The internal geometry of Nejfel’d’s connection. (English. Russian original) Zbl 0608.53016

Sov. Math. 30, No. 2, 97-100 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 2(285), 67-69 (1986).
Let \(M\subset G_{n,k}\) be an open subset of planes in \(E_ n\) and \(f: M\to \tilde M\), \(\tilde M\subset G_{n,n-k}^ a\) \(C^{\omega}\)-mapping such that \(x\oplus f(x)=E_ n\), \(\forall x\in M\). Such a mapping f is called a ”normalization” following A. P. Norden [Sov. Math. 25, No.11, 100-103 (1981); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1981, No.11, 80-83 (1981; Zbl 0481.53012)]; to f one associates the so- called Nejfel’d’s connection whose torsion is null [E. G. Nejfel’d, Izv. Vyssh. Uchebn. Zaved., Mat. 1976, No.11(174), 48-55 (1976; Zbl 0345.53003)].
The choice of a normalization f is equivalent to prescribe a certain covector field \(p_ A\) on \(G_{n,k}\) (Theorem 1). The author establishes, in terms of the covector field \(p_ A\), a necessary and sufficient condition for the Nejfel’d’s connection to be Euclidean or locally symmetric (Theorem 2). By using the results of K. M. Egiazaryan [Sov. Math. 24, No.5, 90-92 (1980); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1980, No.5(216), 76-78 (1980; Zbl 0441.53013), and Tr. Geom. Semin. 12, 27-37 (1980; Zbl 0493.53013)], sufficient conditions for a Nejfel’d’s connection to be the connection of a homogeneous space are studied. Finally a criterion for a connection to be a Nejfel’d’s connection is given.
Reviewer: V.Boju


53B15 Other connections
53C30 Differential geometry of homogeneous manifolds