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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.

##### MSC:
 53B20 Local Riemannian geometry 53B30 Local differential geometry of Lorentz metrics, indefinite metrics
##### References:
  Kowolik J.: On some Riemannian manifolds admitting torse-forming vector fields. Dem. Math. 18, 3 (1985), 885-891. · Zbl 0596.53016  Mikeš J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. New York 78, 3 (1996), 311-333. · Zbl 0866.53028  Mikeš J.: Geodesic Ricci mappings of two-symmetric Riemann spaces. Math. Notes 28 (1981), 622-624. · Zbl 0454.53013  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  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  Rachůnek L., Mikeš J.: On semitorse-forming vector fields. 3rd International Conference APLIMAT 2004, Bratislava, 835-840.  Roter W.: On a class of conformally recurrent manifolds. Tensor N. S. 39 (1982), 207-217. · Zbl 0518.53018  Yano K.: On torse-forming directions in Riemannian spaces. Proc. Imp. Acad. Tokyo 20 (1944), 701-705. · Zbl 0060.39102
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