Speaker
Mr
Felipe Contatto
(Department of Applied Maths and Theoretical Physics - Cambridge)
Description
Given a pseudo-Riemannian manifold, there is a natural notion of geodesics defined by the Levi-Civita connection. But the geodesic equations can be written just in terms of the Christoffel symbols $\Gamma$:
\begin{equation}
\ddot x^a+\Gamma^a_{bc} \dot x^b\dot x^c=0,
\end{equation}
where $x^a$ are local coordinates. Therefore, geodesics can be defined for any symmetric affine connection (not necessarily metric). The same is true for Killing forms $K_a$, which are defined to be solutions to
$$ \nabla_{(a}K_{b)}:=\frac{1}{2}(\nabla_{a}K_{b}-\nabla_{b}K_{a})=0,$$
and correspond to first integrals of the geodesic equations linear in the momenta: $K_a \dot x^a$.
I will present the method of prolongation and Frobenius theorem to determine necessary and sufficient conditions for an affine connection on a two-dimensional manifold (not necessarily endowed with a metric) to admit 0, 1, 2 or 3 Killing forms.
Reference: F. Contatto, M. Dunajski. (2015) First integrals of affine connections and Hamiltonian systems of hydrodynamic type. [ arXiv:1510.01906 ]
Primary author
Mr
Felipe Contatto
(Department of Applied Maths and Theoretical Physics - Cambridge)
Co-author
Dr
Maciej Dunajski
(DAMTP, University of Cambridge)
