We characterize the three-dimensional Riemannian manifolds equipped with a semi-symmetric metric ρ-connection under the assumption that the Riemannian metric is a Yamabe soliton. It is shown that a three-dimensional Riemannian manifold endowed with a semi-symmetric ρ-connection, whose metric is Yamabe soliton, is a manifold of constant sectional curvature −1 and the Yamabe soliton is expanding with

We study geodesics in noncommutative geometry by means of bimodule connections and completely positive maps using the Kasparov, Stinespring, Gel’fand, Naĭmark & Segal (KSGNS) construction. This is motivated from classical geometry, and we also consider examples on the algebras M2(ℂ) and C(Zn), though restricting to classical time t∈R. On the way we have to consider the reality of a noncommutative vector

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate G-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quantities invariant up to a generalized change of variables). One family of invariants

In this paper, non-lightlike ruled surfaces of constant slope in Minkowski 3-space with a non-lightlike base curve c(s)=∫αt+βn+γbds, where t, n, b are tangent, principal normal and binormal vectors of an arbitrary timelike curve Γ(s) are investigated. Non-lightlike ruled surfaces of constant slope parallel to a tangent of a timelike general helix and normal of timelike slant helix are studied. It is

In this article the framework created by Cartan to produce local differential invariants for submanifolds of homogeneous spaces is applied to classify all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds of the nearly Kähler manifold of full flags in ℂ3.

We apply the wall-crossing structure formalism of Kontsevich and Soibelman to Seiberg–Witten integrable systems associated to pure SU(3). This gives an algorithm for computing the Donaldson–Thomas invariants, which correspond to BPS degeneracy of the corresponding BPS states in physics. The main ingredients of this algorithm are the use of split attractor flows and Kontsevich–Soibelman wall-crossing

Let M denote a complete, simply connected n-dimensional Riemannian manifold with sectional curvature KM≤k and Ricci curvature RicM≥(n−1)K, where k,K∈R. Then for a bounded domain Ω⊂M with smooth boundary, we prove that the first nonzero Neumann eigenvalue μ1(Ω)≤Cμ1(Bk(R)). Here Bk(R) is a geodesic ball of radius R>0 in the simply connected space form Mk such that vol(Ω) = vol(Bk(R)), and C is a constant

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m∕S(U2⋅Um) from the equation of Gauss and derive a new formula for the Ricci tensor of M in SU2,m∕S(U2⋅Um). Finally we give a complete classification for Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m∕S(U2⋅Um) with harmonic

In this paper we perform a local regularization of singularities in the restricted (n+1)−body problem on a two-dimensional hyperbolic space using different coordinate and time transformations. The motion of the primary particles is known and their orbit is given by a hyperbolic relative equilibrium. We show some numerical examples of the orbit of the massless particle.

Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.

The geometry of the total space of a principal bundle with regard to the action of the bundle’s structure group is elegantly described by the bundle’s operation, a collection of derivations consisting of the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and satisfying the six Cartan relations. Connections and gauge transformations are defined by the way

It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle’s structure group is codified in the bundle’s operation, a collection of derivations comprising the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and obeying the six Cartan relations. In particular, connections and gauge transformations

In this paper we give an explicit parametrisation of the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere. As Hitchin proved, a harmonic map of a 2-torus is described by its spectral data, which consists of a hyperelliptic curve together with a pair of differentials and a line bundle. The space of spectral data is naturally a fibre bundle over the space of spectral curves. For

In this paper, by defining a new vocabulary for Jacobi theta functions, namely “spectral theta functions”, we show that how remarkably the N=2 superconformal characters are extracted by imposing a residue calculation on sl̂(2|1) affine Lie superalgebra characters in their non-unitary (non-trivial) cases. Therefore, to have more versatility in conformal quantum field theory context, we finalize our

The radial map w(x)=x||x|| is well known as harmonic map into the sphere with a point singularity. Misawa and Nakauchi (0000) gave two examples of harmonic maps with the singularities xixj||x||2 and xixjxk||x||3 at x=0 respectively. In this paper we give a new example of harmonic maps into the spheres. It has the singularity xixjxkxl||x||4 at x=0.

We study the recently introduced ‘critically coupled’ model of magnetic Skyrmions, generalising it to thin films with curved geometry. The model feels keenly the extrinsic geometry of the film in three-dimensional space. We find exact Skyrmion solutions on spherical, conical and cylindrical thin films. Axially symmetric solutions on cylindrical films are described by kinks tunnelling between ‘vacua’

We prove that Chern insulators have topologically protected edge states which not only propagate unidirectionally along a straight line boundary, but also swerve around arbitrary-angled corners and geometric imperfections of the material boundary. This is a physical manifestation of the index theory of certain semigroup operator algebras.

In this paper, we introduce a notion of generalized Killing shape operator (or called the quadratic Killing shape operator) and its geometric meaning on real hypesurfaces in the complex quadric. In addition, we give a non-existence theorem for a Hopf real hypersurface with generalized Killing shape operator in the complex quadric.

