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In this note, we will exploit the classical bijective correspondence between sections of an associated vector bundle and equivariant functions on the underlying principal bundle to revisit a global formula for induced connections on associated vector bundles. Consequently, we give the expression of the curvature in terms of the curvature 2-form of a connection on a principal bundle.

We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over $S^1$.

The problem of extensions of the canonical Lee–Poisson–Kirillov–Kostant symplectic structure of coadjoint orbit of the complex general linear group is considered. The introduced method uses the concept of flag coordinates and does not depend on the Jordan structure of the matrices that form the orbit. The principal bundle associated with the fibration of the orbit over the Grassmanian of flags is constructed.

We give a new construction of the holonomy groupoid of a regular foliation in terms of a partial connection on a diffeological principal bundle of germs of transverse parametrisations, which may be viewed as a systematisation of Winkelnkemper’s original construction using ideas from gauge theory. We extend these ideas to construct a novel holonomy groupoid for any foliated bundle, which we prove sits at the top of a hierarchy of diffeological jet holonomy groupoids associated with the foliated bundle. This shows that while the Winkelnkemper holonomy groupoid is the smallest Lie groupoid that integrates a foliation, it is far from the smallest diffeological groupoid that does so.

We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal bundles and that the case of vector bundles considered earlier is an infinitesimal version of the case of a solution coming from the principal bundle structure.

We investigate the category of regular coverings of a given smooth principal bundle. A covering map of one principal bundle to another principal bundle is understood as a morphism in the category of principal bundles consisting of covering maps of their structural groups, total spaces and bases. The main results are: the construction of an invariant of a regular covering, which is a subsequence of the homotopy sequence of the base bundle; the existence theorem of a covering with a given invariant; criteria for morphisms and isomorphisms; a theorem on the automorphism group of a given covering.

We develop the theory of smooth principal bundles for a smooth group G, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define D-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a D-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor’s spaces EG and BG, show that EG → BG is a D-numerable principal bundle, and prove that it classifies all D-numerable principal bundles over any diffeological space. We deduce analogous classification results for D-numerable diffeological bundles and vector bundles.

We analyze a reaction-diffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the two-species model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of \begin{document}$ N $\end{document} similar phytoplankton species, for any number \begin{document}$ N $\end{document} , generically leads to competitive exclusion. The main tool is the theory of a normalized principal bundle for linear parabolic equations.

A gauge group is the topological group of automorphisms of a principal bundle. We compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension., 8 pages

© 2020 Polish Scientific PublishersIn this work, we define a spinor covariant derivative for degenerate manifolds with 4-dimensions. To perform this, we have found the principal bundle by using a degenerate spin group. Then, we benefit from a covering map to establish a relationship between the local connection forms of principal bundles. After that, we define a covariant derivative on a degenerate spinor bundle which is an associated bundle.

For complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line p in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder-Narasimhan type of the underlying bundle., We thank the referee for helpful comments that have improvedthe exposition. The first-named author is supported by a J. C. Bose Fellowship. Thesecond-named author was partially supported by the Spanish MINECO under ICMATSevero Ochoa project no. SEV-2015-0554, and under grant no. MTM2016-81048-P. Thethird- and fourth-named authors are supported by NSERC Discovery grants. The fourth-named author also acknowledges the University of Queensland for an Ethel RaybouldFellowship held during the preparation of the original manuscript

In this paper, we present a geometric approach to form the governing equations of motion of dynamic systems. A geometric form of the d’Alembert-Lagrange equation on the configuration manifold is first developed and extended to a principal bundle. We can then obtain the explicit form of equations of motion by explicating the geometric form in a coordinate neighborhood on the principal bundle. This approach conveniently permits the choice of quantities to be used which best describe configurations, motions or constraints, and it yields equations of motion in concise forms. Examples are presented to illustrate the use and effectiveness of the approach. The objective of this paper is to provide a general geometric perspective of the governing equations of motion, and explain its suitability for studying complex dynamic systems subject to non-holonomic constraints.

