Theorems concerning certain special tensor fields on Riemannian manifolds

Czeslaw Konopka

Abstract: 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.

Classification (MSC2000): 53B20, 53C20

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