# Who first introduced the notion of Killing vector field?

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

Thanks.

• You mean it wasn't Killing? – Nick Gill Oct 5 '16 at 14:08
• I am guessing it was the physicists... – Igor Rivin Oct 5 '16 at 14:19
• Yes. The Killing field is named after Wilhelm Killing. – C.F.G Oct 5 '16 at 14:21
• This would have been a better question for hsm.stackexchange.com. – Ben Crowell Oct 5 '16 at 14:35

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

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

• If you mean the name: Killing certainly didn't call it "Killing vector field". Eisenhart didn't either, but referred to them as solutions to "the equation of Killing". Bochner's 1946 paper (where he introduced his namesake formula) makes no reference to Killing. He rederives the Killing's equation and references Eisenhart for the discussion. However, by 1951 Yano uses the phrase "Killing vector field" as if everyone is already familiar with what they are. – Willie Wong Oct 5 '16 at 14:28