Generalized Cylinders in Semi-Riemannian and Spin Geometry

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.Comment: 29 pages, 2 figure

A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order

Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explicitly known only to leading order in perturbation theory, but at that order is positive semi-definite, as follows from Perelman's work on the Ricci flow. This gives rise to a monotonicity formula for the flow which is expected to fail only if the beta function perturbation series fails to converge, which can happen if curvatures or their derivatives grow large. We test the validity of the monotonicity formula at next-to-leading order in perturbation theory by explicitly computing the second-order terms in the metric on the space of coupling constants. At this order, this metric is found not to be positive semi-definite. In situations where this might spoil monotonicity, derivatives of curvature become large enough for higher order perturbative corrections to be significant.Comment: 15 pages; Erroneous sentence in footnote 14 removed; this version therefore supersedes the published version (our thanks to Dezhong Chen for the correction

Generic metrics and the mass endomorphism on spin three-manifolds

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $p\in M$ is called the mass endomorphism in $p$ associated to the metric $g$ due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.Comment: 8 page

Compact representation of wall-bounded turbulence using compressive sampling

Compressive sampling is well-known to be a useful tool used to resolve the energetic content of signals that admit a sparse representation. The broadband temporal spectrum acquired from point measurements in wall-bounded turbulence has precluded the prior use of compressive sampling in this kind of flow, however it is shown here that the frequency content of flow fields that have been Fourier transformed in the homogeneous spatial (wall-parallel) directions is approximately sparse, giving rise to a compact representation of the velocity field. As such, compressive sampling is an ideal tool for reducing the amount of information required to approximate the velocity field. Further, success of the compressive sampling approach provides strong evidence that this representation is both physically meaningful and indicative of special properties of wall turbulence. Another advantage of compressive sampling over periodic sampling becomes evident at high Reynolds numbers, since the number of samples required to resolve a given bandwidth with compressive sampling scales as the logarithm of the dynamically significant bandwidth instead of linearly for periodic sampling. The combination of the Fourier decomposition in the wall-parallel directions, the approximate sparsity in frequency, and empirical bounds on the convection velocity leads to a compact representation of an otherwise broadband distribution of energy in the space defined by streamwise and spanwise wavenumber, frequency, and wall-normal location. The data storage requirements for reconstruction of the full field using compressive sampling are shown to be significantly less than for periodic sampling, in which the Nyquist criterion limits the maximum frequency that can be resolved. Conversely, compressive sampling maximizes the frequency range that can be recovered if the number of samples is limited, resolving frequencies up to several times higher than the mean sampling rate. It is proposed that the approximate sparsity in frequency and the corresponding structure in the spatial domain can be exploited to design simulation schemes for canonical wall turbulence with significantly reduced computational expense compared with current techniques

On a spin conformal invariant on manifolds with boundary

On a n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric, a spin structure and a chirality operator, we study some properties of a spin conformal invariant defined from the first eigenvalue of the Dirac operator under the chiral bag boundary condition. More precisely, we show that we can derive a spinorial analogue of Aubin's inequality.Comment: 26 page

The Dirac operator on generalized Taub-NUT spaces

We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact K\"ahler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author.Comment: Final version, 16 page

A characterization of Dirac morphisms

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.Comment: 18 pages; restricted to the even-dimensional cas

Movement Kinematics Dynamically Modulates the Rolandic ~ 20-Hz Rhythm During Goal-Directed Executed and Observed Hand Actions

First Online: 14 February 2018This study investigates whether movement kinematics modulates similarly the rolandic α and β rhythm amplitude during executed and observed goal-directed hand movements. It also assesses if this modulation relates to the corticokinematic coherence (CKC), which is the coupling observed between cortical activity and movement kinematics during such motor actions. Magnetoencephalography (MEG) signals were recorded from 11 right-handed healthy subjects while they performed or observed an actor performing the same repetitive hand pinching action. Subjects’ and actor’s forefinger movements were monitored with an accelerometer. Coherence was computed between acceleration signals and the amplitude of α (8–12 Hz) or β (15–25 Hz) oscillations. The coherence was also evaluated between source-projected MEG signals and their β amplitude. Coherence was mainly observed between acceleration and the amplitude of β oscillations at movement frequency within bilateral primary sensorimotor (SM1) cortex with no difference between executed and observed movements. Cross-correlation between the amplitude of β oscillations at the SM1 cortex and movement acceleration was maximal when acceleration was delayed by ~ 100 ms, both during movement execution and observation. Coherence between source-projected MEG signals and their β amplitude during movement observation and execution was not significantly different from that during rest. This study shows that observing others’ actions engages in the viewer’s brain similar dynamic modulations of SM1 cortex β rhythm as during action execution. Results support the view that different neural mechanisms might account for this modulation and CKC. These two kinematic-related phenomena might help humans to understand how observed motor actions are actually performed.Xavier De Tiège is Postdoctorate Clinical Master Specialist at the Fonds de la Recherche Scientifique (FRS-FNRS, Brussels, Belgium). This work was supported by the program Attract of Innoviris (Grant 2015-BB2B-10 to Mathieu Bourguignon), the Spanish Ministry of Economy and Competitiveness (Grant PSI2016-77175-P to Mathieu Bourguignon), the Marie Skłodowska-Curie Action of the European Commission (grant #743562 to Mathieu Bourguignon), a “Brains Back to Brussels” grant to Veikko Jousmäki from the Institut d’Encouragement de la Recherche Scientifique et de l’Innovation de Bruxelles (Brussels, Belgium), European Research Council (Advanced Grant #232946 to Riitta Hari), the Fonds de la Recherche Scientifique (FRS-FNRS, Belgium, Research Credits: J009713), and the Academy of Finland (grants #131483 and #263800). The MEG project at the ULB-Hôpital Erasme (Brussels, Belgium) is financially supported by the Fonds Erasme

Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar\'e equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle