On the extremal compatible linear connection of a generalized Berwald manifold

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.

PAD Inhibitors as a Potential Treatment for SARS-CoV-2 Immunothrombosis

Since the discovery of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in December 2019, the virus’s dynamicity has resulted in the evolution of various variants, including the delta variant and the more novel mu variant. With a multitude of mutant strains posing as challenges to vaccine efficacy, it is critical that researchers embrace the development of pharmacotherapeutics specific to SARS-CoV-2 pathophysiology. Neutrophil extracellular traps and their constituents, including citrullinated histones, display a linear connection with thrombotic manifestations in COVID-19 patients. Peptidylarginine deiminases (PADs) are a group of enzymes involved in the modification of histone arginine residues by citrullination, allowing for the formation of NETs. PAD inhibitors, specifically PAD-4 inhibitors, offer extensive pharmacotherapeutic potential across a broad range of inflammatory diseases such as COVID-19, through mediating NETs formation. Although numerous PAD-4 inhibitors exist, current literature has not explored the depth of utilizing these inhibitors clinically to treat thrombotic complications in COVID-19 patients. This review article offers the clinical significance of PAD-4 inhibitors in reducing thrombotic complications across various inflammatory disorders like COVID-19 and suggests that these inhibitors may be valuable in treating the origin of SARS-CoV-2 immunothrombosis.

Multiplicative connections and their Lie theory

We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn–Drummond [Lie theory of multiplicative tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connections in the tangent bundle, that we call infinitesimally multiplicative (IM) connections and we prove an integration theorem for IM connections. Finally, we present a few toy examples.

On Invariant Operations on a Manifold with a Linear Connection and an Orientation

We prove a theorem that describes all possible tensor-valued natural operations in the presence of a linear connection and an orientation in terms of certain linear representations of the special linear group. As an application of this result, we prove a characterization of the torsion and curvature operators as the only natural operators that satisfy the Bianchi identities.

Maxwell-modified metric affine gravity

We present a gauge formulation of the special affine algebra extended to include an antisymmetric tensorial generator belonging to the tensor representation of the special linear group. We then obtain a Maxwell modified metric affine gravity action with a cosmological constant term. We find the field equations of the theory and show that the theory reduces to an Einstein-like equation for metric affine gravity with the source added to the gravity equations with cosmological constant $$\mu $$ μ contains linear contributions from the new gauge fields. The reduction of the Maxwell metric affine gravity to Riemann–Cartan one is discussed and the shear curvature tensor corresponding to the symmetric part of the special linear connection is identified with the dark energy. Furthermore, the new gauge fields are interpreted as geometrical inflaton vector fields which drive accelerated expansion.

Conserved Mechanisms, Novel Anatomies: The Developmental Basis of Fin Evolution and the Origin of Limbs

The transformation of paired fins into tetrapod limbs is one of the most intensively scrutinized events in animal evolution. Early anatomical and embryological datasets identified distinctive morphological regions within the appendage and posed hypotheses about how the loss, gain, and transformation of these regions could explain the observed patterns of both extant and fossil appendage diversity. These hypotheses have been put to the test by our growing understanding of patterning mechanisms that regulate formation of the appendage axes, comparisons of gene expression data from an array of phylogenetically informative taxa, and increasingly sophisticated and elegant experiments leveraging the latest molecular approaches. Together, these data demonstrate the remarkable conservation of developmental mechanisms, even across phylogenetically and morphologically disparate taxa, as well as raising new questions about the way we view homology, evolutionary novelty, and the often non-linear connection between morphology and gene expression. In this review, we present historical hypotheses regarding paired fin evolution and limb origins, summarize key aspects of central appendage patterning mechanisms in model and non-model species, address how modern comparative developmental data interface with our understanding of appendage anatomy, and highlight new approaches that promise to provide new insight into these well-traveled questions.

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.

Gauging the higher-spin-like symmetries by the Moyal product

We analyze a novel approach to gauging rigid higher derivative (higher spin) symmetries of free relativistic actions defined on flat spacetime, building on the formalism originally developed by Bonora et al. and Bekaert et al. in their studies of linear coupling of matter fields to an infinite tower of higher spin fields. The off-shell definition is based on fields defined on a 2d-dimensional master space equipped with a symplectic structure, where the infinite dimensional Lie algebra of gauge transformations is given by the Moyal commutator. Using this algebra we construct well-defined weakly non-local actions, both in the gauge and the matter sector, by mimicking the Yang-Mills procedure. The theory allows for a description in terms of an infinite tower of higher spin spacetime fields only on-shell. Interestingly, Euclidean theory allows for such a description also off-shell. Owing to its formal similarity to non-commutative field theories, the formalism allows for the introduction of a covariant potential which plays the role of the generalised vielbein. This covariant formulation uncovers the existence of other phases and shows that the theory can be written in a matrix model form. The symmetries of the theory are analyzed and conserved currents are explicitly constructed. By studying the spin-2 sector we show that the emergent geometry is closely related to teleparallel geometry, in the sense that the induced linear connection is opposite to Weitzenböck’s.