arXiv subjects
Publications and source records attributed to Michael Eastwood.
We present a systematic prolongation procedure and its implementation for Killing two-tensors. We use the resulting machinery to elucidate the natural quadratic mapping from Killing fields to Killing two-tensors in general and on irreducible locally symmetric spaces of compact type.
For semi-Riemannian manifolds of constant sectional curvature, the Calabi operator is a second order linear differential operator that provides local integrability conditions for the range of the Killing operator. In this article, extending earlier results in the Riemannian setting, we identify the Lorentzian locally symmetric spaces on which the Calabi operator is sufficient to identify the range of the Killing operator. Specifically, this is always the case for indecomposable spaces and we identify precisely those products for which it fails. Our method is quite general in that we firstly develop criteria to be in the range of a connection, viewed as a linear differential operator. Then we ascertain how these criteria apply in the case of what we call the Killing connection.
On a Riemannian manifold of constant curvature, the Calabi operator is a second order linear differential operator that provides local integrability conditions for the range of the Killing operator. We generalise this operator to provide linear second order local integrability conditions on Riemannian locally symmetric spaces, whenever this is possible. Specifically, we show that this generalised operator always works in the irreducible case and we identify precisely those products for which it fails.
In conformal differential geometry, there are some distinguished curves, often known as 'conformal circles,' since, on the round sphere, they are the round circles (and these are conformally invariant). But on the two-sphere, the curves of constant compass bearing are also conformally invariant. These 'loxodromes' admit a curved analogue in the realm of Moebius geometry. In this article, these curved analogues are explained and the fifth order invariant ODE that they satisfy is derived.
The Killing tensors of arbitrary rank on complex projective space with its Fubini-Study metric are determined and it is shown that these spaces are generated by the Killing fields.
We linearise the Einstein vacuum equations with a cosmological constant via the Calabi operator from projective differential geometry.
Given a parabolic geometry, it is sometimes possible to find special metrics characterised by some invariant conditions. In conformal geometry, for example, one asks for an Einstein metric in the conformal class. Einstein metrics have the special property that their geodesics are distinguished, as unparameterised curves, in the sense of parabolic geometry. This property characterises the Einstein metrics. In this article we initiate a study of corresponding phenomena for other parabolic geometries, in particular for the hypersurface CR and contact Legendrean cases.
We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on the configuration space of a flying saucer (always a five-dimensional contact manifold).
It is well-known that Lagrange's four-square theorem, stating that every natural number may be written as the sum of four squares, may be proved using methods from the classical theory of modular forms and theta functions. We revisit this proof. In doing so, we concentrate on geometry and thereby avoid some of the tricky analysis that is often encountered. Guided by projective differential geometry we find a new route to Lagrange's theorem.
A successful ground array Radio Frequency (RF)-only self-trigger on 10 high-energy cosmic ray events is demonstrated with 256 dual-polarization antennas of the Owens Valley Radio Observatory Long Wavelength Array (OVRO-LWA). This RF-only capability is predicated on novel techniques for Radio Frequency Interference (RFI) identification and mitigation with an analysis efficiency of 45\% for shower-driven events with a Signal-to-noise ratio $\gtrsim$ 5 against the galactic background noise power of individual antennas. This technique enables more efficient detection of cosmic rays over a wider range of zenith angles than possible via triggers from in-situ particle detectors and can be easily adapted by neutrino experiments relying on RF-only detection. This paper discusses the system design, RFI characterization and mitigation techniques, and initial results from 10 cosmic ray events identified within a 40-hour observing window. A design for a future optimized commensal cosmic-ray detector for the OVRO-LWA is presented, as well as recommendations for developing a similar capability for other experiments -- these designs either reduce data-rate or increase sensitivity by an order of magnitude for many configurations of radio instruments.
Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define certain `manoeuvres' of the saucer, which we call `attacking,' `landing,' or `G2 mode' manoeuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manoeuvre corresponds to the flat Legendrean contact structure, the landing manoeuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1,1), and the most complicated G2 manoeuvre corresponds to the contact Engel structure with split real form of the exceptional Lie group G2 as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic G2 geometries is used to construct a `G2 joystick,' consisting of two balls of radii in ratio 1:3 that transforms the difficult G2 manoeuvre into the pilot's action of rolling one of joystick's balls on the other without slipping nor twisting.
We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of `parabolic geometry') and we provide examples when this geometry is `flat,' meaning that its symmetries comprise the split form of the exceptional Lie algebra G2.
We solve the metrisability problem for generic three-dimensional projective structures.
We use Beltrami's theorem as an excuse to present some arguments from parabolic differential geometry without any of the parabolic machinery.
We use that the $n$-sphere for $n\geq 2$ is simply-connected to prove the Poincaré-Birkhoff-Witt Theorem.
Following the first two annual intensity mapping workshops at Stanford in March 2016 and Johns Hopkins in June 2017, we report on the recent advances in theory, instrumentation and observation that were presented in these meetings and some of the opportunities and challenges that were identified looking forward. With preliminary detections of CO, [CII], Lya and low-redshift 21cm, and a host of experiments set to go online in the next few years, the field is rapidly progressing on all fronts, with great anticipation for a flood of new exciting results. This current snapshot provides an efficient reference for experts in related fields and a useful resource for nonspecialists. We begin by introducing the concept of line-intensity mapping and then discuss the broad array of science goals that will be enabled, ranging from the history of star formation, reionization and galaxy evolution to measuring baryon acoustic oscillations at high redshift and constraining theories of dark matter, modified gravity and dark energy. After reviewing the first detections reported to date, we survey the experimental landscape, presenting the parameters and capabilities of relevant instruments such as COMAP, mmIMe, AIM-CO, CCAT-p, TIME, CONCERTO, CHIME, HIRAX, HERA, STARFIRE, MeerKAT/SKA and SPHEREx. Finally, we describe recent theoretical advances: different approaches to modeling line luminosity functions, several techniques to separate the desired signal from foregrounds, statistical methods to analyze the data, and frameworks to generate realistic intensity map simulations.
On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini-Study form and connection, we can build a series of differential complexes akin to the Bernstein-Gelfand-Gelfand complexes from parabolic differential geometry.
We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.