We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group $G$. As coefficients, we allow $E_\infty$-algebras with $G$-action. For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the $(\infty,1)$-category of cospans of $E_\infty$-algebras., 26 pages, 2 figures, minor changes, comparison to equivariant factorization homology added; accepted for publication in "Homology, Homotopy and Applications"

For any positive integer n and any Lie group G, given a definite symmetric bilinear form on R n and an Ad-invariant scalar product on the Lie algebra of G, we construct a variational problem on fields defined on an arbitrary (n + dimG)-dimensional manifold Y. We show that, if G is compact and simply connected, any global solution of the Euler-Lagrange equations leads to identify Y with the total space of a principal bundle over an n-dimensional manifold X. Moreover X is automatically endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein-Yang-Mills system equation with a cosmological constant.

We argue that the relevant higher gauge group for the non-abelian generalization of the self-dual string equation is the string 2-group. We then derive the corresponding equations of motion and discuss their properties. The underlying geometric picture is a string structure, i.e., a categorified principal bundle with connection whose structure 2-group is the string 2-group. We readily write down the explicit elementary solution to our equations, which is the categorified analogue of the ’t Hooft–Polyakov monopole. Our solution passes all the relevant consistency checks; in particular, it is globally defined on $$\mathbb {R}^4$$ and approaches the abelian self-dual string of charge one at infinity. We note that our equations also arise as the BPS equations in a recently proposed six-dimensional superconformal field theory and we show that with our choice of higher gauge structure, the action of this theory can be reduced to four-dimensional supersymmetric Yang–Mills theory.

In the Hamiltonian formalism, and in the presence of a symmetry Lie group, a variational reduction procedure has already been developed for Hamiltonian systems without constraints. In this paper we present a procedure of the same kind, but for the entire class of the higher order constrained systems (HOCS), described in the Hamiltonian formalism. Last systems include the standard and generalized nonholonomic Hamiltonian systems as particular cases. When restricted to Hamiltonian systems without constraints, our procedure gives rise exactly to the so-called Hamilton-Poincare equations, as expected. In order to illustrate the procedure, we study in detail the case in which both the configuration space of the system and the involved symmetry define a trivial principal bundle.

On the space of a principal bundle, a Lorentzian metric and a time orientation are given that are invariant with respect to the action of the structure group. These objects form a fibered space-time and, in the case of spacelike fibers, induce the same structures on the base. The following causality conditions are discussed: chronology, causality, stable and strong causality, and global hyperbolicity. It is proved that if the base space-time satisfies one of the above conditions, then so does the fibered space-time.

We use augmented commutative differential graded algebra (ACDGA) models to study $G$-representation varieties of fundamental groups $\pi=\pi_1(M)$ and their embedded cohomology jump loci, around the trivial representation 1. When the space $M$ admits a finite family of maps, uniformly modeled by ACDGA morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of ${\rm Hom} (\pi,G)$ and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when $M$ is either a compact K\"ahler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group $G$ is a non-abelian subalgebra of $\mathfrak{sl}_2(\mathbb{C})$., Comment: 44 pages; accepted for publication in Advances in Mathematics

The first author and Moreira Galicia have studied a topological version of Molino's theory. It describes equicontinuous foliated spaces satisfying certain conditions of strong quasi-analyticity, reducing their study to the particular case of G-foliated spaces. That description is sharpened in this paper by introducing a foliated action of a compact topological group on the resulting G-foliated space, like in the case of Riemannian foliations. A C ∞ version is also studied. The triviality of this compact group characterizes compact minimal G-foliated spaces, which are also characterized by their foliated homogeneity in the C ∞ case. We also give an example where the projection of the Molino's description is not a principal bundle, and another example of positive topological codimension where the foliated homogeneity cannot be checked by only comparing pairs of leaves—in the case of zero topological codimension, weak solenoids with this property were given by Fokkink and Oversteegen, and later by Dyer, Hurder and Lukina.