We consider Pimsner algebras that arise from C*-correspondences of finite rank, as dynamical systems with their rotational action. We revisit the Laca-Neshveyev classification of their equilibrium states at positive inverse temperature along with the parametrizations of the finite and the infinite parts simplices by tracial states on the diagonal. The finite rank entails an entropy theory that shapes

In this paper, first we introduce a new notion of generalized Killing Ricci tensor for a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2). Next we give a complete classification for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with generalized Killing Ricci tensor.

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments, lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical

Given a smooth family of massless free fermions parametrized by a base manifold B, we show that the (mathematically rigorous) Batalin–Vilkovisky quantization of the observables of this family gives rise to the determinant line bundle for the corresponding family of Dirac operators.

This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed N=2 supersymmetric Yang–Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a cohomological field theory. The well-known Donaldson–Witten theory is a twisted version of N=2

The 2D Dirac oscillator can be viewed as the Dirac operator for a charged particle in a constant magnetic field. This system admits a non-abelian magnetic translation symmetry. Under a cocycle condition with respect to this symmetry, the 2D Dirac oscillator Hamiltonian is made into a Dirac Hamiltonian defined on the two-torus as a magnetic unit cell. A remarkable feature of this system is the existence

The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory

Inspired by the work of Heller (2013), we show that there exists a DPW potential for the Lawson surface ξk−1,l−1 from which it is possible to reconstruct the minimal immersion f:ξk−1,l−1→S3 via the DPW method. Moreover, we extend the result to surfaces immersed in the 3-sphere with constant mean curvature which satisfy a certain symmetric condition.

We show that split Courant algebroids, i.e., those defined on a Whitney sum A⊕A∗, are in a one-to-one correspondence with multiplicative curved L∞-algebras. This one-to-one correspondence extends to Nijenhuis morphisms and behaves well under the operation of twisting by a bivector.

In this paper we study helix surfaces whose unit normals make a constant angle with a fixed direction. As an application of the singularity theory, we classify the generic singularities of helix surfaces, which are cuspidaledges and swallowtails. These singularities are deeply related to the order of contact between the generating curves of helix surfaces and the affine tangent bundles over unit spheres

We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein Riemannian compact manifolds using T. Kobayashi’s conjecture as a guiding principle.

We consider Finsler submanifolds of nonnegative Ricci curvature in a Minkowski space which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag

In this paper, we study the standard binary Darboux transformation of the multi-component short pulse (SP) equation. For this purpose, we construct the Darboux matrix from the particular eigenvector solutions to the Lax pair not only in the direct space but also in the adjoint space and obtain the binary Darboux matrix. By iterating its standard binary Darboux transformation, we obtain quasi-Grammian

Consider a Leibniz superalgebra L additionally graded by an arbitrary set I (set grading). We show that L decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case of L being of maximal length, the simplicity of L is also characterized in terms of connections.

We explore the relationship between mechanical systems describing the motion of a particle with the mechanical systems describing a continuous medium. More specifically, we will study how the so-called intermediate integrals or fields of solutions of a finite dimensional mechanical system (a second order differential equation) are simultaneously Euler’s equations of fluids and conversely. This will

DeTurck and Yang have shown that in the neighbourhood of every point of a 3-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, whith respect to which the metric has diagonal form). We show that this property does not generalize to higher dimensions. In particular, the complex projective spaces CPm and the quaternionic projective spaces HPq, endowed with their

In the first part of this paper, using an optimization method on Riemannian submanifolds, we prove that for any Legendrian submanifold of a Sasakian space form M̄2n+1(c) of constant ϕ-sectional curvature c, we have two sharp inequalities relating some basic extrinsic and intrinsic invariants of the immersion, namely the ϕ-sectional curvature, the normalized scalar curvature, the mean curvature and

The present paper is to study estimation of inequalities for warped product semi-slant submanifolds of Kaehler manifolds. Next we obtain several fundamental results of pointwise semi-slant submanifolds and their warped products in Kaehler manifolds.

Let M1n+1 be a light-like geodesically complete Lorentzian (n+1)-manifold satisfying the null energy condition. We show that null hypersurfaces properly immersed in M1n+1 are totally geodesic.

We generalise a classical argument for deducing algebraic models of Riemann surfaces from the Riemann–Roch theorem. The method involves counting arguments based on equivariant resolutions.

The equivalence problem for linear differential operators of the second order, acting in vector bundles, is discussed. The field of rational invariants of symbols is described and connections, naturally associated with differential operators, are found. These geometrical structures are used to solve the problems of local as well as global equivalence of differential operators.

