arXiv research

Introduces a new geometric framework for non-perturbative BV-theory.

We explain how to translate several recent results in derived algebraic geometry to derived differential geometry. These concern shifted Poisson structures on NQ-manifolds, Lie groupoids, smooth stacks and derived generalisations, and include existence and classification of various deformation quantisations.

Extends differential geometry concepts to manifolds with super tangent bundles.

Develops derived differential geometry for supermanifolds.

New theory proves representability of PDE solutions without complex machinery.

The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions.…

The study sets limits on the complexity of Klein geometries.

This paper proves equivalence between derived manifolds and differential graded manifolds.

Foundations of derived geometry in smooth settings.

Study of nonlinear PDEs using derived geometry and BV formalism.

Quaternionic differential geometry expands geometric concepts using quaternions.

Study derived Lie ∞-groupoids and algebroids in higher differential geometry.

Develops derived differential geometry theory.

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a gen…

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

Introduces new connections in higher geometry.

Derived geometry can be defined as the universal way to adjoin finite homotopical limits to a given category of manifolds compatibly with products and glueing. The point of this paper is to show that a construction closely resembling existing approaches to derived geometry in fact produces a geometry with this universa…

Curved loxodromes on spheres are explained and their ODE derived.

We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the …

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…

This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algeb…

This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general o…

In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of nonc…

String backgrounds and D-branes do not possess the structure of Lorentzian manifolds, but that of manifolds with area metric. Area metric geometry is a true generalization of metric geometry, which in particular may accommodate a B-field. While an area metric does not determine a connection, we identify the appropriate…

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition of the corresponding coboundary rel…

This paper compiles formulas involving differential operators and interior products.

If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation $Φ_f:C^\infty(X)^n\to C^\infty(X)$ acting by $Φ_f:(c_1,\ldots,c_n)\mapsto f(c_1,...,c_n)$, a…

In commutative differential geometry the Frölicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Frölicher-Nijenhuis bracket was developped for universal differential forms of non-commutative algebras, and several applications were given. In this paper this bracke…

The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …

Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

New formula for instantaneous frequency in unbalanced systems.

Recently Berman and Perry constructed a four-dimensional M-theory effective action which manifests SL(5) U-duality. Here we propose an underlying differential geometry of it, under the name `SL(5) U-geometry' which generalizes the ordinary Riemannian geometry in an SL(5) compatible manner. We introduce a `semi-covarian…

This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such…

Study of time-dependent metrics and connections in geometry.

Study of $H$-eigenvalues for complex tensors and their applications in differential geometry.

Introduces infinite-dimensional differential geometry using Bastiani calculus.

Optimization on manifolds is a class of methods for optimization of an objective function, subject to constraints which are smooth, in the sense that the set of points which satisfy the constraints admits the structure of a differentiable manifold. While many optimization problems are of the described form, technicalit…

Geometric theory of integration developed in SDG.

A framework for precontact geometry using pairs of differential forms.

Algebraic geometry replaces manifolds in differential geometry.

The paper simplifies FLRW photon propagators using geometric embeddings.

New geometries derived from symplectic Monge-Ampère structures.

We discuss some differential geometry pertaining to continuum mechanics and the route recently taken by D.N. Arnold, R.S. Falk, and R. Winther in deriving new improved finite element schemes in linear elasticity from constructions in projective geometry.

New concept of coisotropic structures for differentiable stacks defined.

If $X$ is a manifold then the set $C^\infty(X)$ of smooth functions $f:X\to\mathbb R$ is a $C^\infty$-ring, a rich algebraic structure with many operations. $C^\infty$-schemes are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry. They include smooth manifolds, but also…

The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential in…