Normal frames for general connections on differentiable fibre bundles
Bozhidar Z. Iliev
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24 LaTeX pages. The packages AMS-LaTeX and amsfonts are required. In version 2 some results are generalized and proved under weaker conditions. For other papers on the same topic view the "publication" pages at http://theo.inrne.bas.bg/~bozho/
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizontal path is proved. A necessary and sufficient condition of existence of bundle coordinates normal along injective horizontal mappings is derived.
The paper contains a review on the general connection theory on differentiable fibre bundles. Particular attention is paid to (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic and arbitrary frames is considered.
A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the former.
Transports along path in fibre bundles are axiomatically introduced. Their general functional form and some their simple properties are investigated. The relationships of the transports along paths and lifting of paths are studied.
We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $λ$ of the fibre-wise part of $H$, satisfying a local gap condition, and every $N\in \mathbb{N}$ there exists a subspace of $L^2(M)$ that is invariant under $H$ up to errors of order $\varepsilon^{N+1}$. The dynamical and spectral features of $H$ on this subspace can be described by an effective operator on the fibre-wise $λ$-e…
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way.
The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional conditions, are constructed. In particular, the equivalence between the concepts "connection" and "(axiomatically defined) parallel transport" is established. Some results which are specific for topological and vector bundles are presented.
We introduce the notion of a nonlinear splitting on a fibre bundle as a generalization of an Ehresmann connection. We present its basic properties and we pay attention to the special cases of affine, homogeneous and principal nonlinear splittings. We explain where nonlinear splittings appear in the context of Lagrangian systems and Finsler geometry and we show their relation to Routh symmetry reduction, submersive second-order differential equations and unreduction. We define a curvature map for a nonlinear splitting, and we indicate where this concept appears in the context of nonholonomic sy…
In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result is based on a tensoriality argument and extends to nonlinear connections on fibre bundles a well-known formula for linear connections on vector bundles.
Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$-class. In this note I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for p…
We put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid and groupoid action in the focus of fibre bundle theory in general, and in connection theory in particular.
The general problem for consistency between arbitrary transports along paths in fibre bundles and bundle morphisms between them is formulated and investigated. The special case of one fibre bundle, its morphism and transport along paths acting in it is considered. The consistency between linear transports along paths in a vector bundle and a Hermitian structure in it is studied.
We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. This work unifies the notion of topological fault tolerance with fault tolerance in other kinds of quantum error-correcting codes. In particular, we use fibre bundles with a natural flat projective connection to study the transformation of codewords under unitary fault-tolerant evolutions. We show that the fault-tolerant logical operations are given by the monodromy group for either of two bundles, both of which have flat projective connection…
We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there is a principal fibre bundle isomorphism between pull-backs obtained from homotopic maps. This enables simple proofs of several results on the structure of principal fibre bundles. No new results are obtained---this is simply an accessible presentation of an important idea.
We construct an explicit example of a stable bundle on the twistor space $\mathrm{Tw}(M)$ of a hyperkähler manifold $M$ whose restrictions to all the fibres of the natural twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$ are nonstable. We also describe the relationship between bundles on $\mathrm{Tw}(M)$ that do not have subsheaves of strictly lower rank and bundles that stably restrict to the fibres of $π$, and announce a result whose proof will appear in a forthcoming paper.
The paper introduces the notion of Γ-linear connection \nabla on the 1-jet fibre bundle J^1(T,M), and presents its local components. We also describe the local Ricci and Bianchi identities of $\nabla$.
Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperkähler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $π^{-1}(I)$ of the twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship between stability of $E$ and its fibrewise stability. We verify that the argument of Teleman establishing the Zariski openness of stability and semi-stability in families of bundles applies in the case of the family $\left\{E_I\right\}$. We prove a partial converse to a result of Kaledin and Verbitsky, showing t…
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. Its evolution is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The menti…
Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case and the case when the map's domain is a Cartesian product of two sets. Also considered are the consistency with bundle morphisms and a number of special cases.
We define the secondary invariants L^2- eta and -rho forms for families of generalized Dirac operators on normal coverings of fibre bundles. On the covering family we assume transversally smooth spectral projections, and Novikov--Shubin invariants bigger than 3(dim B+1) to treat the large time asymptotic for general operators. In the particular case of a bundle of spin manifolds, we study the L^2- rho class in relation to the space of positive scalar curvature vertical metrics.
We study $S^1$-bundles and $S^1$-gerbes over differentiable stacks in terms of Lie groupoids, and construct Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvature.
This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way. It starts off with a review of the mathematical tools needed and then moves on to build the objects necessary.
We propose a version of the non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over the space-time. There evolution is governed via (a kind of) a parallel transport in this bundle. Some problems concerning observables are considered. There are derived the equations of motion for the state sections and observables. We show that up to a constant the matrix of the coefficients of the evolution operator (transport) coincides with the matrix of the Hamiltonian of the investigated quan…
This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results may be extended to Finsler spaces and homogeneous nonlinear connections in general.
We describe spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications we study irreducible components of singular Hitchin fibre…
We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural only in this extended sense. We classify fibre bundles with this property, assuming a finite dimensional structure group. This includes all spin structures, but only some generalised spin structures. This classification links the gauge group G to the topology of space-time.
This is a survey paper, starting from the general notion of coordinate bundle taken from Steenrod. Its aim is to provide a motivation for the introduction of cyclic homology (and the closely related noncommutative de Rham cohomology) by Connes, Tsygan and the author. The bridge is made through a generalization of Chern-Weil theory, explained in a very concrete manner through the transition functions of the bundles involved. The final result is a "Chern character" introduced by Connes and the author in the framework of K-theory and functional analysis.
The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given.
Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can be used to model the motion of a particle in a gravitational field. In subsequent papers, Elie Cartan extended these concepts for other types of connections on a manifold: Euclidean, Galilean and Minkows…
The fibre derivative of a bundle map is studied in detail. In the particular case of a real function, several constructions useful to study singular lagrangians are presented. Some applications are given; in particular, a geometric construction useful to solve the Euler-Lagrange equation of a singular lagrangian. A free particle in a curved space-time is studied as an example.
We study the behaviour of Laplace-type operators H on a complex vector bundle E $\rightarrow$ M in the adiabatic limit of the base space. This space is a fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/$ε$. Under a gap condition on the fibre-wise eigenvalues we prove existence of effective operators that provide asymptotics to any order in $ε$ for H (with Dirichlet boundary conditions), on an appropriate almost-invariant subspace of L${}^2$(E).
In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of the structure group of the principle fibre bundle and we also derive a straightforward generalisation of the Reduction Theorem.
Let $(P, Y)$ be a bundle gerbe over a fibre bundle $Y \to M$. We show that if $M$ is simply-connected and the fibres of $Y \to M$ are connected and finite-dimensional then the Dixmier-Douady class of $(P, Y)$ is torsion. This corrects and extends an earlier result of the first author.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regularisable bundles have well defined, both heat-kernel and zeta function regularised volumes. We introduce two notions of minimality (which extend the finite dimensional one) for these orbits, using both heat-kernel and zeta function regularisation methods and show they coincide. For …
We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with Bismut. Under additional conditions on F, the torsion T is related to Igusa's higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles p_i: M -> B with diffeomorphic fibres that are homeomorphic but not …
