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Journal articles on the topic 'Cartan's equivalence method'

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Published: 6 July 2024

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1

KOILLER, JAIR, PAULO R. RODRIGUES, and PAULO PITANGA. "Non-holonomic connections following Élie Cartan." Anais da Academia Brasileira de Ciências 73, no. 2 (June 2001): 165–90. http://dx.doi.org/10.1590/s0001-37652001000200003.

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In this note we revisit E. Cartan's address at the 1928 International Congress of Mathematicians at Bologna, Italy. The distributions considered here will be of the same class as those considered by Cartan, a special type which we call strongly or maximally non-holonomic. We set up the groundwork for using Cartan's method of equivalence (a powerful tool for obtaining invariants associated to geometrical objects), to more general non-holonomic distributions.

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2

Gallo, Emanuel, Mirta Iriondo, and Carlos Kozameh. "Cartan's equivalence method and null coframes in general relativity." Classical and Quantum Gravity 22, no. 9 (April 14, 2005): 1881–901. http://dx.doi.org/10.1088/0264-9381/22/9/025.

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3

Thompson, G. "Cartan's method of equivalence and second-order equation fields." Journal of Physics A: Mathematical and General 18, no. 16 (November 11, 1985): L1009—L1015. http://dx.doi.org/10.1088/0305-4470/18/16/003.

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4

Grissom, Charles, Gerard Thompson, and George Wilkens. "Linearization of second order ordinary differential equations via Cartan's equivalence method." Journal of Differential Equations 77, no. 1 (January 1989): 1–15. http://dx.doi.org/10.1016/0022-0396(89)90154-x.

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5

Morozov, O. I., and C. Wafo Soh. "The equivalence problem for the Euler–Bernoulli beam equation via Cartan's method." Journal of Physics A: Mathematical and Theoretical 41, no. 13 (March 17, 2008): 135206. http://dx.doi.org/10.1088/1751-8113/41/13/135206.

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6

Mansouri, Abdol-Reza. "An extension of Cartan's method of equivalence to immersions: I. Necessary conditions." Differential Geometry and its Applications 27, no. 5 (October 2009): 635–46. http://dx.doi.org/10.1016/j.difgeo.2009.03.004.

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7

Kogan, Irina A. "Two Algorithms for a Moving Frame Construction." Canadian Journal of Mathematics 55, no. 2 (April 1, 2003): 266–91. http://dx.doi.org/10.4153/cjm-2003-013-2.

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AbstractThe method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.

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8

Hwang. "An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle." Annals of Mathematics 189, no. 3 (2019): 979. http://dx.doi.org/10.4007/annals.2019.189.3.8.

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9

Hsu, L., and N. Kamran. "Symmetries of second-order ordinary differential equations and Elie Cartan's method of equivalence." Letters in Mathematical Physics 15, no. 2 (February 1988): 91–99. http://dx.doi.org/10.1007/bf00397829.

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10

Atkins, Richard. "The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces." Canadian Mathematical Bulletin 49, no. 2 (June 1, 2006): 170–84. http://dx.doi.org/10.4153/cmb-2006-018-7.

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AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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11

Blaom, Anthony D. "Lie algebroids and Cartan’s method of equivalence." Transactions of the American Mathematical Society 364, no. 6 (June 1, 2012): 3071–135. http://dx.doi.org/10.1090/s0002-9947-2012-05441-9.

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12

Imsatfia, Moheddine. "APPLICATION OF CARTAN’S EQUIVALENCE METHOD TO DISTRIBUTION OF PLANES." Advances in Differential Equations and Control Processes 26 (January 1, 2022): 113–30. http://dx.doi.org/10.17654/0974324322008.

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Imsatfia, Moheddine, and Anouar Houmia. "APPLICATION OF CARTAN’S EQUIVALENCE METHOD TO DISTRIBUTION OF PLANES." Advances in Differential Equations and Control Processes 26 (January 1, 2022): 123–40. http://dx.doi.org/10.17654/0974324322007.

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14

Ehlers, Kurt, and Jair Koiller. "Cartan meets Chaplygin." Theoretical and Applied Mechanics 46, no. 1 (2019): 15–46. http://dx.doi.org/10.2298/tam190116006e.

