Nonassociative algebras: a framework for differential geometry

A First Extension of Geometric Control Theory to Underwater Vehicles

A rigidity theorem for pre-Lie algebras

Color algebras and affine connections on S6

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

We study groups with “differential structure” in the framework of Abstract Differential Geometry, an abstraction of the classical differential geometry of manifolds, via sheaf-theoretic methods, without ordinary calculus; the basic tool is the notion of a differential triad. First, we consider pre-Lie groups, i.e., semi-topological groups with compatible differential triads and we prove that such groups have “left-invariant vector fields” and “left-invariant derivations”, behaving like the classical ones. Next, for every pre-Lie group, we define an appropriate Lie algebra and prove the existence of a naturally associated adjoint representation of the initial group into the latter.

Within the framework of differential geometry, Yang's parallel-displacement gauge theory is considered with respect to “pure” gravitational fields. In afour-dimensional Riemannian manifold it is shown that thedouble self-dual solutions obey Einstein's vacuum equations with the cosmological term, whereas the doubleanti-self-dual configurations satisfy the Rainich conditions of Wheeler'sgeometrodynamics. Conformal methods reveal that the gravitational analog of the “instanton” or pseudoparticle solution of Yang-Mills theory was already known to Riemann.

We study the formulation of bond-orientational order in an arbitrary two dimensional geometry. We find that bond-orientational order is properly formulated within the framework of differential geometry with torsion. The torsion reflects the intrinsic frustration for two-dimensional crystals with arbitrary geometry. Within a Debye-Huckel approximation, torsion may be identified as the density of dislocations. Changes in the geometry of the system cause a reorganization of the torsion density that preserves bond-orientational order. As a byproduct, we are able to derive several identities involving the topology, defect density and geometric invariants such as Gaussian curvature. The formalism is used to derive the general free energy for a 2D sample of arbitrary geometry, both in the crystalline and hexatic phases. Applications to conical and spherical geometries are briefly addressed.

Introduction. In this series of papers we will discuss homotopy invariants of differentiable maps f: M -? N in various situations within the framework of differential geometry. We shall be particularly interested in the case where f is an immersion. Our principle is simple. We use the fact that the pullbackf*W of c is a differentiable homotopy invariant where w is an arbitrary cohomology class always over the real number field, R. We note if F= M x I -1 N is a differentiable map with f=f, thenf*w belongs to the same cohomology class asfJ*w where I is the interval [0, 1] and ft is defined by ft(x)=F(x, t), x E M, t EI L There are many known examples beside various characteristic classes. To quote a few of them, letf be an immersion of the two-torus T2 into the complement of the diagonal set of the six dimensional euclidean space R6 considered as R3 x R3; f: T2 R3 x R3 A, where A is the diagonal set. The space R3 x R 3A is diffeomorphic with R4 x S2 where S2 is the two-sphere. Thus, if a) denotes the volume element of 52, a becomes a 2-dimensional cohomology class of R4 x S2 by pulling back with the projection onto S2 and gives us a homotopy invariantf*co. Consider T2 interpreted as the direct product S1 x S 1 of circles andf as the pair of two closed curves ci: SI --> R3 without intersection. Then f *w is nothing but the linking number of two closed -curves cl and c2 (up to a universal constant multiple) according to Gauss [8]. Another (but somewhat more extraneous) example, due to J. H. C. Whitehead [12], is the Hopf invariant, H(f), for a differentiable map f: S2n-1 _? Sn. Again, denoting the volume element of the n-sphere Sn by c, H(f) is given by the closed form 0 Af*w where 0 is any (n l)-form with dO =f*U. In this paperf will be an isometric immersion of a compact oriented Riemannian manifold M into a euclidean space Rn + v. Since the cohomology groups of N=Rn + v are then trivial, our principle does not apply tof directly, but we construct a manifold B from M and replace f by a certain map fB: M -? B. To do this, let us recall Hirsch's theorem [5] to the effect that the regular homotopy classes of the immersionsf: M -? Nofany manifold Minto another manifold N with dim N> dim M are in a one-to-one correspondence with the homotopy classes of the crosssections fB: M -->B of a certain bundle B over M, where a regular homotopy F= M x I -* N means one for which each ft is an immersion. Despite the triviality of the cohomology groups of N= Rn+ v in the case above, we can expect to obtain homotopy invariants fB*0 corresponding to a cohomology class 0 of B. In ?4, we

Control Contraction Metrics (CCMs) can be described as a non-linear control technique based indifferential dynamics on the framework of differential geometry. This work presents preliminaryresults on applying CCMs to the control of a Blended Wing Body (BWB) aircraft configuration,in particular, regarding flight conditions involving high angle of attack and medium Reynoldsnumber regimes. These preliminary results show that in absence of conventional stabilityequilibrium points, proper CCMs may allow effective control on aircraft dynamics given theadequate operational conditions. These conditions and configuration are expected to be presentin new concepts of Remotely Piloted Aircraft Systems (RPAS), extending their applicationsfrom planetary exploration to support the development of new transportation concepts incommercial airliners. Methods described, data, and results, are expected to be of applicationfor EXperimental Aircraft for European Leadership in Aviation (EXAELIA) project, a HorizonEurope international effort to de-risk and develop new concepts of passsenger aircraft to faceaerospace industry challenges by 2050.This work has been funded by the European Union under GA No. 101191922. Viewsand opinions expressed are however those of the author only and do not necessarily reflect thoseof the European Union or the European Climate, Infrastructure and Environment ExecutiveAgency. Neither the European Union nor the granting authority can be held responsible forthem.

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in X** forms a linear basis of the free Pre-Lie algebra PLie(X) generated by the set X. For completeness, we give the details of the proof of Shirshov’s Composition-Diamond lemma for non-associative algebras.

Nonassociative quaternion algebras were first discovered over the real numbers independently by Dickson and Albert and provided some of the first examples of nonassociative division algebras. They were later classified completely by Waterhouse. Cyclic algebras can be seen as a natural generalisation of the classical quaternion algebras. With this in mind we generalise nonassociative quaternion algebras and introduce nonassociative cyclic algebras. These provide new examples of nonassociative central division algebras with Nucleus a separable field extension of degree n.

In brain-computer interfaces based on motor imagery, covariance matrices are widely used through spatial filters computation and other signal processing methods. Covariance matrices lie in the space of Symmetric Positives-Definite (SPD) matrices and therefore, fall within the Riemannian geometry domain. Using a differential geometry framework, we propose different algorithms in order to classify covariance matrices in their native space.

Prime Lie algebras satisfying the standard Lie identity of degree 5

We deal with a 2m‐dimensional Riemannian manifold (M, g) structured by an affine connection and a vector field 𝒯, defining a 𝒯‐parallel connection. It is proved that 𝒯 is both a torse forming vector field and an exterior concurrent vector field. Properties of the curvature 2‐forms are established. It is shown that M is endowed with a conformal symplectic structure Ω and 𝒯 defines a relative conformal transformation of Ω.

Stable solutions of symmetric systems involving hypoelliptic operators