Let $X$ be a torus manifold with locally standard action of a compact torus $T$ of half the dimension and orbit space a homology polytope. Smooth complete complex toric varieties and quasi-toric manifolds are examples of torus manifolds. Consider a principal bundle with total space $E$ and base $B$ with fibre and structure group $T$. Let $E(X)$ denote the total space of the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as an algebra over the singular cohomology ring of $B$ and a presentation of the topological $K$-ring of $E(X)$ as an algebra over the topological $K$-ring of $B$. These are relative versions of the results of M. Masuda and T. Panov [13] on the cohomology ring of a torus manifold and P. Sankaran [14] on the topological $K$-ring of a torus manifold. Further, they extend the results due to P. Sankaran and V. Uma [15] on the cohomology ring and topological $K$-ring of toric bundles with fibre a smooth projective toric variety, to a toric bundle with fibre any smooth complete toric variety., Comment: 14 pages

Let $X$ be a smooth irreducible projective curve. Recently, Pauly and Pe\'on-Nieto shows that a vector bundle over $X$ is very stable if and only if the Hitchin map on the vector space of Higgs field on that vector bundle is proper. In this notes, we generalize this result to principal $G-$bundles for any semisimple linear algebraic group $G$. We also study the relation between very stability and other stability conditions in the case of $\text{SL}_2-$bundles., Comment: 10 pages

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincare equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincare equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

The classical astrophysical theory of Riemann ellipsoids and the quantum nuclear theory of Bohr and Mottelson share a common mathematical foundation in terms of the differential geometry of a principal bundle ${\cal P}$ and its associated vector bundle E, respectively. The bundle ${\cal P}=GL_+(3,R)$ is the connected component of the general linear group, the structure group G=SO(3) is the vorticity group, and the base manifold is the space of positive-definite real $3\times 3$ symmetricmatrices, identified geometrically with the space of inertia ellipsoids. The bundle is a Riemannian manifold whose metric is inherited from three-dimensional Euclidean space. A nonholonomic constraint force, like irrotational flow, determines a connection on the bundle.Wave functions of the Bohr-Mottelson model are sections of the associated vector bundle E = ${\cal P}\times_\rho$V, where $\rho$ denotes an irreducible representation of the vorticity group on the vector space V. Using the de Rham Laplacian $\triangle = \star d_\nabla \star d_\nabla$ for the kinetic energy introduces a "magnetic" term due to the connection between base manifold rotational and fiber vortex degrees of freedom. A class of Ehresmann connections creates a new model of nuclear rotation that predicts moments of inertia in agreement with experiment.

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$ -deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

The Dunkl operators associated with a necessarily finite Coxeter group acting on a Euclidean space are generalized to any finite group using the techniques of non-commutative geometry, as introduced by the authors to view the usual Dunkl operators as covariant derivatives in a quantum principal bundle with a quantum connection. The definitions of Dunkl operators and their corresponding Dunkl connections are generalized to quantum principal bundles over quantum spaces which possess a classical finite structure group. We introduce cyclic Dunkl connections and their cyclic Dunkl operators. Then, we establish a number of interesting properties of these structures, including the characteristic zero-curvature property. Particular attention is given to the example of complex reflection groups, and their naturally generalized siblings called groups of Coxeter type.

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples illustrating these constructions include: Galois objects of Taft algebras, a monopole bundle over a quantum spheres and a not faithfully flat Hopf--Galois extension of commutative algebras. The latter two examples have in fact a structure of Hopf algebroid for a suitable invertible antipode., Comment: 33 pages. Sect. 5.1 expanded. Few minor changes. arXiv admin note: substantial text overlap with arXiv:2002.06097

