arXiv subjects

Keegan J. Flood

Publications and source records attributed to Keegan J. Flood.

7 recordsLinked to original sources

arXiv2026

Canonical differential calculi via functorial geometrization

Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$ such that $\mathcal{E}$ admits a canonical functor to the category of first order differential calculi in $\mathcal{V}$. Generalizing the procedure of extending a first order differential calculus to its maximal prolongation to this setting, we obtain a canonical functor from $\mathcal{E}$ to the category of differential calculi in $\mathcal{V}$. This yields a simultaneous generalization of the de Rham complex on $C^{\infty}$-rings, the Kähler differentials on commutative algebras, and the universal differential calculus on associative algebras. As a consequence, such categories $\mathcal{E}$ admit natural analogues of the notions of smooth map and diffeomorphism, as well as a functorial de Rham theory. Moreover, whenever two such faithful isofibrations to $\operatorname{Mon}(\mathcal{V})$ factor suitably, their corresponding de Rham functors are related via a comparison map. Developing this theory requires first extending the noncommutative geometry formalism of differential calculi from associative algebras to the setting of monoids internal to monoidal additive categories.

math.CT↗

arXiv2025

Flat extensions of principal connections and the Chern-Simons $3$-form

We introduce the notion of a flat extension of a connection $θ$ on a principal bundle. Roughly speaking, $θ$ admits a flat extension if it arises as the pull-back of a component of a Maurer-Cartan form. For trivial bundles over closed oriented $3$-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern-Simons invariant associated with $θ$. As an application, we recover the obstruction of Chern-Simons for the existence of a conformal immersion of a Riemannian $3$-manifold into Euclidean $4$-space. In addition, we obtain corresponding statements for a Lorentzian $3$-manifold, as well as a global obstruction for the existence of an equiaffine immersion into $\mathbb{R}^4$ of a $3$-manifold that is equipped with a torsion-free connection preserving a volume form.

math.DG↗

arXiv2025

Higher Order Connections in Noncommutative Geometry

We prove that, in the setting of noncommutative differential geometry, a system of higher order connections is equivalent to a suitable generalization of the notion of phase space quantization (in the sense of Moyal star products on the symbol algebra). Moreover, we show that higher order connections are equivalent to (ordinary) connections on jet modules. This involves introducing the notion of natural linear differential operator, as well as an important family of examples of such operators, namely the Spencer operators, generalizing their corresponding classical analogues. Spencer operators form the building blocks of this theory by providing a method of converting between the different manifestations of higher order connections. A system of such higher order connections then gives a quantization, by which we mean a splitting of the quotient projection that defines symbols as classes of differential operators up to differential operators of lower order. This yields a notion of total symbol and of star product, the latter of which corresponds, when restricted to the classical setting, to phase space quantization in the context of quantum mechanics. In this interpretation, we allow the analogues of the position coordinates to form a possibly noncommutative algebra.

math.QA↗

arXiv2025

Symbols in Noncommutative Geometry

In this paper we prove that the classical Lie bracket of vector fields can be generalized to the noncommutative setting by antisymmetrizing (in a suitable noncommutative sense) their compositions. This construction turns out to depend on the representability of linear differential operators, as it relies on the interpretation of vector fields as differential operators. In particular we provide necessary and sufficient conditions for (noncommutative) jet modules to be representing objects for differential operators. Furthermore, the primary ingredient for guaranteeing the closure of a bracket operation is a treatment of symbols, which classically represent, in an intrinsic way, the highest-order term of a differential operator. Thus, we provide an extensive theory of symbols herein.

math.QA↗

arXiv2025

Jet Functors in Noncommutative Geometry

In this article we construct three infinite families of endofunctors $J_d^{(n)}$, $J_d^{[n]}$, and $J_d^n$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring $\mathbb{k}$, equipped with an exterior algebra $Ω^\bullet_d$. We prove that these functors generalize the corresponding classical notions of nonholonomic, semiholonomic, and holonomic jet functors, respectively. Our functors come equipped with natural transformations from the identity functor to the corresponding jet functors, which play the rôles of the classical prolongation maps. This allows us to define the notion of linear differential operators with respect to $Ω^{\bullet}_d$. We show that if $Ω^1_d$ is flat as a right $A$-module, the semiholonomic jet functor satisfies the semiholonomic jet exact sequence $0 \rightarrow \bigotimes^n_A Ω^1_d \rightarrow J^{[n]}_d\rightarrow J^{[n-1]}_d \rightarrow 0$. Moreover, we construct a functor of symmetric (in a suitable noncommutative sense) forms $S^n_d$ associated to $Ω^\bullet_d$, and proceed to introduce the corresponding noncommutative analogue of the Spencer $δ$-complex. We give necessary and sufficient conditions under which the holonomic jet functor $J_d^n$ satisfies the (holonomic) jet exact sequence, $0\rightarrow S^n_d \rightarrow J_d^n \rightarrow J_d^{n-1} \rightarrow 0$. In particular, for $n=1$ the sequence is always exact, for $n=2$ it is exact for $Ω^1_d$ flat as a right $A$-module, and for $n\ge 3$, it is sufficient to have $Ω^1_d$, $Ω^2_d$, and $Ω^3_d$ flat as right $A$-modules and the vanishing of the Spencer $δ$-cohomology $H^{\bullet,2}_{δ_d}$.

math.QA↗

arXiv2021

Geometry of solutions to the c-projective metrizability equation

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a non-degenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of $\mathbb{CP}^m$. We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric.

math.DG↗

arXiv2018

Metrics in projective differential geometry: the geometry of solutions to the metrizability equation

Pseudo-Riemannian metrics with Levi-Civita connection in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the maximal rank solutions of a certain overdetermined projectively invariant differential equation often called the metrizability equation. Dropping this rank assumption we study the solutions to this equation given less restrictive generic conditions on its prolonged system. In this setting we find that the solution stratifies the manifold according to the strict signature (pointwise) of the solution and does this in way that locally generalizes the stratification of a model, where the model is, in each case, a corresponding Lie group orbit decomposition of the sphere. Thus the solutions give curved generalizations of such embedded orbit structures. We describe the smooth nature of the strata and determine the geometries of each of the different strata types; this includes a metric on the open strata that becomes singular at the strata boundary, with the latter a type of projective infinity for the given metric. The approach reveals and exploits interesting highly non-linear relationships between different linear geometric partial differential equations. Apart from their direct significance, the results show that, for the metrizability equation, strong results arising for so-called normal BGG solutions, and the corresponding projective holonomy reduction, extend to a far wider class of solutions. The work also provides new results for the projective compactification of scalar-flat metrics.

math.DG↗