Pawel Nurowski

Differential equations and conformal structures

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of third order ODEs considered modulo contact transformations of variables and (local) three-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of third order ODEs satisfying the Wuenschmann and the Cartan conditions define a three-dimensional Lorentzian–Einstein–Weyl geometry. The third example associates to each point equivalence class of third order ODEs a six-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of second order ODEs and four-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature (3, 2). The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group G2. All the examples are deeply rooted in Elie Cartan’s works on exterior differential systems.

Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations

The equivalence problem for second-order ordinary differential equations (ODEs) given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate three-dimensional Cauchy–Riemann structures. This approach enables an analogue of the Fefferman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second-order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs.

Ambient Metrics for n-Dimensional pp-Waves

We provide an explicit formula for the Fefferman-Graham ambient metric of an n-dimensional conformal pp-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional pp-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional pp-waves have vanishing critical Q-curvature.

Covariant definition of inertial forces

We present a covariant definition of inertial forces in general relativity (gravitational, centrifugal, Euler and Coriolis-Lense-Thirring) which is valid in all spacetimes including ones with no symmetry.

Projective connections associated with second-order ODEs

We show that every second-order ODE defines a four-parameter family of projective connections on its two-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of connections always has a preferred representative. This preferred representative turns out to be identical to the projective connection described in Cartans classic paper (Cartan E 1924 Bull. Soc. Math. France 52 205–41, 1955 Oeuvres III 1 825–62).

Non-vacuum twisting type-N metrics

We present a number of results for twisting type-N metrics. (a) A maximally reduced system of equations corresponding to the twisting type-N Einstein metrics is given. When the cosmological constant λ→0 they reduce to the standard equations for the vacuum twisting type Ns. (b) All the metrics which are conformally equivalent to the twisting type-N metrics and which admit three-dimensional conformal group of symmetries are presented. (c) In the Feferman class of metrics an example is given of a twisting type-N metric which satisfies Bachs equations but is not Einstein.

Optical geometries and related structures

Abstract Two natural optical geometries on the space p of all null directions over a four-dimensional Lorentzian manifold M are defined and studied. One of this geometries is never integrable and the other is integrable iff the metric of M is conformally flat. Sections of p forming a zero set of integrability conditions for the latter optical geometry are interpreted as principal null directions on m . Certain well-defined conditions on p are shown to be equivalent to the vanishing of the traceless part of the Ricci tensor of m . Sections of p forming a zero set for these new conditions correspond to the eigendirections of the Ricci tensor of m . An analogy between optical and Hermitian geometries is discussed. Existing (or possible to exist) mutual counterparts between facts from optical and Hermitian geometries are listed. In this analogy, construction of the optical geometries on p constitutes a Lorentzian counterpart of the Atiyah-Hitchin-Singer construction of two natural almost Hermitian structures on the twistor space of four-dimensional Euclidean manifold.

Conformal Einstein equations and Cartan conformal connection

Necessary and sufficient conditions for a spacetime to be conformal to an Einstein spacetime are interpreted in terms of curvature restrictions for the corresponding Cartan conformal connection.

Cartan normal conformal connections from differential equations

We explore and show a natural relationship between all third-order ordinary differential equations that possess a vanishing Wunschmann invariant, with conformal metrics on 3-manifolds and Cartans normal O(3, 2) conformal connections. The generalization to pairs of second-order PDEs and their relationship to Cartans normal O(4, 2) conformal connections on four-dimensional manifolds is discussed.

Locally Sasakian manifolds

We show that every Sasakian manifold in dimensions 2k + 1 is locally generated by a free real function of 2k variables. This function is a Sasakian analogue of the Kahler potential for the Kahler geometry. It is also shown that every locally Sasakian-Einstein manifold in 2k + 1 dimensions is generated by a locally Kahler-Einstein manifold in dimension 2k.