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In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide in the following cases: i) Rank two distributions. This implies for instance that in Cartan?s example of a sphere rolling on a plane without slipping or twisting, a (2,3,5) distribution, the two notions of equivalence coincide; ii) For a rank 3 or higher distribution, the corank of D in D+[D,D] must be at least 3 in order to find examples where the two notions of equivalence do not coincide. This rules out the possibility of finding examples on (3,5) distributions such as Chaplygin?s marble sphere. Therefore the beautiful (3,6) example by Oliva and Terra is minimal. 1.

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15

Bakhshandeh-Chamazkoti, Rohollah. "The geometry of fourth-order differential operator." International Journal of Geometric Methods in Modern Physics 12, no. 05 (May 2015): 1550055. http://dx.doi.org/10.1142/s0219887815500553.

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In the present paper, the equivalence problem for fourth-order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth-order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.

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16

Romanovskii, Yu R. "Computation of local symmetries of second-order ordinary differential equations by the Cartan equivalence method." Mathematical Notes 60, no. 1 (July 1996): 56–67. http://dx.doi.org/10.1007/bf02308880.

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17

YANG, YUN, and YANHUA YU. "AFFINE MAURER–CARTAN INVARIANTS AND THEIR APPLICATIONS IN SELF-AFFINE FRACTALS." Fractals 26, no. 04 (August 2018): 1850057. http://dx.doi.org/10.1142/s0218348x18500573.

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In this paper, we define the notion of affine curvatures on a discrete planar curve. By the moving frame method, they are in fact the discrete Maurer–Cartan invariants. It shows that two curves with the same curvature sequences are affinely equivalent. Conditions for the curves with some obvious geometric properties are obtained and examples with constant curvatures are considered. On the other hand, by using the affine invariants and optimization methods, it becomes possible to collect the IFSs of some self-affine fractals with desired geometrical or topological properties inside a fixed area. In order to estimate their Hausdorff dimensions, GPUs can be used to accelerate parallel computing tasks. Furthermore, the method could be used to a much broader class.

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18

Atanov, Artem, and Alexander Loboda. "On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra." Mathematical Physics and Computer Simulation, no. 2 (June 2019): 5–20. http://dx.doi.org/10.15688/mpcm.jvolsu.2019.2.1.

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This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

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19

CRAMPIN, M., and D. J. SAUNDERS. "Fefferman-type metrics and the projective geometry of sprays in two dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 3 (May 2007): 509–23. http://dx.doi.org/10.1017/s0305004107000047.

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AbstractA spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy–Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.

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20

Blaga, Cristina, Paul Blaga, and Tiberiu Harko. "Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole." Symmetry 15, no. 2 (January 24, 2023): 329. http://dx.doi.org/10.3390/sym15020329.

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We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.

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21

Fonn, Sharon, Jia Hu, Jude Ofuzinim Igumbor, Duncan Gatoto, Adamson Muula, and Alex Ezeh. "Quantifying the cost of in-kind contributions to a multidonor-funded health research capacity-building programme: the case of the Consortium for Advanced Research Training in Africa." BMJ Global Health 5, no. 6 (June 2020): e002286. http://dx.doi.org/10.1136/bmjgh-2020-002286.

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IntroductionThere are significant investments in health research capacity development in the ‘global-south’. The monetary value of contributions from institutions running these programmes is not known.MethodsUsing the Consortium for Advanced Research Training in Africa (CARTA) as a case study we estimate in-kind contributions made by consortium members. We measured unpaid hours of labour contributed by consortium members and converted this to full-time equivalents. We assigned a monetary value to the time contributed by staff based on salaries by seniority and region. We estimated the monetary value of the contribution made by the African institutions that hosted CARTA events by comparing the difference in cost between university-hosted events with those held in commercial venues. We calculated the foregone overhead costs associated with hosting the CARTA secretariat. We excluded many costs where data were difficult to verify.ResultsAnnually, CARTA member institutions committed a minimum of 4.3 full-time staff equivalents that are not funded by the grants. CARTA’s annual in-kind contribution represents at least 20% of total annual donor expenditure. African institutions accounted for 82.9% of the in-kind labour contribution and 91.6% of total in-kind contribution.ConclusionThe consortium’s institutions and academic and non-academic staff make significant contributions to ensure the effective implementation of donor-funded programmes. This is not unique to CARTA. These contributions are usually not counted, often not recognised at institutional level nor remunerated through grants. Knowing these costs would allow for sustainability appraisals and cost-benefit assessments. This paper offers a method of how to measure these contributions and begins a discussion around this.

