First integrals of affine connections and Hamiltonian systems of hydrodynamic type

Abstract

We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence of a Hamiltonian formulation of Dubrovin-Novikov type for a given one-dimensional system of hydrodynamic type. We give several examples including Zoll connections, and Hamiltonian systems arising from two-dimensional Frobenius manifolds.

Metrisability of Painlevé equations

“…This implies that the corresponding projective class [∇] contains a representative ∇ which has symmetric Ricci tensor, and admits a linear first integral. In [5] it was shown that for such affine connections ν 5 = 0, where ν 5 is a point invariant for (1.1) defined by Liouville [21]. This is in agreement with [13], where it was stated that ν 5 vanishes for all Painlevé equations.…”

Vortices, Painlevé integrability and projective geometry

“…Chapters 2, 4 and 5 are based on (but are not the same as) my papers in collaboration with my supervisor [12,13,11], respectively, except for Section 4.4. Chapter 3 is essentially the same as my individual paper [9].…”

“…In section 4.3, we show that all Painlevé equations admit a special representative admitting a Killing form (following Theorem 4.4). The problem of existence of Killing forms for two-dimensional affine connections was solved in [11] (c.f. also Chapter 5), where it was also shown that the semi-invariant ν 5 defined in [31] necessarily vanishes for projective structures having special representatives admitting Killing forms.…”

“…The vanishing of both L 1 and L 2 is invariant under point transformation, but not the vanishing of each one individually. This is because L 1 and L 2 can be interpreted as the components of a 1-form recovered in [11] (c.f. also Chapter 5).…”