zbMATH — the first resource for mathematics

On tensor fields semiconjugated with torse-forming vector fields. (English) Zbl 1092.53016
Let \((\mathbb{V}_{n}, g)\), \(\nabla\) and \(R\) be the \((n)\)-dimensional Riemannian space with metric \( g\), an affine connection and the Riemannian tensor of \(V_{n}\), respectively. A vector field \(\xi\) is called torse-forming, if \(\nabla_{X}\xi =\rho X+a(X)\xi\) for all \(X\in TV_{n}\) (space of all tangent vector fields). \(\xi\) is called recurrent and semitorse-forming if \(\rho =0\) and if \(R(X,\xi)\xi=0\) for each \(X\in TV_{n}\), respectively. If \(a\) is a gradient (i.e., there exists a function \(\varphi (x)\) such that \(a=\partial_{i}\varphi (x)dx^{i}\)) \(\xi\) is called concircular; if in addition \(\rho = \text{const}\exp (\varphi)\), \(\xi\) is called convergent. Let \(T\) be a tensor field of type: \((1,q)\), \(T\) is called semiconjugated with the vector field \(\xi\), if \(R(X,\xi)\cdot T=0\) for each \(X\in TV_{n}\), where \(R(X,\xi)\cdot T (X_{1},X_{2},\dots, X_{q})=\sum_{s=1}^{q}T (X_{1},X_{2},\dots, X_{s-1},R(X,\xi)X_{s},\dots X_{q})\) and \(X_{1},X_{2},\dots, X_{q}\) are vector fields of \(TV_{n}\). In this paper the authors study tensor fields which are semiconjugated with torse-forming vector fields. They give existence results for semitorse-forming vector fields and for convergent vector fields.

53B20 Local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
[1] Kowolik J.: On some Riemannian manifolds admitting torse-forming vector fields. Dem. Math. 18, 3 (1985), 885-891. · Zbl 0596.53016
[2] Mikeš J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. New York 78, 3 (1996), 311-333. · Zbl 0866.53028
[3] Mikeš J.: Geodesic Ricci mappings of two-symmetric Riemann spaces. Math. Notes 28 (1981), 622-624. · Zbl 0454.53013
[4] Mikeš J., Rachůnek L.: \(T\)-semisymmetric spaces and concircular vector fields. Supplemento ai Rendiconti del Circolo Matematico di Palermo, II. Ser. 69 (2002), 187-193. · Zbl 1023.53014
[5] Rachůnek L., Mikeš J.: Torse-forming vector fields in \(T\)-semisymmetric Riemannian spaces. Steps in differential geometry. Proceedings of the colloquium on differential geometry, Debrecen, Hungary, July 25-30, 2000. Univ. Debrecen, Institute of Mathematics and Informatics, 2001, 219-229. · Zbl 0994.53009
[6] Rachůnek L., Mikeš J.: On semitorse-forming vector fields. 3rd International Conference APLIMAT 2004, Bratislava, 835-840.
[7] Roter W.: On a class of conformally recurrent manifolds. Tensor N. S. 39 (1982), 207-217. · Zbl 0518.53018
[8] Yano K.: On torse-forming directions in Riemannian spaces. Proc. Imp. Acad. Tokyo 20 (1944), 701-705. · Zbl 0060.39102
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.