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22

Cabral, Francisco, Francisco S. N. Lobo, and Diego Rubiera-Garcia. "Fundamental Symmetries and Spacetime Geometries in Gauge Theories of Gravity—Prospects for Unified Field Theories." Universe 6, no. 12 (December 11, 2020): 238. http://dx.doi.org/10.3390/universe6120238.

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Gravity can be formulated as a gauge theory by combining symmetry principles and geometrical methods in a consistent mathematical framework. The gauge approach to gravity leads directly to non-Euclidean, post-Riemannian spacetime geometries, providing the adequate formalism for metric-affine theories of gravity with curvature, torsion and non-metricity. In this paper, we analyze the structure of gauge theories of gravity and consider the relation between fundamental geometrical objects and symmetry principles as well as different spacetime paradigms. Special attention is given to Poincaré gauge theories of gravity, their field equations and Noether conserved currents, which are the sources of gravity. We then discuss several topics of the gauge approach to gravitational phenomena, namely, quadratic Poincaré gauge models, the Einstein-Cartan-Sciama-Kibble theory, the teleparallel equivalent of general relativity, quadratic metric-affine Lagrangians, non-Lorentzian connections, and the breaking of Lorentz invariance in the presence of non-metricity. We also highlight the probing of post-Riemannian geometries with test matter. Finally, we briefly discuss some perspectives regarding the role of both geometrical methods and symmetry principles towards unified field theories and a new spacetime paradigm, motivated from the gauge approach to gravity.

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23

Syromyatnikov, A. G. "Electro-gravity spin density waves." International Journal of Geometric Methods in Modern Physics 14, no. 10 (September 13, 2017): 1750146. http://dx.doi.org/10.1142/s0219887817501468.

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It is known that some string models predict that strong bursts of gravitational radiation which should be detectable by LIGO, VIRGO and LISA detectors are accompanied by cosmologic gamma-ray bursts (GRBs). GRBs of low-energy gamma ray are associated with core-collapse supernovae (SN). However, measurements of the X-ray afterglow of very intense GRBs (allow a critical test of GRB theories) disagree with that predicted by widely accepted fireball internal–external shocks models of GRBs. It is also known that in a system of a large number of fermions, pairs of gravitational interaction occur on spontaneous breaking of the vacuum spatial symmetry, accompanied by gravitational mass defect. In another side, the space rays generation mechanism on a method of direct transformation of intergalactic gamma-rays to the proton current on spin shock-waves ensures precise agreement between generated proton currents (spin shock waves theory) with the angular distribution data of Galactic gamma-rays as well as for the individual pulses of gamma-/X-ray bursts. There is a precise confirmation of the generated currents (theory) with the burst radiation data characterized by the standard deviation of [Formula: see text] in intensity in relative units within the sensitivity of the equipment. Thus, it was found that the spin angular momentum conservation law (equation of dynamics of spin shock waves) in the X-ray/gamma ranges is fulfilled exactly in real time. The nature of gamma bursts is largely determined by the influence of powerful external sources. The angular distributions anisotropy of Galactic gamma rays and pulsars are determined by the paradoxes way, so this can only take place under conditions of the isotropy of space–time. In this regard, promising gravity in a Finsler space can have the selected direction in flat Minkowski space metric with torsion as in the Einstein–Cartan theory. Considering the induction of torsion in conformal transformations of tetrades (N-ades in arbitrary dimension N) under the Conformal Gauge Theory of Gravity (CGTG), here is considered an exact cosmological solution with Friedman’s asymptotic in the form of conformal flat Fock’s metrics at large times, describing the stage of decay on a cold dust-like medium of do-not-interacting-among-themselves particles and a light-like isotropic radiation. It is shown that at high times, indeed, the process of enlarging the space–time in the model metrics Friedman conformal is equivalent to Minkowski space with a gradient torsion trace in the CGTG Newtonian limit, accompanied by a polarization effect separation of electric charges induced by an electric field [Formula: see text] is manifested in the formation of plasma-like medium with a zero complete electric charge, that in the later stages of evolution is identical to the Fock’s model of a cold dust-like medium of do-not-interacting-among-themselves particles moving here with the same speed. The trace of torsion on the CGTG formula is freezing into an electromagnetic field spin tensor trace density and [Formula: see text] defined inside a spherical surface, moving at the speed of light, on which experiencing a gap. Therefore, this decision takes the form of an electro-gravity spin density wave, as performed in kinematic and dynamic close connection conditions for theorems on spin shock waves with spin flip at the front of the wave, moving at the speed of light in a vacuum. The theoretical dependence of electro-gravity wave energy output from the size of the emitting object is received. When applied to GRBs, this can give a new mechanism of nonthermal gamma rays production.