In this short paper, we will review the proposal of a correspondence between the doubled geometry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theory is T-duality covariant formulation of the supergravity limit of String Theory, which generalises Kaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space. In light of the proposed correspondence, this doubled geometry is interpreted as an atlas description of the higher geometry of bundle gerbes. In this sense, Double Field Theory can be interpreted as a field theory living on the total space of the bundle gerbe, just like Kaluza-Klein theory is set on the total space of a principal bundle. This correspondence provides a higher geometric interpretation for para-Hermitian geometry which opens the door to its generalisation to the other Extended Field Theories. This review is based on, but not limited to, my talk at the workshop Generalized Geometry and Applications at Universit\"at Hamburg on 3rd of March 2020., Comment: Prepared for submission to Complex Manifolds, special issue: Generalized Geometry. 35 pages, 2 figures. Published version

In this work we study discrete analogues of an exact sequence of vector bundles introduced by M. Atiyah in 1957, associated to any smooth principal G-bundle π : Q → Q / G . In the original setting, the splittings of the exact sequence correspond to connections on the principal bundle π. The discrete analogues that we consider here can be studied in two different categories: the category of fiber bundles with a (chosen) section, Fbs, and the category of local Lie groupoids, l L G p d . In Fbs we find a correspondence between a) (semi-local) splittings of the discrete Atiyah sequence (DAS) of π, b) discrete connections on the same bundle π, and c) isomorphisms of the DAS with certain fiber product extensions in Fbs. We see that the right splittings of the DAS (in Fbs) are not necessarily right splittings in l L G p d : we use this obstruction to define the discrete curvature of a discrete connection. Then, there is a correspondence between the right splittings of the DAS in l L G p d and discrete connections with trivial discrete curvature, that is, flat discrete connections. We also introduce a semidirect product between (some) local Lie groupoids and prove that there is a correspondence between semidirect product extensions and right splittings of the DAS in l L G p d .

In the present paper the model for growing oriented solid is developed. Such solids have no global stress-free shape in Euclidean physical space due to the distributed defects, "recorded" into the solid during growing process. Nonetheless, in the framework of geometric approach in continuum mechanics the desired stress-free reference shape can be found in non-Euclidean space with specific (material) connection. For oriented solids, whose particles have to be identified with positions and orientations, such space can be represented by a submanifold in total space of principal bundle, defined over basic (conventional) material manifold. In such a case the deformation of growing solid can be generalized as smooth embedding from such space to total space of physical principal bundle. Invariants of connection on material principal bundle characterize the incompatibility of local deformations and can serve as cost functions in optimization problems.

This chapter examines curvature on a principal bundle. The curvature of a connection on a principal G-bundle is a g-valued 2-form that measures, in some sense, the deviation of the connection from the Maurer-Cartan connection on a product bundle. The Maurer-Cartan form Θ‎ on a Lie group G satisfies the Maurer-Cartan equation. Let M be a smooth manifold. The chapter then pulls the Maurer-Cartan equation back and uses Proposition 14.3 to get the Maurer-Cartan connection. It also considers the second structural equation; the first structural equation is discussed in a previous chapter. Finally, the chapter derives some properties of the curvature form.

This chapter discusses connections on a principal bundle. Throughout the chapter, G will be a Lie group with Lie algebra g. One possible definition of a connection on a principal G-bundle P is a C∞ right-invariant horizontal distribution on P. Equivalently, a connection on P can be given by a right-equivariant g-valued 1-form on P that is the identity on vertical vectors. The chapter shows the equivalence of these two definitions of a connection. A connection is one of the most basic notions of differential geometry. It is essentially a way of differentiating sections. From a connection, the notions of curvature and geodesics follow.

