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


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