zbMATH — the first resource for mathematics

Almost CR structures, \(f\)-structures, almost product structures and associated connections. (English) Zbl 0806.53030
Let \(CTM\) be the complexification of the tangent bundle \(TM\) of the manifold \(M\). An almost CR-structure is a complex subbundle \(H \subset CTM\) such that \(H \cap \overline{H} = 0\). In [Geometry of CR- submanifolds, Dordrecht, D. Reidel Publ. Comp. (1986; Zbl 0605.53001)] A. Bejancu gave some relations between \(f\)-structures and almost CR-structures. Annihilating forms and frames defined in the present paper are used to define two higher-codimensional analogues of pseudo-hermitian structures and several examples are given. The notion of partial connection is introduced here to study affine connections associated to almost product structures. The main result states that a nondegenerate annihilating frame for a partially integrable almost CR-structure determines an affine connection. A simple nondegenerate annihilating form determines a pair of partial connections.
Reviewer: C.Tiba (Iaşi)

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
32V40 Real submanifolds in complex manifolds
Zbl 0605.53001
Full Text: DOI
[1] A. Bejancu, Geometry of CR-submanifolds , D. Reidel, Dordrecht, 1986. · Zbl 0605.53001
[2] S. Kobayashi and N. Nomizu, Foundations of differential geometry , II, Interscience, New York, 1969. · Zbl 0175.48504
[3] R. Mizner, CR structures of codimension 2, J. Differ. Geometry 30 (1989), 167-190. · Zbl 0675.32017
[4] R. Stong, The rank of an \(f\)-structure , Kōdai Math. Sem. Rep. 29 (1977), 207-209. · Zbl 0409.53028
[5] N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds , Kinokuniya Book-Store, Tokyo, 1975. · Zbl 0331.53025
[6] S. Tanno, Variational problems on contact Riemannian manifolds , preprint. JSTOR: · Zbl 0677.53043
[7] A. Walker, Almost-product structures , Proc. Symp. Pure Math. III (1961), 94-100. · Zbl 0103.38801
[8] S. Webster, Pseudo-hermitian structures on a real hypersurface , J. Differ. Geometry 13 (1978), 25-41. · Zbl 0379.53016
[9] T. Willmore, Connexions for systems of parallel distributions , Quart. J. Math. 7 (1956), 269-276. · Zbl 0074.38001
[10] ——–, Parallel distributions on manifolds , Proc. London Math. Soc. 6 (1956), 191-204. · Zbl 0070.38804
[11] ——–, Systems of parallel distributions , J. London Math. Soc. 32 (1957), 153-156. · Zbl 0091.35201
[12] K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds , Birkhäuser, Boston, 1983. · Zbl 0496.53037
[13] ——–, Structures on manifolds , World Scientific, Singapore, 1984. · Zbl 0557.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.