Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal O_{X_k}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically -- in a sense to be specified later) only on the curvature of $V$ and on the structure of the fibration $X_k\to X$. When $X$ is a surface and $V=T_X$, we give explicit formulae to write down the above curvature as a product of matrices. As an application, we obtain a new proof of the existence of global invariant jet differentials vanishing on an ample divisor, for $X$ a minimal surface of general type whose Chern classes satisfy certain inequalities, without using a strong vanishing theorem of Bogomolov.
In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by primary and secondary structures belong to the same atlas. We prove that double jet bundles can be considered as a quotient of second order jet bundle. We show that there exists a natural involution that interchanges between primary and secondary vector bundle structures on double…
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geometry of higher-order total differentials, to appear in Cahiers de Topologie et Géométrie Différentielle Catégoriques] we have introduced another synthetic approach to jet bundles in terms of infinitesimal spaces D_{n}'s, and have compared it with the former approach both in the …
We identify and study a matrix algebra consisting of Pascal-type matrices. The generator of the matrix algebra is shown to well define a canonical bundle map, called the Pascal map on jet bundles, and we use it to give an intrinsic definition of point-wise contact between Hermitian vector bundles in terms of unitary equivalence of the Pascal maps.
We discuss the Morse estimates for the curvature of several metrics on Semple weighted projective bundle over a projective variety. Following Demailly works on holomorphic Morse inequalities we show an analogue of his results along the Green-Griffiths conjecture for invariant jets.
The jet bundle $J^kG$ of $k$-jets of curves in a Lie group $G$ has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie group extension $\mathfrak{g}\to J^{k}G\to J^{k-1}G$.
Using Caputo fractional derivative of order $α$ we build the fractional jet bundle of order $α$ and its main geometrical structures. Defined on that bundle, some fractional dynamical systems with applications to economics are studied.
Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle is passed down to Jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure between the groups of automorphisms of related Weil algebras
Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
We study natural lifting operations from a bundle E over R to the dual bundle of its first-jet bundle. The main purpose is to define a complete lift of a type (1,1) tensor field on E and to understand all features of its construction. Various other lifting operations of tensorial objects on E are needed for that purpose. We prove that the complete lift of a type (1,1) tensor with vanishing Nijenhuis torsion gives rise to a Poisson-Nijenhuis structure on the dual of the first-jet bundle, and discuss in detail how the construction of associated Darboux-Nijenhuis coordinates can be carried out.
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of t…
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 $\nabla$.
A family of holomorphic vector bundles is constructed on a complex manifold $X$. The space of the holomorphic sections of these bundles are calculated in certain cases. As an application, if $X$ is an $N$-dimensional compact Kähler manifold with holonomy group $SU(N)$, the space of holomorphic vector fields on its jet scheme $J_m(X)$ is calculated. We also prove that the space of the global sections of the chiral de Rham complex of a K3 surface is the simple $N=4$ superconformal vertex algebra with central charge $6$.
We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.
The goal of this work is to establish a proof of the Gromov convergence in Hoelder spaces for curves with a totally real boundary condition following the original geometric idea of Gromov. We use a local reflection principle in neighbourhoods of the totally real submanifold as developed by Ivashkovich and Shevchishin and existence results for special connections on spaces of jet bundles to obtain higher regularity and Gromov-Schwarz estimates along the boundary.
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems of PDEs, including schemes based on differential algebra and geometric approaches including the theory of exterior differential systems and the so-called ``formal theory'' built on the jet bundle formalism. This paper is a brief introduction to jet bundles focusing on the comple…
A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a nonlinear system of partial differential equations (PDE). This leads to the concept of formal integrability. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, modelled as a section of a natural bundle. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of the…
This paper is the sequel to our previous paper (Differetial Geometry of Microlinear Frolicher spaces IV-1), where three approaches to jet bundles are presented and compared. The first objective in this paper is to give the affine bundle theorem for the second and third approaches to jet bundles. The second objective is to deal with the three approaches to jet bundles in the context where coordinates are available. In this context all the three approaches are shown to be equivalent.
The aim of this paper is to open the problem of construction of a nonlinear connection $Γ=(M^{(i)}_{(α)β}, N^{(i)}_{(α)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(α)(β)}_{(i)(j)}=h^{αβ}g_{ij}$, possibly provided by multi-time dependent quadratic Lagrangians coming from various branches of theoretical physics: bosonic strings theory, electrodynamics or elasticity.
We review our construction of star-products on Poisson manifolds and discuss some examples. In particular, we work out the relation with Fedosov's original construction in the symplectic case.
