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Using multi-frequency Very Long Baseline Interferometer (VLBI) observations, we probe the jet size in the optically thick hard state jets of two black hole X-ray binary (BHXRB) systems, MAXI J1820+070 and V404 Cygni. Due to optical depth effects, the phase referenced VLBI core positions move along the jet axis of the BHXRB in a frequency dependent manner. We use this "core shift" to constrain the physical size of the hard state jet. We place an upper limit of $0.3$\,au on the jet size measured between the 15 and 5 GHz emission regions of the jet in MAXI J1820+070, and an upper limit of $1.0$\,au between the $8.4$ and $4.8$\,GHz emission regions of V404 Cygni. Our limit on the jet size in MAXI J1820+070 observed in the low-hard state is a factor of $5$ smaller than the values previously observed in the high-luminosity hard state (using time lags between multi-frequency light curves), thus showing evidence of the BHXRB jet scaling in size with jet luminosity. We also investigate whether motion of the radio-emitting region along the jet axis could affect the measured VLBI parallaxes for the two systems, leading to a mild tension with the parallax measurements of Gaia. Having mitigated the impact of any motion along the jet axis in the measured astrometry, we find the previous VLBI parallax measurements of MAXI J1820+070 and V404 Cygni to be unaffected by jet motion. With a total time baseline of $8$ years, due to having incorporated fourteen new epochs in addition to the previously published ones, our updated parallax measurement of V404 Cygni is $0.450 \pm 0.018$\,mas ($2.226 \pm 0.091$\,kpc)., Comment: Accepted in MNRAS. 7 Figures and 3 Tables

Tracking the motions of transient jets launched by low-mass X-ray binaries (LMXBs) is critical for determining the moment of jet ejection, and identifying any corresponding signatures in the accretion flow. However, these jets are often highly variable and can travel across the resolution element of an image within a single observation, violating a fundamental assumption of aperture synthesis. We present a novel approach in which we directly fit a single time-dependent model to the full set of interferometer visibilities, where we explicitly parameterise the motion and flux density variability of the emission components, to minimise the number of free parameters in the fit, while leveraging information from the full observation. This technique allows us to detect and characterize faint, fast-moving sources, for which the standard time binning technique is inadequate. We validate our technique with synthetic observations, before applying it to three Very Long Baseline Array (VLBA) observations of the black hole candidate LMXB MAXI J1803-298 during its 2021 outburst. We measured the proper motion of a discrete jet component to be $1.37\pm0.14$ mas/hr, and thus we infer an ejection date of MJD $59348.08_{-0.06}^{+0.05}$, which occurs just after the peak of a radio flare observed by the Australia Telescope Compact Array (ATCA) and the Atacama Large Millimeter/Sub-Millimeter Array (ALMA), while MAXI J1803-298 was in the intermediate state. Further development of these new VLBI analysis techniques will lead to more precise measurements of jet ejection dates, which, combined with dense, simultaneous multi-wavelength monitoring, will allow for clearer identification of jet ejection signatures in the accretion flow., Comment: 15 pages, 9 figures, 4 tables; Accepted for publication in MNRAS

During a 2018 outburst, the black hole X-ray binary MAXI J1820+070 was comprehensively monitored at multiple wavelengths as it underwent a hard to soft state transition. During this transition a rapid evolution in X-ray timing properties and a short-lived radio flare were observed, both of which were linked to the launching of bi-polar, long-lived relativistic ejecta. We provide detailed analysis of two Very Long Baseline Array observations, using both time binning and a new dynamic phase centre tracking technique to mitigate the effects of smearing when observing fast-moving ejecta at high angular resolution. We identify a second, earlier ejection, with a lower proper motion of $18.0\pm1.1$ mas day$^{-1}$. This new jet knot was ejected $4\pm1$ hours before the beginning of the rise of the radio flare, and $2\pm1$ hours before a switch from type-C to type-B X-ray quasi-periodic oscillations (QPOs). We show that this jet was ejected over a period of $\sim6$ hours and thus its ejection was contemporaneous with the QPO transition. Our new technique locates the original, faster ejection in an observation in which it was previously undetected. With this detection we revised the fits to the proper motions of the ejecta and calculated a jet inclination angle of $(64\pm5)^\circ$, and jet velocities of $0.97_{-0.09}^{+0.03}c$ for the fast-moving ejecta ($\Gamma>2.1$) and $(0.30\pm0.05)c$ for the newly-identified slow-moving ejection ($\Gamma=1.05\pm0.02$). We show that the approaching slow-moving component is predominantly responsible for the radio flare, and is likely linked to the switch from type-C to type-B QPOs, while no definitive signature of ejection was identified for the fast-moving ejecta., Comment: 11 pages including 5 figures. Accepted for publication in MNRAS

The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on 3-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric., Comment: 32 pages

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to correlate harmonic vector fields on the quadrics of neutral signature., Comment: 20 pages

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vector fields that are harmonic. In particular, the harmonic Killing fields and conformal gradient fields are classified, a loop of non-congruent harmonic conformal fields on the hyperbolic plane constructed, and the 2-dimensional classification achieved for conformal fields. A classification is then given of all harmonic quadratic gradient fields on spheres., Comment: 29 pages. This is such a complete overhaul of arXiv:math/0703060 that we submit it as new article

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre., Comment: 21 pages

We study the geometry of the tangent bundle equipped with a two-parameter family of Riemannian metrics. After deriving the expression of the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. Specializing to the case of space forms, we characterise the metrics giving positive sectional curvature and show that one can always find parameters ensuring positive scalar curvature on the tangent space. Under some curvature conditions, this extends to general base manifolds., Comment: Improved version with a larger class of parameters. A more detailed version is available on request

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves their non-existence on S^2, obtains rigidity results for harmonic gradient vector fields on the two-sphere, classifies spherical quadratic gradient fields in all dimensions and determines the tension field, concluding with the discovery of a family of metrics making Hopf vector fields harmonic maps on S^{2n+1}., Comment: A detailed version is available at: http://maths2.univ-brest.fr/~loubeau/articles/harm-long.pdf

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the harmonic unit vector field tangent to a parallel family of hyperbolic geodesics is shown to be unstable, by constructing a class of compactly supported energy-decreasing variations. All examples considered have infinite total bending., Comment: 18 pages

An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations reduce to those for the characteristic field to be a harmonic section of the unit tangent bundle. These include trans-Sasakian structures, and certain nearly cosymplectic structures. On the other hand, we obtain examples where the characteristic field is harmonic but the almost contact structure is not. Many of our examples are obtained by considering hypersurfaces of almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both structures., Comment: 23 pages

We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class., Comment: 25 pages

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of the 3-dimensional Hopf vector field is compared and contrasted with that of the closely-related Hopf map. Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted to unit vector fields on the 3-sphere., Comment: AMSTeX, 20 pages, to appear in J Geom Phys