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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 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.

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

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.

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

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

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r&lt, k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

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).

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.

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.

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

We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $\Gamma$-structured elliptic curves with marked points, where $\Gamma=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$, and $M,N\geq1$ are two integers. It restricts to a flat connection on $\Gamma$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical $r$-matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras., Comment: 50 pages. Main changes in v3 (final version): updated biblio (unused refs deleted), shift in numbering in Section 3 (to make it agree with the published version), and minor change in glossary of notation (to make it consistent with the body of the text) Also available at https://rdcu.be/b822g

Let Y be a smooth projective surface defined over an algebraically closed field k with char k ≠ 2, and let τ: X → Y be a double covering branched along a smooth divisor. We show that IX is stable with respect to τ*H if the tangent bundle IY is semi-stable with respect to some ample line bundle H on Y.

A discrete dynamical system generated by a diffeomorphism f on a compact manifold is considered. The Morse spectrum is the limit set of Lyapunov exponents of periodic pseudotrajectories. It is proved that the Morse spectrum coincides with the set of averagings of the function ϕ(x, e) = ln|Df(x)e| over the invariant measures of the mapping induced by the differential Df on the projective bundle.

This article discusses necessary and sufficient conditions for the existence of global attractors for transformation semigroups on principal bundles. Since the global attractor is a compact set, the discussion involves the compactness of the fibres. A compact structure group is a necessary condition for the existence of the global attractor. In specific situations, the global attractor exists if the structure group is compact.

We consider a set of gauge-theoretic equations on closed oriented four-manifolds, which was introduced by Vafa and Witten. The equations involve a triple consisting of a connection and extra fields associated to a principal bundle over a closed oriented four-manifold. They are similar to Hitchin's equations over compact Riemann surfaces, and as part of the resemblance, there is no $L^2$-bound on the curvature without an $L^2$-bound on the extra fields. In this article, however, we observe that under the particular circumstance where the curvature does not become concentrated and the limiting connection is not locally reducible, one obtains an $L^2$-bound on the extra fields.

We build a concrete pseudo-Anosov bundle on which there is no transversely projective foliation without compact leaf that is not transversely affine.

Let $R$ be a ring spectrum and $ E\to X$ an $R$-module bundle of rank $n$. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, $hAut^R(E)$. This will generalize the result regarding $R$-line bundles previously proven by the authors. The main application is the calculation of the homotopy type of $BGL_n(End ((L))$ where $L \to X$ is any $R$-line bundle, and $End (L)$ is the ring spectrum of endomorphisms. In the case when such a bundle is the fiberwise suspension spectrum of a principal bundle over a manifold, $G \to P \to M$, this leads to a description of the $K$-theory of the string topology spectrum in terms of the mapping space from $M$ to $BGL (\Sigma^\infty (G_+))$., Comment: 10 pages

Two holomorphic fields of geometric objects on a transverse Weil bundle are called equivalent if there exists a holomorphic diffeomorphism of this bundle onto itself which induces the identity transformation of the base manifold and maps one of these fields into the other. In terms of transverse Lie jets, we establish necessary and sufficient conditions for a holomorphic field of geometric objects on a transverse Weil bundle to be equivalent to the prolongation of a field of foliated geometric objects given on the base manifold. As an example, we consider a holomorphic linear connection on a transverse bundle.

Let $G$ be a real reductive Lie group, and $H^{\mathbb{C}}$ the complexification of its maximal compact subgroup $H\subset G$. We consider classes of semistable $G$-Higgs bundles over a Riemann surface $X$ of genus $g\geq2$ whose underlying $H^{\mathbb{C}}$-principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of $G$-Higgs bundles over $X$ to the moduli space of $H^{\mathbb{C}}$-bundles over $X$, in contrast with the situation when $g=1$, and to show reducibility of the nilpotent cone of the moduli space of $G$-Higgs bundles, for $G$ complex., 14 pages

We present some lifts (associated to a product preserving gauge bundle functor on vector bundles) of sections of the dual bundle of a vector bundle, some derivations and linear connections on vector bundles.

