Smooth Vector Bundle Research Articles

Deformation spaces, rescaled bundles, and the generalized Kirillov formula

In this paper, we construct a smooth vector bundle over the deformation to the normal cone \mathrm{DNC}(V,M) through a rescaling of a vector bundle E\to V , which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham–Dirac operator using the deformation to the normal cone obtained recently by O. Mohsen.

Perturbations of principal submodules in the Drury–Arveson space

We study the geometry in the perturbations of principal submodules in the Drury–Arveson space. We show that the perturbations give rise to smooth vector bundles of Hilbert spaces which are equipped with natural Hermitian connections. We compute the associated parallel transport operators and explore properties of the monodromy.

Intrinsic Riemannian functional data analysis for sparse longitudinal observations

A new framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from sparsely observed Riemannian functional data, while the covariance vector bundle provides a rigorous mathematical foundation for formulating such smoothing problems. The parallel transport and the bundle metric together make it possible to measure fidelity of fit to the covariance function. They also play a critical role in quantifying the quality of estimators for the covariance function. As an illustration, based on the proposed framework, we develop a local linear smoothing estimator for the covariance function, analyze its theoretical properties and provide numerical demonstration via simulated and real data sets. The intrinsic feature of the framework makes it applicable to not only Euclidean submanifolds but also manifolds without a canonical ambient space.

An open mapping theorem for nonlinear operator equations associated with elliptic complexes

Let be the elliptic complex on an n-dimensional smooth closed Riemannian manifold X with the first-order differential operators and smooth vector bundles over X. We consider nonlinear operator equations, associated with the parabolic differential operators , generated by the Laplacians of the complex , in special Bochner–Sobolev functional spaces. We prove that under reasonable assumptions regarding the nonlinear term the Frechét derivative of the induced nonlinear mapping is continuously invertible and the map is open and injective in chosen spaces.

Generic Dynamical Properties of Connections on Vector Bundles

Abstract Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla ^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted conformal Killing tensors (CKTs) are generically trivial when $\dim (M) \geq 3$, answering an open question of Guillarmou–Paternain–Salo–Uhlmann [ 14]. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations, which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla ^{\textrm{End}({\operatorname{{\mathcal{E}}}})}$ on the endomorphism bundle $\textrm{End}({\operatorname{{\mathcal{E}}}})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e., the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely that generically there are no non-trivial subbundles of $\mathcal{E}$ that are preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott–Ruelle resonances in the Anosov case).

On the stability phenomenon of the Navier-Stokes type equations for elliptic complexes

ABSTRACT Let be a Riemannian n-dimensional smooth closed manifold, , be smooth vector bundles over and be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with on the scale of anisotropic Hölder spaces over the layer with finite time T>0. Using the properties of the differentials and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form , where K is a compact continuous operator. It appears that the Fréchet derivative is continuously invertible at every point of each Banach space under consideration and the map is open and injective in the space.

Jacobi geometry and Hamiltonian mechanics: The unit-free approach

We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function [Formula: see text]. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

Elliptic operators and K-homology

If a differential operator D on a smooth Hermitian vector bundle S over a compact manifold M is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If D is also elliptic, then the Hilbert space of square integrable sections of S with the canonical left C(M)-action and the operator χ(D) for χ a normalizing function is a Fredholm module, and its K-homology class is independent of χ. In this expository article, we provide a detailed proof of this fact following the outline in the book “Analytic K-homology” by Higson and Roe.

Invariants of families of flat connections using fiber integration of differential characters

Let $E\to B$ be a smooth vector bundle of rank $n$, and let $P \in I^p(GL(n,\mathbb{R}))$ be a $GL(n,\mathbb{R})$-invariant polynomial of degree $p$ compatible with a universal integral characteristic class $ u \in H^{2p}(BGL(n,\mathbb{R}),\mathbb{Z})$. Cheeger-Simons theory associates a rigid invariant in $H^{2p-1}(B,\mathbb{R}/\mathbb{Z})$ to any flat connection on this bundle. Generalizing this result, Jaya Iyer (\textit{Letters in Mathematical Physics}, 2016, 106 (1) pp. 131-146) constructed maps $H_r(\mathcal{D}(E)) \to H^{2p-r-1}(B,\mathbb{R}/\mathbb{Z})$ for $p>r+1$ where $\mathcal{D}(E)$ is the simplicial set of relatively flat connections, thereby associating invariants to families of flat connections. In this article we construct such maps for the cases $p<r$ and $p>r+1$ using fiber integration of differential characters. We find that for $p>r+1$ case, the invariants constructed here coincide with those obtained by Jaya Iyer, and that in the $p<r$ case the invariants are trivial. We further compare our construction with other results in the literature.

Conjugacy classes of commuting nilpotents

We consider the space M q , n \mathcal {M}_{q,n} of regular q q -tuples of commuting nilpotent endomorphisms of k n k^n modulo simultaneous conjugation. We show that M q , n \mathcal {M}_{q,n} admits a natural homogeneous space structure, and that it is an affine space bundle over P q − 1 {\mathbb {P}}^{q-1} . A closer look at the homogeneous structure reveals that, over C {\mathbb {C}} and with respect to the complex*1pt topology, M q , n \mathcal {M}_{q,n} is a smooth vector bundle over P q − 1 {\mathbb {P}}^{q-1} . We prove that, in this case, M q , n \mathcal {M}_{q,n} is diffeomorphic to a direct sum of twisted tangent bundles. We also prove that M q , n \mathcal {M}_{q,n} possesses a universal property and represents a functor of ideals, and we use it to identify M q , n \mathcal {M}_{q,n} with an open subscheme of a punctual Hilbert scheme. Using a result of A. Iarrobino’s, we show that M q , n → P q − 1 \mathcal {M}_{q,n} \to {\mathbb {P}}^{q-1} is not a vector bundle, hence giving a family of affine space bundles that are not vector bundles.

