## 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:

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