Este artículo se encuentra disponible en la siguiente URL: https://link.springer.com/article/10.1007/s10543-021-00884-x Este es el postprint del siguiente artículo: Billaud-Friess, M., Falcó, A. & Nouy, A. (2022). A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds. Bit Numerical Mathematics, vol. 62, i. 2 (jun.), pp. 387?408, que se ha publicado de forma definitiva en https://doi.org/10.1007/s10543-021-00884-x This is the peer reviewed version of the following article: Billaud-Friess, M., Falcó, A. & Nouy, A. (2022). A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds. Bit Numerical Mathematics, vol. 62, i. 2 (jun.), pp. 387?408, which has been published in final form at https://doi.org/10.1007/s10543-021-00884-x In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of the set of fixed rank matrices. We first introduce a geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be exact in some particular case. Numerical experiments confirm this result and illustrate the robustness of the proposed algorithm. Postprint

We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity., Comment: Version augmented with a new appendix. Correction of many misprints in formulae and in text. 81 pages

This chapter covers the theory of quantum principal bundles, where the fibre is a Hopf algebra or quantum group. In the case of the universal calculus, this reduces to a notion in algebra of a Hopf-Galois extension. We provide the general theory of associated bundles and bimodule connections on them induced by a connection-form. We also study differential fibrations more generally. The last section is an application to quantum framed spaces, i.e., differential algebras appearing as the base of a quantum principal bundle and data such that the 1-forms form an associated bundle. The chapter includes bundles and bimodule q-monopole connections on the q- sphere, which combines with our quantum framing theory to provide our first encounter with a ’quantum Levi-Civita connection’ as a torsion free metric compatible bimodule connection on the cotangent bundle.

In this paper, we define an isometry on a total space of a vector bundle E by using a given isometry on the base manifold M. For this definition, we assume that the total space of the bundle is equipped with a special metric which has been introduced in one of our previous papers. We prove that the set of these derived isometries on E form an imbedded Lie subgroup Ge of the isometry group I E . Using this new subgroup, we construct two different principal bundle structures based one on E and the other on the orbit space E/Ge. Key words: Fiber bundles, isometry group, vector bundles, principal bundles

A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Mauer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also consider the critical points of some related functionals. Throughout, we see how some of the known properties of $G_{2}$-structures can be seen from this more general setting., Comment: 101 pages, 3 figures

A gauge transformation of categorical principal bundles arises from a functorial isomorphism between such bundles. We determine the geometric nature of such gauge transformations. For a twisted-product categorical principal bundle whose structure group is given by a pair of Lie groups G and H we show that a pair consisting of a traditional gauge transformation θ , given by a G -valued function, and an L ( H ) -valued 1-form Λ H determine a categorical gauge transformation. More general gauge transformations are also studied.

Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, i.e., actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples that naturally arise from non-integrable Lie algebroids., Comment: 80 pages. v.2: new introduction. A shortened version of this paper is accepted at IMRN

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge. We also show how our results from sub-Riemannian geometry can give improved approximations for the case of studying expansions of holonomy of principal bundles over the Euclidean space.

principal G-bundle with singularities is a principal bundle π: $$\bar P$$ → M with structure group $$\bar G$$ which reduces to a subgroup G ⊂ $$\bar G$$ on the set M \ Σ, where M is an n-dimensional compact manifold and Σ ⊂ M is a k-dimensional submanifold. For example, a vector field on an n-dimensional Riemannian manifold M defines reduction of the orthonormal frame bundle of M to the subgroup O(n − 1) ⊂ O(n) on the set M \ Σ, where Σ is the set of zeros of this vector field. The aim of this paper is to construct topological invariants of principal bundles with singularities. To do this we apply the obstruction theory to the sectionM → $$\bar P$$ /Gcorresponding to the reduction and obtain the topological invariant as a class in Hn−k(M,M \ Σ; πn−k−1( $$\bar G$$ /G)). We study the properties of this invariants and, in particular, consider cases k = 0 y k = n − 1.

For a Riemann surface X and the moduli of regularly stable G-bundles M, there is a naturally occuring “adjoint” vector bundle over X × M. One can take the determinant of this vector bundle with respect to the projection map onto M. Our aim here is to study the curvature of the determinant bundle as the conformal structure on X varies over the Teichmuller space.