Heat coefficient [formula omitted] for non minimal Laplace type operators

Given a smooth hermitean vector bundle V of fiber ℂN over a compact Riemannian manifold and ∇ a covariant derivative on V, let P=−(|g|−1∕2∇μ|g|1∕2gμνu∇ν+pμ∇μ+q) be a non minimal Laplace type operator acting on smooth sections of V where u,pν,q are MN(ℂ)-valued functions with u positive and invertible. For any a∈Γ(End(V)), we consider the asymptotics Trae−tP∼t↓0∑r=0∞ar(a,P)t(r−d)∕2 where the coefficients ar(a,P) can be written as an integral of the functions ar(a,P)(x)=tr[a(x)ℛr(x)].This paper revisits the previous computation of ℛ2 by the authors and is mainly devoted to a computation of ℛ4. The results are presented with u-dependent operators which are universal (i.eP-independent) and which act on tensor products of u, pμ, q and their derivatives via (also universal) spectral functions which are fully described.

Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori

Let $P$ be a Laplace type operator acting on a smooth hermitean vector bundle $V$ of fiber $\mathbb{C}^N$ over a compact Riemannian manifold given locally by $P= - [g^{\mu\nu} u(x)\partial_\mu\partial_\nu + v^\nu(x)\partial_\nu + w(x)]$ where $u,\,v^\nu,\,w$ are $M_N(\mathbb{C})$-valued functions with $u(x)$ positive and invertible. For any $a \in \Gamma(\text{End}(V))$, we consider the asymptotics $\text{Tr} (a e^{-tP}) \underset{t \downarrow 0^+}{\sim} \,\sum_{r=0}^\infty a_r(a, P)\,t^{(r-d)/2}$ where the coefficients $a_r(a, P)$ can be written locally as $a_r(a, P)(x) = \text{tr}[a(x) \mathcal{R}_r(x)]$. The computation of $\mathcal{R}_2$ is performed opening the opportunity to calculate the modular scalar curvature for noncommutative tori.

Derivations and Linear Connections on Clifford Vector Bundles

The goal of this article is to introduce a concept of Clifford structures on vector bundles as natural extensions of the standard complex and quaternionic structures, and to determine the derivations and linear connections on smooth Clifford vector bundles compatible with their Clifford structures. The basic object used to get such descriptions is an involution on the space of derivations of a Clifford vector bundle explicitly defined in terms of the specific Clifford structure. That involution is actually derived from an operation called the Clifford conjugation relative to a Clifford structure, which is defined in a purely algebraic setting as an involution on the space of derivations of a Euclidean Clifford algebra. Its definition essentially relies on the use and a complete description of the geometric concept of tangent Clifford structures of a Euclidean Clifford algebra.

Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them

Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in [8]. We give more details regarding the behavior of this operation with respect to gluing, also providing some details omitted from [8], and pay more attention to the relations with the spaces of smooth maps. We also show that a usual smooth vector bundle over a manifold that admits a finite atlas can be seen as a result of a diffeological gluing, and thus deduce that its usual dual bundle is the same as its diffeological dual. We then consider the notion of a pseudo-metric, the fact that it does not always exist (which seems to be related to non-local-triviality condition), construction of an induced pseudo-metric on a pseudo-bundle obtained by gluing, and finally, the relation between the spaces of all pseudo-metrics on the factors of a gluing, and on its result. We conclude by commenting on the induced pseudo-metric on the pseudo-bundle dual to the given one.

Differential Calculus onN-Graded Manifolds

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.

On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles

Let $(M,\sigma)$ be a compact Klein surface of genus $g \geq 2$ and let $E$ be a smooth Hermitian vector bundle on $M$. Let $\tau$ be a Real or Quaternionic structure on $E$ and denote respectively by $\mathcal{G}^{\tau}_{\mathbb{C}}$ and $\mathcal{G}^{\tau}_{E}$ the groups of complex linear and unitary automorphisms of $E$ that commute to $\tau$. In this paper, we study the action of $\mathcal{G}^{\tau}_{\mathbb{C}}$ on the space $\mathcal{A}^{\tau}_{E}$ of $\tau$-compatible unitary connections on $E$ and show that the closure of a semi-stable $\mathcal{G}^{\tau}_{\mathbb{C}}$-orbit contains a unique $\mathcal{G}^{\tau}_{E}$-orbit of projectively flat connections. We then use this invariant-theoretic perspective to prove a version of the Narasimhan–Seshadri correspondence in this context: $S$-equivalence classes of semi-stable Real and Quaternionic vector bundes are in bijective correspondence with equivalence classes of certain appropriate representations of orbifold fundamental groups of Real Seifert manifolds over the Klein surface $(M,\sigma)$.

On smooth Lie algebra bundles

The existence of a class of smooth Lie algebra bundles is proved and examples are given. Further we construct smooth vector bundles which have the structure of non trivial smooth Lie algebra bundles.

∂¯-tangential invariants of certain vector bundles over complex foliations

The Dolbeault cohomology plays an important role in the study of some ∂ ¯ -invariants of complex and holomorphic vector bundles as ∂ ¯ -Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for F -holomorphic vector bundles and ∂ ¯ -tangential Chern classes for tangentially smooth vector bundles over a manifold M endowed with a complex foliation F . Also, ∂ ¯ -tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces.

Yang–Mills theory and jumping curves

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

The Kato Square Root Problem on Vector Bundles with Generalised Bounded Geometry

We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.