Geometric interpretation of Killing-Yano fields?

[ajb]

To my knowledge there are two natural generalisations of Killing vectors on a (even) Riemannian manifold.

 

1) Killing--Stackel fields, which are understood as symmetric fields.

 

2) Killing--Yano fields, which are antisymmetric fields.

 

To my knowledge there is no clear geometric interpretation of Killing--Yano fields (either as differential forms or multivetor fields).

 

Let me just explain the symmetric case of Killing--Stackel fields. Let [math](M,g)[/math] be a Riemannian manifold. (What I say will also hold on even Riemannian supermanifolds). I will understand a Killing--Stackel field to be a function on [math]T^{*}M[/math]. Similarly we can understand the inverse of the metric as a function [math]g^{-1} \in C^{\infty}(T^{*}M)[/math]. The manifold [math]T^{*}M[/math] comes with a canonical Poisson bracket. It is just the one we are all used to from classical mechanics.

 

Definition

A Killing--Stackel field is a function [math]K \in C^{\infty}(T^{*}M)[/math] such that

 

[math]\{K, g^{-1} \}=0[/math],

 

where the bracket is the canonical Poisson bracket on the cotangent bundle.

 

The above is not the standard definition, but can be shown to be equivalent to the various definitions out there.

 

Then we see that Killing--Stackel fields have a clear geometric origin and we can describe the geometric variations associated with them. Specifically we see that Killing--Stackel fields generate infinitesimal canonical transformations that preserve the inverse of the metric.

 

The "attitude" to take is

 

[math]Diff(M) \subset Can(T^{*}M)[/math].

 

We can then think of the Killing--Stackel fields as acting on the inverse of the metric via a Lie derivative with respect to symmetric tensors (just like in Hamiltonian mechanics)

 

[math]L_{K}g^{-1}= \{ K , g^{-1}\}=0[/math]

 

Then it is clear that the Poisson bracket of two Killing--Stackel fields is itself a Killing--Stackel field. (Consult any text on geometric mechanics.)

 

Now for the Killing-Yano tensors, which I will think of as multivector fields, that is functions on [math]\Pi T^{*}M[/math] (look up my earlier posts if you are unsure about this). Again we have a canonical bracket, known as the Schouten--Nijenhuis bracket. This too we can think of in terms of a Lie derivative acting on multivector fields

 

[math]L_{X}Y = [X,Y][/math].

 

 

Now we have a problem. There is no way to define a Lie derivative acting on symmetric tensors along a multivector field. So, Killing--Yano tensors cannot have a simple interpretation in terms of Lie derivatives (geometric variations). Nor do we a theorem that states closure of the Killing--Yano tensors under the Schouten--Nijenhuis bracket in general.

 

Question Does anyone know a nice way to think of Killing--Yano tensors?

 

Now, this ties in with my other thread on odd and even structures. I think the notion of a Killing--Yano multivector is clear for odd Riemannian geometry. The inverse of the odd metric is a multivector field. But now the notion of a Killing--Stackel field is going to unclear geometrically.

 

One possibility is to think about higher order natural bundles. However, I have yet to make this at all clear in my mind.

 

Thank for reading this.

[ajb]

So, for completeness let me give you the standard definition of a Killing-Yano form. Let us consider a standard (pseudo) Riemannian manifold [math](M,g)[/math].

 

A differential form is said to be a Killing-Yano form if and only if

 

[math] \nabla_{X}\alpha \propto i_{X}\alpha[/math],

 

for all [math]X \in Vect(M)[/math]. Here we are thinking about the standard canonical connection on a Riemannian manifold, the Levi-Civita connection. I have been a bit slack with the proportionality here as I am sure this will depend on conventions.

 

Maybe I should also mention why one is interested in Killing--Stackel and Killing--Yano tensors.

 

The main reason is that they provide constants of motion along geodesic flows. You can formulate geodesics in terms of a Hamiltonian ([math]H = g^{-1}[/math]). If there is enough constants of motion them then we have an integrable system in the sense of Liouville. Initially, this meant searching for Killing vectors, but soon it was realised other objects can play a similar role.

 

Now, as we have seen only Killing--Stackel fields (including Killing vectors) can be thought of as "symmetries of the metric and geodesics". Killing--Yano are only "symmetries of the geodesics".