In this paper, we introduce the representation theory of δ-Hom-Jordan Lie conformal superalgebras and discuss the case of adjoint representations. Furthermore, we develop the cohomology theory of δ-Hom-Jordan Lie conformal superalgebras and discuss some applications to the study of deformations of regular δ-Hom-Jordan Lie conformal superalgebras. Finally, we introduce derivations of multiplicative

In this note we develop the theory of double brackets in the sense of van den Bergh (2008) in Kontsevich’s non-commutative “Lie World”. These double brackets can be thought of as Poisson structures defined by formal expressions only involving the structure maps of a quadratic Lie algebra. The basic example is the Kirillov–Kostant–Souriau (KKS) Poisson bracket. We introduce a notion of non-degenerate

In the present paper we introduce a semi-quasi-Einstein manifold from a semi symmetric metric connection. Among others, the popular Schwarzschild and Kottler spacetimes are shown to possess this structure. Certain curvature conditions are studied in such a manifold with a Killing generator.

In generalization of the classical Atiyah–Bott Poisson brackets on the moduli spaces of surfaces we define quasi-Poisson brackets on the moduli spaces of quasi-surfaces.

In this paper we deal with Bernoulli equations dy∕dx=A(x)yn+B(x)y, where A(x) and B(x) are real polynomials with A(x)⁄≡0 and n≥3. We prove that these Bernoulli equations can have at most 2 rational limit cycles if n is odd and at most one rational limit cycle if n is even. We also provide examples of Bernoulli equations with these numbers of rational limit cycles. Moreover we deal with the Riccati

There are several different, but equivalent definitions of geodesics in a Riemannian manifold, based on two characteristic properties: geodesics as shortest curves and geodesics as straightest curves. They are generalized to sub-Riemannian manifolds, but become non-equivalent. We give an overview of different approaches to the definition, study and generalization of sub-Riemannian geodesics and discuss

Ricci-like solitons with potential Reeb vector field are introduced and studied on almost contact B-metric manifolds. The cases of Sasaki-like manifolds and torse-forming potentials have been considered. In these cases, it is proved that the manifold admits a Ricci-like soliton if and only if the structure is Einstein-like. Explicit examples of Lie groups as 3- and 5-dimensional manifolds with the

This paper proves local gradient estimates on positive solutions to the following nonlinear elliptic equation Δfu+au(logu)α=0,where a and α are real constants, on complete weighted manifolds with Bakry–Émery Ricci tensor bounded from below. For applications, global estimates and some Liouville type theorems are derived.

In the present paper we consider a problem of separation of variables for Lax-integrable hamiltonian system generated by general non-skew-symmetric gl(2)⊗gl(2)-valued classical r-matrix r(u,v) with spectral parameters. We specify the general separability condition of Dubrovin and Skrypnyk (2018) on the components of the r-matrix and consider two new classes of examples of classical non-skew-symmetric

In this paper we give new examples of QCH Kähler surfaces whose opposite almost Hermitian structure is Hermitian and not locally conformally Kähler. In this way we give also a large class of examples of Hermitian surfaces with J-invariant Ricci tensor which are not l.c.k.

For any positive natural number r∈N+ we construct new explicit proper r-harmonic functions on the celebrated 3-dimensional Thurston geometries Sol, Nil, SL˜2(R), H2×R and S2×R.

We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analysed. Using Connes’ pseudodifferential calculus for the noncommutative tori, we explicitly

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie–Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished

We consider magnetic billiards under a strong constant magnetic field. The purpose of this paper is two-folded. We examine the question of existence of polynomial integral of billiard magnetic flow. As in our previous paper (Bialy and Mironov, 2016) we succeed to reduce this question to algebraic geometry test on existence of polynomial integral, which shows polynomial non-integrability for all but

In the context of K3 mirror symmetry, the Greene–Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in P3, starting from a special one-parameter family of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian

We study an equivariant extension of the Batalin–Vilkovisky formalism for quantizing gauge theories. Namely, we introduce a general framework to encompass failures of the quantum master equation, and we apply it to the natural equivariant extension of AKSZ solutions of the classical master equation (CME). As examples of the construction, we recover the equivariant extension of supersymmetric Yang–Mills

We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type A and X over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered. In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson

For a proper, cocompact action by a locally compact group of the form H×G, with H compact, we define an H×G-equivariant index of H-transversally elliptic operators, which takes values in KK∗(C∗H,C∗G). This simultaneously generalises the Baum–Connes analytic assembly map, Atiyah’s index of transversally elliptic operators, and Kawasaki’s orbifold index. This index also generalises the assembly map to

We give a detailed and unified survey of equivariant KK-theory over locally compact, second countable, locally Hausdorff groupoids. We indicate precisely how the “classical” proofs relating to the Kasparov product can be used almost word-for-word in this setting, and give proofs for several results which do not currently appear in the literature.

We extend the notion of r-minimality of a submanifold in arbitrary codimension to u-minimality for a multi-index u∈Nq, where q is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of u-minimality comes from the variation of the σu–symmetric function obtained

In this paper, group analysis of the time-fractional Harmonic Oscillator equation with Riemann–Liouville derivative is performed and its reduced fractional ordinary differential equations are determined. With the aid of the concept of nonlinear self-adjoint and the fractional generalization of the Noether operators are obtained conserved vector for this equation. Furthermore, by using Laplace transform