Theorems Concerning Certain Special Tensor Fields on Riemannian Manifolds

Czeslaw Konopka

Publications de l'Institut Mathématique, _N_S_53 (1993) no. 67, p. 115

Résumé

Let $M$ be an $n$-dimensional Riemannian manifold and $F$ a symmetric $(0,2)$-tensor field on $M$, which satisfies the condition $R\cdot F=0$. Let, additionally, $H$, $A$ and $B$ be symmetric $(0,2)$-tensor fields on $M$. If the tensor $B$ commutes with $F$ (cf. (1.3)) and $H$ satisfies the condition $R\cdot H=Q(A,B)$, then $ (A_{jk} - \frac{\tr(A)}{\tr(B)}B_{jk}) (B_{ir} F^r_{ m} - \frac{\tr(B,F)}{\tr(B)}B_{im}) = 0 $ on the open subset of $M$ on which $\tr(B)\ne 0$. It is also proved that, in certain separately Einstein manifolds, null geodesic collineation and projective collineations reduce to motions.

Détail

Classification : 53B20 53C20