hyper-Hermitian Metrics Research Articles

Non-Berwaldian Randers metrics of Douglas type on four-dimensional hypercomplex Lie groups

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the sectional curvature of the hyper-Hermitian metrics have the same sign.

ON SOME HYPERCOMPLEX 4-DIMENSIONAL LIE GROUPS OF CONSTANT SCALAR CURVATURE

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.

Randers metrics of Berwald type on four-dimensional hypercomplex Lie groups

In the present paper we study Randers metrics of Berwald type on simply connected four-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulae for computing the flag curvature of these metrics. By this study we construct two four-dimensional Berwald spaces, one of them with a non-negative flag curvature and the other one with a non-positive flag curvature.

Einstein–Weyl geometry, the dKP equation and twistor theory

It is shown that Einstein–Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev–Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector then it is locally given by h= dy 2−4 dx dt−4u dt 2 , ν=−4u x dt , where u= u( x, y, t) satisfies the dKP equation ( u t − uu x ) x = u yy . Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-Kähler metrics in signature (++−−) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP 1 -bundle over a Weyl space. It is proved that complex solutions to the dKP equations, modulo a certain coordinate freedom, are in a one-to-one correspondence with mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2) ) that admit a section of κ −1/4, where κ is the canonical bundle of Z . Real solutions are obtained if the mini-twistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ −1/4 that are invariant under the involution.

The twisted photon associated to hyper-Hermitian four-manifolds

The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a pair of potentials that generalises Plebanski's second heavenly equation for hyper-Kähler four-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a combination of the Nonlinear Graviton Construction with the Ward transform for anti-self-dual Maxwell fields.

Hyper-Hermitian metrics with symmetry

We show that any four-dimensional hyper-Hermitian manifold admitting a non-trivial triholomorphic Killing vector field is locally determined by the solution of a monopole-like equation on a three-dimensional Einstein-Weyl space of a special type. Conversely, any four-dimensional hyper-Hermitian manifold admitting a non-trivial tri-holomorphic Killing vector field arises in this way.