We study the properties of the normal bundle defined by the bundle of the G-orbits of the action of a semisimple Lie group G on a pseudo Riemannian manifold M, as a consequence we obtain that the foliation induced by the normal bundle is integrable and totally geodesic.

In this paper we develop a theory of Lyapunov stability for generalized flows on principal and associated bundles. We present a study of Lyapunov stability and attraction in the total space of a principal bundle by means of the action of the structure group.We also relate limit sets, prolongations, prolongational limit sets, attracting sets and stable sets in the total space of an associated bundle to the corresponding concepts in the fibers.

This paper is devoted to rigidity of smooth bundles which are equipped with fiberwise geometric or dynamical structure. We show that the fiberwise associated sphere bundle to a bundle whose leaves are equipped with (continuously varying) metrics of negative curvature is a topologically trivial bundle when either the base space is simply connected or, more generally, when the bundle is fiber homotopically trivial. We present two very different proofs of this result: a geometric proof and a dynamical proof. We also establish a number of rigidity results for bundles which are equipped with fiberwise Anosov dynamical systems. Finally, we present several examples which show that our results are sharp in certain ways or illustrate necessity of various assumptions.

We describe a geometric theory of Virasoro constraints in generalized Drinfeld-Sokolov hierarchies. Solutions of Drinfeld-Sokolov hierarchies are succinctly described by giving a principal bundle on a complex curve together with the data of a Higgs field near infinity. String solutions for these hierarchies are defined as points having a big stabilizer under a certain Lie algebra action. We characterize principal bundles coming from string solutions as those possessing connections compatible with the Higgs field near infinity. We show that tau-functions of string solutions satisfy second-order differential equations generalizing the Virasoro constraints of 2d quantum gravity., 28 pages

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of two-dimensional sigma models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a `formal loop group action' on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said `formal loop group actions' and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of `formal loop spaces'. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus., 45 pages (minor changes, added references to related works)

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly K\"ahler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-K\"ahler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion., Comment: 42 pages; thoroughly revised version, including a simpler definition of the geometry with parallel curvature determined by a geometry with parallel skew-symmetric torsion, and an appendix discussing 3-(\alpha,\delta)-Sasakian structures in our framework

In this paper we import the theory of “Calabi–Yau” algebras and categories from symplectic topology and topological field theories, to the setting of spectra in stable homotopy theory. Twistings in this theory will be particularly important. There will be two types of Calabi–Yau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. The main application of twisted compact Calabi–Yau ring spectra that we will study is to describe, prove, and explain a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas and Sullivan (1999) and the Lie group string topology of Chataur and Menichi (2012). This will extend and generalize work of Gruher (2007). Then, generalizing work of Cohen and Jones (2017), we show how the gauge group of the principal bundle acts on this compact Calabi–Yau structure, and we compute some explicit examples. We then extend the notion of the Calabi–Yau structure to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles over the loop space, [math] , have this structure. In the case when [math] is a sphere, we will use these twisted smooth Calabi–Yau ring spectra to study Lagrangian immersions of the sphere into its cotangent bundle. We recast the work of Abouzaid and Kragh (2016) to show that the topological Hochschild homology of the Thom ring spectrum induced by the [math] -principle classifying map of the Lagrangian immersion detects whether that immersion can be Lagrangian isotopic to an embedding. We then compute some examples. Finally, we interpret these Calabi–Yau structures directly in terms of topological Hochschild homology and cohomology.

The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to \({\mathbb{P}_1}\) of fixed numerical type with simple branch points, are extensively studied in the literature.We apply deformation theory, and present a study of the Kahler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil–Petersson Kahler form on the Hurwitz space. This form turns out to be the curvature of a Quillen metric on a determinant line bundle. Alternatively, the generalized Weil–Petersson Kahler form can be characterized as the curvature form of the hermitian metric on the Deligne pairing of the relative canonical line bundle and the pull back of the anti-canonical line bundle on \({\mathbb{P}_1}\). Replacing the projective line by an arbitrary but fixed curve Y, we arrive at a generalized Hurwitz space with similar properties. The determinant line bundle extends to a compactification of the (generalized) Hurwitz space as a line bundle, and the Quillen metric yields a singular hermitian metric on the compactification so that a power of the determinant line bundle provides an embedding of the Hurwitz space in a projective space.

