Does anybody know who that first introduced the notion of Killing vector field?


Thanks.



Who first introduced the notion of Killing vector field?
Well, this is a point of contention depending on what precisely you call a Killing vector field. 


You can argue that implicitly in the work of Sophus Lie on his namesake groups and algebras the idea of infinitesimal symmetries, and hence the Killing vector fields associated to the bi-invariant metric, are already present. 


But if you take the definition of Killing vector field to be "A vector field $V$ on some (pseudo)Riemannian manifold $(M,g)$ such that the symmetric part of $\nabla V$ vanishes", then the condition 


$$ \nabla\_a V\_b + \nabla\_b V\_a = 0 $$


(which is called Killing's equation, by the way) was indeed introduced by Wilhelm Killing. 


L.P. Eisenhart in his *Riemannian Geometry* gives citation to page 167 of the following article



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> W. Killing, *Über die Grundlagen der Geometry*, Crelle's Journal, v109 (1892) pps 121--186
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