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$$T$$-semisymmetric spaces and concircular vector fields. (English) Zbl 1023.53014
Slovák, Jan (ed.) et al., The proceedings of the 21th winter school “Geometry and physics”, Srní, Czech Republic, January 13-20, 2001. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 69, 187-193 (2002).
In this paper a concircular vector field $$\xi$$ on a (pseudo-)Riemannian manifold is defined by the equation $$\nabla_X\xi= \rho X$$ for any $$X$$ where $$\rho$$ is a scalar function. In the literature, such a vector field is also called a closed conformal vector field since it is locally a gradient field and since the equation $${\mathcal L}_\xi g= 3\rho g$$ of a conformal Killing field is satisfied. A convergent concircular field in the sense of this paper (i.e., the case of constant $$\rho$$) is often called a closed homothetic field. The paper gives local restrictions on the existence of such vector fields on $$T$$-semisymmetric spaces. It should be mentioned that concircular vector fields in a different meaning were studied by Y. Tashiro [Trans. Am. Math. Soc. 117, 251-275 (1965; Zbl 0136.17701)] including some global results.
For the entire collection see [Zbl 0994.00029].

##### MSC:
 53B20 Local Riemannian geometry 53B30 Local differential geometry of Lorentz metrics, indefinite metrics