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 $EG \to BG$ is a $D$-numerable principal bundle, and prove that it classifies all $D$-numerable principal bundles over …
We express the first jet bundle of curves in Euclidean space as homogeneous spaces associated to a Galilean-type group. Certain Cartan connections on a manifold with values in the Lie algebra of the Galilean group are characterized as geometries associated to systems of second order ordinary differential equations. We show these Cartan connections admit a form of normal coordinates, and that in these normal coordinates the geodesic equations of the connection are second order ordinary differential equations. We then classify such connections by some of their torsions, extending a classical the…
In this paper we propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total m…
Let K be a a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.
The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we give a basis of constraint functions in the Lagrangian and Hamiltonian sides. An additional description of constraints is also given considering at the same time compatibility, stability and second-order condition problems. Finally, a classification of the constraints in first and second class is obtained using a cosymplectic geometry setting. Using the second …
The geometric Lagrangian theory (of arbitrary order) is based on the analysis of some basic mathematical objects such as: the contact ideal, the (exact) variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin forms, etc. In this paper we give new and much simpler proofs for the whole theory using Fock space methods. Using these results we give the most general expression for a variationally trivial Lagrangian and the generic expression for a locally variational system of partial differential equation.
We obtain the cohomology of the variational bicomplex on the infinite order jet space of a smooth fiber bundle in the class of exterior forms of finite jet order. This provides a solution of the global inverse problem of the calculus of variations of finite order on fiber 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 an…
A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentiall…
A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth free loop space of the manifold. First, a Clifford algebra bundle over the loop space is constructed from the vector bundle. A polarization class bundle is constructed, choosing continuously over each point of the loop space a polarization class of Lagrangian subspaces of the comple…
In this paper, we study a new operation named pushforward on diffeological vector pseudo-bundles, which is left adjoint to the pullback. We show how to pushforward projective diffeological vector pseudo-bundles to get projective diffeological vector spaces, producing many concrete new examples, together with application to smooth splittings of some projective diffeological vector spaces related to geometry. This brings new objects to diffeology from classical vector bundle theory.
We show that for every nonelementary representation of a surface group into $SL(2,{\mathbb C})$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.
A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.
There is a natural filtration on the space of degree-$k$ homogeneous polynomials in $n$ independent variables with coefficients in the algebra of smooth functions on the Grassmannian $\mathrm{Gr}(n,s)$, determined by the tautological bundle. In this paper we show that the space of $s$-dimensional integral elements of a Cartan plane on $J^{k-1}(E,n)$, with $\mathrm{dim}\, E=n+m$, has an affine bundle structure modeled by the the so-obtained bundles over $\mathrm{Gr}(n,s)$, and we study a natural distribution associated with it. As an example, we show that a third-order nonlinear PDE of Monge-Am…
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. We prove fundamental results about these tangent spaces, compute them in many examples, and observe that while they agree for smooth manifolds and many of the exam…
In this review paper we discuss the different interpretations of the concept of connection in a fiber bundle and in a jet bundle, and relate it with first and second-order systems of partial differential equations (PDE's) and multivector fields. As particular cases we analyze the concepts of linear connections and connections in a manifold.
The consequences of the splitting of higher jet sequences of vector bundles on a compact complex (Kaehler) manifold are studied, proving an infinitesimal convers to a result of I.Biswas.
We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.
Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic…
We prove vanishing results for the generalized Miller-Morita-Mumford classes of some smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. We also find, under some extra conditions, that the vertical tangent bundle is topologically rigid.
We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not support strictly 1/4-pinched positively curved Riemannian metrics on their fibers.
We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of…
Given a smooth complex projective variety $M$ and a smooth closed curve $X \subset M$ such that the homomorphism of fundamental groups $π_1(X) \rightarrow π_1(M)$ is surjective, we study the restriction map of Higgs bundles, namely from Higgs bundles on $M$ to those on $X$. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on $M$ and $X$. We also consider the set-up where a finite group is acting on $M$ via holomorphic automorphisms or anti-holomorphic involutions, and the curve $X$ is preserved …
We provide a general, homotopy-theoretic definition of string group models within an $\infty$-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is a presheaf of $\infty$-groupoids on the category of cartesian spaces. The key to our definition and construction of smooth string group models is a version of the singular complex functor, which assigns to a smooth space an underlying ordinary space. We provide new characterisations of principal $\infty$-bundles and group extensions in $\infty$-topoi, building on work of Nikolaus, Schreiber, and S…
Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an specific class of local coordinate transformations is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit $k$-jet approximations and are invariant under arbitr…
