Does anybody know who that first introduced the notion of Killing vector field?
Thanks.
$\begingroup$
$\endgroup$
4
-
6$\begingroup$ You mean it wasn't Killing? $\endgroup$– Nick GillCommented Oct 5, 2016 at 14:08
-
$\begingroup$ I am guessing it was the physicists... $\endgroup$– Igor RivinCommented Oct 5, 2016 at 14:19
-
$\begingroup$ Yes. The Killing field is named after Wilhelm Killing. $\endgroup$– C.F.GCommented Oct 5, 2016 at 14:21
-
1$\begingroup$ This would have been a better question for hsm.stackexchange.com. $\endgroup$– user21349Commented Oct 5, 2016 at 14:35
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
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
answered Oct 5, 2016 at 14:21
-
5$\begingroup$ 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. $\endgroup$ Commented Oct 5, 2016 at 14:28