We define Hitchin’s moduli space for a principal bundle P, whose structure group is a compact semisimple Lie group K, over a compact non-orientable Riemannian manifold M. We use the Donaldson–Corlette correspondence, which identifies Hitchin’s moduli space with the moduli space of flat connections, which remains valid when M is non-orientable. This enables us to study Hitchin’s moduli space both by gauge theoretical methods and algebraically by using representation varieties. If the orientable double cover M~ of M is a Kahler manifold with odd complex dimension and if the Kahler form is odd under the non-trivial deck transformation τ on M~, Hitchin’s moduli space of the pull-back bundle has a hyper-Kahler structure and admits an involution induced by τ. The fixed-point set is symplectic or Lagrangian with respect to various symplectic structures on the moduli space. We compare the gauge theoretical constructions with the algebraic approach using representation varieties.

To any flat section equation of the form $$\nabla _0\Psi =\Phi \Psi $$ in a principal bundle over a Riemann surface ( $$\nabla _0$$ is a reference connection), we associate an infinite sequence of “correlators”, symmetric n-differentials on $$\Sigma $$ that we denote $$\{W-n\}_{n \in \mathcal {N}}$$ . The goal of this article is to prove that these correlators are solutions to “loop equations,” the same ones satisfied by correlation functions in random matrix models, or equivalently Ward identities of Virasoro or $${\mathcal {W}}$$ -symmetric CFT.

We define a diffeology on the Milnor classifying space of a diffeological group $G$, constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori., 29 pages -- v3 fixes another error, and clarifies some comments on differential forms

We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.

Let $P$ be a parabolic subgroup of a connected simply connected complex semisimple Lie group $G$. Given a compact K\"ahler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduction of holomorphic principal bundles over $X\times G/P$. The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of $G$-equivariant principal bundles over $X\times G/P$, and to establish a Hitchin--Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that $X$ is a complex projective manifold., Comment: 23 pp. Dedicated to Ugo Bruzzo on his 60th birthday. v2: minor corrections; to appear in the Journal of Geometry and Physics

We construct a codimension 3 completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp ( 2 ) -principal bundle with the structure group Sp ( 1 ) . The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a ( 4 n + 3 ) -dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere S 7 .

This paper introduces a large class of differential-geometric distances between finite-dimensional linear dynamical systems, collectively called the alignment distance. Contrary to the existing distances, the alignment distance is based on the state-space description of dynamical systems, and is defined on the manifolds of systems of fixed order and fixed input–output dimension under a matrix rank constraint (e.g., minimality, controllability, or observability). While the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory, distances natural to this structure have not been studied. The starting point for defining such a distance is to identify a linear system of order $n$ with its equivalence class of state-space realizations, all related by the so-called similarity action, i.e., state-space change of basis under $GL(n)$ , the Lie group of nonsingular $n \times n$ matrices. The main idea of the alignment distance is to first find the best state-space change of basis that brings a realization of a system “as close as possible” to a realization of another system (the alignment step), and then compare the aligned realizations. A direct implementation of this idea, due to noncompactness of $GL(n)$ , is complicated. However, using the notion of “reduction of the structure group” of a principal bundle, we show that the change of basis can be restricted to an orthogonal change of basis, provided one uses realizations in a reduced subbundle. This key observation brings about significant computational benefits. As a technical contribution (possibly of independent interest), we show that several forms of realization balancing available in the control literature have differential-geometric significance, and are, indeed, examples of reducing the structure group from $GL(n)$ to its subgroup of orthogonal matrices $O(n)$ . The alignment distance can be defined for stable and unstable systems, discrete or continuous-time, and stochastic systems.

The paper is a survey with modifications on the research of the so-called Newton–Nelson equation (the equation of motion in Nelson’s stochastic mechanics) on the total space of a bundle in two cases: where the base of the bundle is a Riemannian manifold and the bundle is real and where the base of the bundle is a Lorentz manifold and the bundle is complex. In the latter case, we describe the relations with the equation of motion of the quantum particle in the classical gauge field (the above-mentioned connection). Moreover, a certain second-order ordinary differential equation on the bundle with connection that is interpreted as the equation of motion of the classical particle in the classical gauge field is described.