arXiv research

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 …

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.

Given a generic Lagrangian system, its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condi…

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.

On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is proposed and a method of finding some of satisfied by them identities is given in the c…

As the fourth paper of our series of papers concerned with axiomatic differential geometry, this paper is devoted to the general Jacobi identity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.

Study local commutation relation on almost complex manifolds.

A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-rec…

We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …

Abstract: Study Kähler identities on almost complex manifolds.

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framewor…

Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.

We investigate the problems of identity and closeness testing over a discrete population from random samples. Our goal is to develop efficient testers while guaranteeing Differential Privacy to the individuals of the population. We describe an approach that yields sample-efficient differentially private testers for the…

The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.

The paper proves positivity of characteristic forms for certain vector bundles.

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamen…

We give a new proof of the generalized Minkowski identities relating the higher degree mean curvatures of orientable closed hypersurfaces immersed in a given constant sectional curvature manifold. Our methods rely on a fundamental differential system of Riemannian geometry introduced by the author. We develop the notio…

We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.

Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and in particular the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new op…

The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential …

This paper introduces a probability density estimator based on Green's function identities. A density model is constructed under the sole assumption that the probability density is differentiable. The method is implemented as a binary likelihood estimator for classification purposes, so issues such as mis-modeling and …

Connections are an important tool of differential geometry. This paper investigates their definition and structure in the abstract setting of tangent categories. At this level of abstraction we derive several classically important results about connections, including the Bianchi identities, identities for curvature and…

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

Study on opers over complex manifolds of dimension one.

A method makes particle filters differentiable without altering their forward pass.

The study examines how brokers' identity affects their trading strategies on the Toronto Stock Exchange.

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore,…

Study rigidity on CR Yamabe equation on Sasakian manifolds.

No time-periodic Majorana fermions found in Kerr-Newman spacetimes with nontrivial charge.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

The study classifies differentiable structures on a 'Y' shape manifold.

We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of principal series representations of the Lie group $O(n+1,1)$. Further, we provide…

We extend to a scheme-theoretic context the notion of a combinatorial differential form, due to A.Kock in the framework of synthetic differential geometry. We show that group-valued combinatorial forms on a scheme may be identified, under very general hypotheses, with traditional Lie algebra-valued differential forms, …

This paper compiles formulas involving differential operators and interior products.

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor $\mathcal{M}_{ab}$ which solves the…

New Spencer complexes for Lie groupoids developed.

The Killing operator on a Riemannian manifold is a linear differential operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as aris…

Smooth contact mappings in a flat (2,3,5)-distribution are shown to be smoother.

New formulas compare total mean curvatures of nested hypersurfaces.

We present in this article a family of new combinatorial identities via purely differential/complex geometry methods, which include as a speical case a unified and explicit formula for Chern numbers of all complex flag manifolds. Our strategy is to construct concrete circle actions with isolated fixed points on these m…

We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction. (Based on a talk at XXII Workshop on Geometric Methods in Physi…

New privacy framework tailored to specific data distributions.

The (twice-contracted) second Bianchi identity is a differential curvature identity that holds on any smooth manifold with a metric. In the case when such a metric is Lorentzian and solves Einstein's equations with an (in this case inevitably smooth) energy-momentum-stress tensor of a "matter field" as the source of sp…

Let M be a compact Riemannian manifold equipped with a parallel differential form ω. We prove a version of Kaehler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient $(H^*_c(M), d)$, called {\bf the pseudocohomology} of M. When M is compact and Kaehler…

We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…

If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the ident…

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps …

Constructing $3$-Lie algebroids via connections