Open AccessCurvature of Cr Manifolds
Author(s) -
Aurel Bejancu,
Hani Reda Farran
Publication year - 2013
Publication title -
analele ştiinţifice ale universităţii "al.i. cuza" din iaşi. matematică
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.144
H-Index - 12
eISSN - 2344-4967
pISSN - 1221-8421
DOI - 10.2478/v10157-012-0021-z
Subject(s) - connection (principal bundle) , riemann curvature tensor , scalar curvature , ricci curvature , uniqueness , curvature , mathematics , pure mathematics , manifold (fluid mechanics) , einstein tensor , einstein , mathematical physics , mathematical analysis , metric tensor , differential geometry , torsion (gastropod) , tensor field , exact solutions in general relativity , geometry , geodesic , mechanical engineering , medicine , surgery , engineering
We prove the existence and uniqueness of a torsion-free and h -metric linear connection ▽ ( CR connection) on the horizontal distribution of a CR manifold M . Then we define the CR sectional curvature of M and obtain a characterization of the CR space forms. Also, by using the CR Ricci tensor and the CR scalar curvature we define the CR Einstein gravitational tensor field on M . Thus, we can write down Einstein equations on the horizontal distribution of the 5-dimensional CR manifold involved in the Penrose correspondence. Finally, some CR differential operators are defined on M and two examples are given to illustrate the theory developed in the paper. Most of the results are obtained for CR manifolds that do not satisfy the integrability conditions