We present a multisymplectic formulation of the Yang- Mills equations. The connections are represented by normalized equivariant 1-forms on the total space of a principal bundle, with values in a Lie alge- bra. Within the multisymplectic framework we realize that, under reasonable hypotheses, it is not necessary to assume the equivariance condition a priori, since this condition is a consequence of the dynamical equations. The motivation of the following work was at first to provide a Hamiltonian formulation of the Yang-Mills system of equations which would be as much covariant as possible. This means that we look for a formulation which does not depend on choices of space-time coordinates nor on the trivialization of the principal bundle. Among all possible frameworks (covariant phase space, etc.) we favor the multisymplectic formalism which takes automatically into account the locality of fields theories. Following this approach we have been led to discover a new variational formulation of the Yang-Mills equations with nice geometrical features. The origin of the multisymplectic formalism goes back to the discovery by V. Volterra in 1890 (28, 28) of generalizations of the Hamilton equations for variational problems with several variables. These ideas were first developped mainly around 1930 (4, 7, 30, 24) and in 1950 (6). After 1968 this theory was geometrized in a way analogous to the construction of symplectic geometry by several mathematical physicists (10, 12, 23) and in particular by a group

We give a criterion for filtered vector bundles over curves to admit a filtration preserving meromorphic connection that induces a given meromorphic connection on the corresponding graded bundle.

In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W r FX of FX with structure group G nr. For a structure group of J r FX we find a suitable subgroup of G nr. We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r FX corresponds with the standard actions of differential groups on tensor spaces.

We study operators associated with a bundle with compact base and fiber. We construct an algebra of such operators. For elliptic elements of the algebra, we prove the finiteness theorem and derive an index formula.

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover CP^1 \times CP^1 \to CP^2, is naturally a co-Higgs bundle, with the twisted endomorphism, or "Higgs field", also descending from the double cover. Allowing the branch conic to vary, we find that Schwarzenberger bundles give rise to an 8-dimensional moduli space of co-Higgs bundles. After studying the deformation theory for co-Higgs bundles on complex manifolds, we conclude that a co-Higgs bundle arising from a Schwarzenberger bundle with nonzero Higgs field is rigid, in the sense that a nearby deformation is again Schwarzenberger., Comment: 25 pages, 2 tables

The space Π of centered m-planes is considered in projective space P n . A principal bundle is associated with the space Π and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space Π . Semi-normalized spaces Π 1 , Π 2 and normalized space Π 1 , 2 are investigated. By virtue of the Cartan–Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space Π to the normalized space Π 1 , 2 .

Topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable. Some connection is established between finite dimensionality or separability of the stalks of a bundle and the analogous properties of the stalks of the ample hull of the bundle. We obtain a new criterion for existence of a dual bundle.

The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S2d for 2 ≤ d ≤ p − 1 and n ≤ 2p − 1, we give a concrete classification of their p-local homotopy types.

For a product preserving gauge bundle functor on vector bundles, we present some lifts of smooth functions that are constant or linear on fibers, and some lifts of projectable vector fields that are vector bundle morphisms.

We describe some general geometric properties of the fiber product preserving bundle functors. Special attention is paid to the vertical Weil bundles. We discuss namely the flow natural maps and the functorial prolongation of connections.

Earlier, one of the authors have introduced the concept of generalized bundle spaces [2, 7]. This term refers to a structure similar to the principal bundle in which the group acting in a leaf depends on the leaf. In the paper, we develop this idea as applied to G -structures and find the structure equations of a K - generalized G -structure.

In this paper we present a bundle of pairs of volume forms V2. We describe horizontal lift of a tensor of type (1; 1) and we show that horizontal lift of an almost complex structure on a manifold M is an almost complex structure on the bundle V2. Next we give conditions under which the almost complex structure on V 2 is integrable. In the second part we find horizontal lift of vector fields, tensorfields of type (0; 2) and (2; 0), Riemannian metrics and we determine a family of a t-connections on the bundle of pairs of volume forms. At the end, we consider some properties of the horizontally lifted vector fields and certain infinitesimal transformations.