N$. Sufficient conditions for the holomorphic distribution $D$ in $M$ to be integrable are derived. We also show that $D$ is minimal. It follows that an (almost) complex submanifold of a quasi-Kaehler manifold is minimal, this generalizes the well known result that a complex submanifold of a Kaehler manifold is minimal.

scholarly journals On some 3-dimensional CR submanifolds in S6

1999 ◽  
Vol 156 ◽  
pp. 171-185 ◽  
Author(s):  
Hideya Hashimoto ◽  
Katsuya Mashimo
Keyword(s):  
Irreducible Representation ◽  
Parameter Family ◽  
Dimensional Sphere ◽  
3 Dimensional ◽  
Cr Submanifold ◽  
Lie Subgroup ◽  
Cr Submanifolds

We give two types of 3-dimensional CR-submanifolds of the 6-dimensional sphere. First we study whether there exists a 3-dimensinal CR-submanifold which is obtained as an orbit of a 3-dimensional simple Lie subgroup of G2. There exists a unique (up to G2) 3-dimensional CR-submanifold which is obtained as an orbit of reducible representations of SU(2) on R7. As orbits of the subgroup which corresponds to the irreducible representation of SU(2) on R7, we obtained 2-parameter family of 3-dimensional CR-submanifolds. Next we give a generalization of the example which was obtained by K. Sekigawa.

scholarly journals Warped product skew CR-submanifolds of Kenmotsu manifolds and their applications

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3505-3528 ◽  
Author(s):  
Monia Naghi ◽  
Ion Mihai ◽  
Siraj Uddin ◽  
Falleh Al-Solamy
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kenmotsu Manifold ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Cr Submanifold ◽  
Kenmotsu Manifolds ◽  
Cr Submanifolds

In this paper, we introduce the notion of warped product skew CR-submanifolds in Kenmotsu manifolds. We obtain several results on such submanifolds. A characterization for skew CR-submanifolds is obtained. Furthermore, we establish an inequality for the squared norm of the second fundamental form of a warped product skew CR-submanifold M1 x fM? of order 1 in a Kenmotsu manifold ?M in terms of the warping function such that M1 = MT x M?, where MT, M? and M? are invariant, anti-invariant and proper slant submanifolds of ?M, respectively. Finally, some applications of our results are given.

scholarly journals TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A KAEHLER MANIFOLD

1993 ◽  
Vol 24 (1) ◽  
pp. 43-49
Author(s):  
S. M. KHURSHEED HAIDER ◽  
V. A. KHAN ◽  
S . I. HUSAIN
Keyword(s):  
Classification Theorem ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
Cr Submanifold ◽  
Cr Submanifolds

In the present paper we study totally umbilical CR-submanifolds of a Kaehler manifold. A classification theorem for a $D^\perp$-totally umbilical CR-submanifold is proved. The conditions under which a CR- submanifold becomes a CR-product are obtained, and finally a theorem for a CR-submanifold to be a proper CR-product is also established.

scholarly journals TOTALLY UMBILICAL SEMI-INVARIANT SUBMANIFOLDS AND CR-SUBMANIFOLDS OF A SASAKIAN MANIFOLD

1993 ◽  
Vol 24 (2) ◽  
pp. 161-172
Author(s):  
S. M. KHURSEED HAIDER ◽  
V. A. KHAN ◽  
S. I. HUSAIN
Keyword(s):  
Space Form ◽  
Sasakian Manifold ◽  
Classification Theorem ◽  
Sasakian Space Form ◽  
Invariant Submanifold ◽  
Totally Umbilical ◽  
Cr Submanifold ◽  
Invariant Submanifolds ◽  
Cr Submanifolds

In the present paper, a classification theorem for totally um- bilical semi-invariant submanifold is established. CR-submanifolds of a Sasakian space form are studied in detail, and finally a theorem for a CR- submanifold of a Sasakian manifold to be a proper contact CR-product is proved.

scholarly journals Contact co-isotropic CR submanifolds of a pseudo-Sasakian manifold

1984 ◽  
Vol 7 (2) ◽  
pp. 339-350 ◽  
Author(s):  
Vladislav V. Goldberg ◽  
Radu Rosca
Keyword(s):  
Vertical Distribution ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Totally Geodesic ◽  
Isotropic Submanifold ◽  
Cr Submanifold ◽  
Necessary And Sufficient ◽  
Cr Submanifolds

It is proved that any co-isotropic submanifoldMof a pseudo-Sasakian manifoldM˜(U,ξ,η˜,g˜)is a CR submanifold (such submanfolds are called CICR submanifolds) with involutive vertical distributionν1. The leavesM1ofD1are isotropic andMisν1-totally geodesic. IfMis foliate, thenMis almost minimal. IfMis RicciD1-exterior recurrent, thenMreceives two contact Lagrangian foliations. The necessary and sufficient conditions forMto be totally minimal is thatMbe contactD1-exterior recurrent.

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