arXiv research

We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and highe…

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …

Defines new Roe algebras for cylindrical spaces, solving metric curvature problems.

The abstract aims to generalize classical curve concepts to uniquely define complex curves.

We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension $n=4$, and complete characterization for a dense open subset of the space of operators in dimension $4$. We also briefly examine higher-dimentional curvature operators.

The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for t…

Combines higher complex structures with flat connections to link to $W$-algebras.

Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p<q$.

Defines a bundle map for currents on manifolds using higher covariant derivatives.

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochne…

Geometric formalism views optimization algorithms as discrete connections, revealing their algebraic curvature and flatness properties.

In this article we are interested in the differential geometric properties of certain higher direct images of exterior powers of the sheaf of relative differentials twisted with a line bundle. We obtain explicit curvature formulas, especially in case where the said line bundle satisfies a natural curvature assumption. …

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe a…

We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6…

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric. The group $P(M)$ measures the size of the space of positive scalar curvature metr…

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvat…

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is…

Develops connections between operator K-theory and positive scalar curvature.

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external p…

The Bianchi identities for bosonic fluxes in supergravity can receive higher derivative quantum and string corrections, the most well known being that of Heterotic theory $d H = \tfrac{1}{4}α'(\text{tr } F^2 - \text{tr } R^2)$. Less studied are the modifications at order $R^4$ that may arise, for example, in the Bianch…

We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loo…

We solve higher-order morphisms for twisted Courant algebras.

It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order `structure constants…

Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.

Let $J(π)$ be the higher order Jacobi operator. We study algebraic curvature tensors where $J(π)J(π^{\perp})=J(π^{\perp})J(π)$. In the Riemannian setting, we give a complete characterization of such tensors; in the pseudo-Riemannian setting, partial results are available. We present non-trivial geometric examples of Ri…

We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of sections with an $L_{\infty}$-algebra, which we refer to as a homotopy Kirillov …

Functoriality proved for higher rho invariants of elliptic operators.

Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a ma…

New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.

After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra cohomology: a brief introduction 4. Symmetric polynomials and higher order cocycles 5…

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define co…

Expands differential geometry to higher-order infinitesimals.

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…

Study on deformation cohomology for braided commutative structures.

Superintegrable systems on surfaces are classified geometrically.

Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cu…

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such mani…

New minimal surfaces in 4D space derived from parametric equations.

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $Δ$. Given this, we introduce an infinite sequence of higher brackets on the image of the project…

Action of loop groups on Cuntz algebras constructs geometric twists.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

The paper studies properties of stated SL(n)-skein algebras and their centers.

Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.

Recalls and refines the concept of algebraically rectifiable curves.

After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra $Σ$ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming u…

We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally g…

Defines and characterizes operators on Lie ∞-algebras with respect to actions.