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24

Morozov, Oleg. "A recursion operator for the universal hierarchy equation via Cartan’s method of equivalence." Open Mathematics 12, no. 2 (January 1, 2014). http://dx.doi.org/10.2478/s11533-013-0345-2.

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AbstractWe apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.

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25

Bor, Gil, Omid Makhmali, and Paweł Nurowski. "Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions." Geometriae Dedicata 216, no. 1 (January 4, 2022). http://dx.doi.org/10.1007/s10711-021-00665-4.

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AbstractWe study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

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26

Gregoris, Daniele, Yen Chin Ong, and Bin Wang. "Curvature invariants and lower dimensional black hole horizons." European Physical Journal C 79, no. 11 (November 2019). http://dx.doi.org/10.1140/epjc/s10052-019-7423-y.

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AbstractIt is known that the event horizon of a black hole can often be identified from the zeroes of some curvature invariants. The situation in lower dimensions has not been thoroughly clarified. In this work we investigate both ($$2+1$$2+1)- and ($$1+1$$1+1)-dimensional black hole horizons of static, stationary and dynamical black holes, identified with the zeroes of scalar polynomial and Cartan curvature invariants, with the purpose of discriminating the different roles played by the Weyl and Riemann curvature tensors. The situations and applicability of the methods are found to be quite different from that in 4-dimensional spacetime. The suitable Cartan invariants employed for detecting the horizon can be interpreted as a local extremum of the tidal force suggesting that the horizon of a black hole is a genuine special hypersurface within the full manifold, contrary to the usual claim that there is nothing special at the horizon, which is said to be a consequence of the equivalence principle.

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27

Meskhidze, Helen, and James Owen Weatherall. "Torsion in the Classical Spacetime Context." Philosophy of Science, October 27, 2023, 1–12. http://dx.doi.org/10.1017/psa.2023.136.

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Abstract Teleparallel gravity, an empirically equivalent counterpart to General Relativity, represents the influence of gravity using torsional forces. It raises questions about theory interpretation and underdetermination. To better understand the torsional forces of Teleparallel gravity, we consider a context in which forces are better understood: classical spacetimes. We propose a method of incorporating torsion into the classical spacetime context that yields a classical theory of gravity with a closed temporal metric and spacetime torsion. We then prove a result analogous to the Trautman degeometrization theorem, that every model of Newton-Cartan theory gives rise, non-uniquely, to a model of this theory.

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28

Massa, Enrico, and Enrico Pagani. "On the Herglotz variational problem." Journal of Mathematical Physics 64, no. 10 (October 1, 2023). http://dx.doi.org/10.1063/5.0165641.

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A geometric approach to the Herglotz problem is developed, based on the bundle of affine scalars on the configuration manifold of the given system. The environment, originally introduced to formalize the gauge structure of Lagrangian Mechanics [E. Massa, E. Pagani, and P. Lorenzoni, Transp. Theory Stat. Phys. 29, 69 (2000)], provides the natural setting for the representation of the Herglotz functional as well as for the study of its extremals. Various aspects of the problem are considered: the Lagrangian approach, leading to a generalization of the Poincaré-Cartan algorithm; the parametric approach, involving the introduction of an appropriate super-Lagrangian; the corresponding Hamiltonian and super-Hamiltonian counterparts; the relationship between the Herglotz problem and a constrained variational problem; the evaluation of the abnormality index [Massa et al., Int. J. Geom. Methods Mod. Phys. 12, 1550061 (2015)] of the resulting extremals; the gauge structure of the theory and the consequent existence of Herglotz’s functionals gauge-equivalent to ordinary action functionals.

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29

Damon, James. "Characteristic cohomology II: Matrix singularities." Journal of Topology 17, no. 3 (June 27, 2024). http://dx.doi.org/10.1112/topo.12330.

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AbstractFor a germ of a variety , a singularity of “type ” is given by a germ , which is transverse to in an appropriate sense, such that . In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for a hypersurface), and complement and link (for the general case). It captures the cohomology of inherited from and is given by subalgebras of the cohomology for for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences for Milnor fibers and for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology.In this paper, we apply these methods in the case denotes any of the varieties of singular complex matrices, which may be either general, symmetric, or skew‐symmetric (with even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact “model submanifolds” for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general complex matrices.Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of “kite map germ of size ” based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ containing such a kite map germ of size . Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices.

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