We study hypersurfaces of the homogeneous nearly Kähler manifold S3×S3 which have P-isotropic normal vector field. We describe the immersion of such hypersurfaces in S3×S3 and we give one example. We prove that they cannot be either Hopf or minimal hypersurfaces.

Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto–Witten mechanism of supersymmetry enhancement N3=3→N3=4 from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto–Witten identities to be satisfied by the

We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott’s spectral sequence which converges to the cohomology of the classifying space of

In this paper, we introduce a new notion of quadratic Killing normal Jacobi operator (simply, Killing normal Jacobi operator) and its geometric meaning for real hypersurfaces in the complex Grassmannians of rank two G2m+2(c), c≠0. In addition, we give two classification theorems for Hopf real hypersurfaces with quadratic Killing normal Jacobi operator in complex two-plane Grassmannians of compact type

We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a k-degree form 0≤k≤4, those modified with multi-degree (0,k)-form 0≤k≤3 and the horizon Dirac operators which

This paper presents a unified method upon Hamiltonian description of one dimensional second order dissipative systems. In this method, the first order partial differential equation is constructed by the first integral. As long as the corresponding characteristic equation has an explicit solution, the dissipative system has an explicit Hamiltonian function. The Hamiltonians for three types of representative

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space forms have constant mean curvature. Our result gives the positive answer to the conjecture proposed by Balmuş-Montaldo-Oniciuc in 2008 for four dimensional hypersurfaces.

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a

This paper mainly discusses the high-order Landau–Lifshitz equation. The author probes both the existence and the uniqueness of global smooth solutions into the Cauchy problem in the Sobolev space by adopting methods of making a priori integral estimation and utilizing continuity.

In this paper we have investigated the curvature restricted geometric properties of the generalized Kantowski-Sachs (briefly, GK-S) spacetime metric, a warped product of 2-dimensional base and 2-dimensional fibre. It is proved that GK-S metric describes a generalized Roter type, 2-quasi Einstein and Ein(3) manifold. It also has pseudosymmetric Weyl conformal tensor as well as conharmonic tensor and

We study left-invariant foliations F on Riemannian Lie groups G generated by a subgroup K. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations F when the subgroup K is one of the important SU(2)×SU(2), SU(2)×SL2(R), SU(2)×SO(2) or SL2(R)×SO(2). By this we yield new multi-dimensional families of Lie groups G carrying such foliations

We construct an A∞-structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal A∞-products are obtained as derivatives of those of transversal A∞-products.

We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields over a differentiable manifold, each of them with vanishing Haantjes torsion. Also, they satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the

Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly (1,2). More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. (1,3), up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation

The “odd transgression” introduced in Bressler and Rengifo (2018) is applied to construct and study the inverse image functor in the theory of Courant algebroids.

A Dirac bundle is a euclidean bundle over a riemannian manifold M which is a compatible left Cℓ(M)-module, together with a metric connection also compatible with the Clifford action in a natural way. We prove some vanishing theorems and introduce the twistor equation within this framework. In particular, we exhibit a characterization of solutions for this equation in terms of the Dirac operator D and

We consider nonlinear electrical circuits for which we derive a port-Hamiltonian formulation. After recalling a framework for nonlinear port-Hamiltonian systems, we model each circuit component as an individual port-Hamiltonian system. The overall circuit model is then derived by considering a port-Hamiltonian interconnection of the components. We further compare this modeling approach with standard

The object of the present paper is to introduce and investigate the P-curvature tensor that generalizes projective, conharmonic, M-projective and the set of Wi curvature tensors introduced by Pokhariyal and Mishra. It is proven that pseudo-Riemannian manifolds admitting a traceless P-curvature tensor are Einstein and those admitting flat P-curvature tensor has a constant curvature. Classification theorems

Suppose f,g are homogeneous polynomials of degree d defining smooth hypersurfaces Xf=V(f)⊂Pm−1 and Xg=V(g)⊂Pn−1. Then the sum of f and g defines a smooth hypersurface X=V(f⊕g)⊂Pm+n−1 with an action of μd scaling the g variables. Motivated by the work of Orlov, we construct a semi-orthogonal decomposition of the derived category of coherent sheaves on [X∕μd] provided d≥max{m,n}.

A formulation of quaternionic quantum mechanics (HQM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After a brief discussion on unitary operators in this formalism, we conclude that this quantum theory is indeed consistent, and can be a valuable tool in the search for new physics

The moduli spaces of symplectic vector bundles of arbitrary rank on projective space P3 are far from being well-understood. By now the only type of such bundles having satisfactory description are the so-called tame symplectic instantons. It is shown by U.Bruzzo, D.Markushevich and the first author in two papers from 2012 and 2016 that the moduli space of tame symplectic instantons are irreducible

A longstanding open problem in mathematical physics has been that of finding an action principle for the Einstein-Weyl (EW) equations. In this paper, we present for the first time such an action principle in three dimensions in which the Weyl vector is not exact. More precisely, our model contains, in addition to the Weyl nonmetricity, a traceless part. If the latter is (consistently) set to zero,

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler–Lagrange equations on the 2nd order tangent group from the 1st order Euler–Lagrange equations on the iterated tangent group. We also present in detail the 2nd order Lagrangian

We prove that all geodesics of homogeneous Gödel-type metrics are homogeneous. This result makes natural to ask whether these spaces are naturally reductive, and a positive answer is provided for all of them through the study of their homogeneous Lorentzian structures.

Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or antisymmetric representations of SU(N). Some of the mutant

In this paper we introduce exotic twisted T-equivariant K-theory of loop space LZ depending on the (typically non-flat) holonomy line bundle LB on LZ of a gerbe with connection on Z. We define an exotic twisted T-equivariant Chern character on the exotic twisted T-equivariant K-theory of LZ that maps to the exotic twisted T-equivariant cohomology of LZ as previously defined in Han and Mathai (2015)

In this work, the generalized Robin–Regge problem with complex coefficients is investigated. Two new uniqueness theorems are proved for the inverse spectral problems, and the corresponding algorithms are also provided. For one of the inverse spectral problems, the local stability and solvability are discussed by using a constructive algorithm related to the property of Riesz basis.

In this paper, we obtain some basic inequalities of curvature invariants for submanifolds of quaternionic space forms with a quarter-symmetric connection.

It is still a long-standing open problem in Finsler geometry, is there any regular Landsberg metric which is not Berwaldian. However, there are non-regular Landsberg metrics which are not Berwaldian. The known examples are established by G. S. Asanov and Z. Shen. In this paper, we use the Maple program to study some explicit examples of non-Berwaldian Landsberg metrics. In fact, such kinds of examples

We explore the geometric implications of introducing a spectral cut-off on compact Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are truncated by spectral projections of Dirac-type operators. We associate a metric space of ‘localized’ states to each truncation. The Gromov–Hausdorff limit of these spaces is then

We study the bigraded vector space structure of Landau–Ginzburg orbifolds with the permutation symmetries of variables. We calculate the formula of the Poincaré polynomial for a certain loop polynomial orbifoldized by a semidirect product of a diagonal symmetry group and cyclic permutation group and check that the generalization of Berglund–Hübsch transpose gives mirror symmetric pairs for these cases

Two-sided conformally recurrent 4-dimensional self-dual spaces are considered. It is shown that such spaces are equipped with nonexpanding congruences of null strings. The general structure of weak nonexpanding hyperheavenly spaces is given. Finally, the general metrics of Petrov–Penrose type [D]⊗[−] spaces are presented.

In this paper, we study a fourth order curve flow ∂X∂t=−Δ2X for a smooth closed curve X in R2, which means that a fourth order nonlinear parabolic partial differential equation arises naturally in the field of biharmonic maps. We first derive the evolution equations for different geometric quantities along this flow. Utilizing the evolution equations, we prove that for any smooth closed initial curve

In noncommutative geometry, Connes’s spectral distance is an extended metric on the state space of a C∗-algebra generalizing Kantorovich’s dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a supremum. We present a dual formula – as an infimum – generalizing Beckmann’s “dual of the dual” formulation of the Wasserstein distance. We then discuss some examples

Let k≥1 be an integer and μ be the half of a nonzero integer. It is reported here that the phase space of the (2k+1)-dimensional magnetized Kepler problem with magnetic charge μ can be identified with the elliptic co-adjoint orbit of the real Lie algebra so(2,2k+2) that corresponds to the dominant weight (|μ|,…,|μ|︸k+1,μ). As a corollary, one has the following result: the aforementioned elliptic co-adjoint

In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together with noncommutative embeddings. We show that basic concepts, such as the second fundamental form and the Weingarten map, translate into the noncommutative setting and, in particular, we prove a noncommutative analogue of Gauss’ equations for the curvature of a submanifold. Moreover, the mean curvature

This paper studies geometric structures on noncommutative hypersurfaces within a module-theoretic approach to noncommutative Riemannian (spin) geometry. A construction to induce differential, Riemannian and spinorial structures from a noncommutative embedding space to a noncommutative hypersurface is developed and applied to obtain noncommutative hypersurface Dirac operators. The general construction

Motivated by generalizing Szemerédi’s theorem, we define the elements in a discrete quantum group asymptotically fixing a sequence of finite subsets and prove that the set of these elements is a quantum subgroup. Using this we obtain a version of mean ergodic theorem for discrete quantum groups.

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (Helleland and Hervik, 2018; Helleland and Hervik, 2019). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group (a Lie group with a left-invariant metric) can be Wick-rotated to a Riemannian Lie group. We define a Cartan involution of a general Lie algebra

We study linear Jacobi structures on vector spaces and their non-degenerate version, the linear contact structures. We touch several aspects, like transitivity, Poissonization, coisotropy, product structures, transversals. We consider the linear version of contact dual pairs and we point out some of their properties.

We present a classification theorem for separable amenable simple stably projectionless C∗-algebra s with finite nuclear dimension whose K0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C∗-algebra s in the class is denoted by Z0 which has a unique tracial state, K0(Z0)=Z and K1(Z0)={0}. Let A and B be two separable simple C∗-algebras satisfying the UCT and have finite nuclear

We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus of possibly higher order band crossings aka. degeneracies, we add the study of local models. These give more detailed information including that of the local Chern classes

In this article, we construct supersymmetric C type Kadomtsev–Petviashvili (CKP) hierarchy and six-reduced supersymmetric CKP hierarchy and their additional symmetries. These additional flows of the supersymmetric CKP hierarchy constitute a C type SW1+∞ Lie algebra. Meanwhile, we find the six-reduced supersymmetric CKP hierarchy contains the super Sawada–Kotera equation as a primary equation. Lastly

We prove localization and integration formulas for the equivariant basic cohomology of Riemannian foliations. As a corollary we obtain a Duistermaat–Heckman theorem for transversely symplectic foliations.

First we introduce a new notion of Ricci-soliton and pseudo-Einstein for real hypersurfaces in the noncompact complex hyperbolic quadric SO2,m0∕SO2SOm. Next as an application we will prove non-existence properties of Ricci soliton and pseudo-Einstein real hypersurfaces in the complex hyperbolic quadric SO2,m0∕SO2SOm.

In this paper we are focusing on functional inequalities on compact simple edge spaces. More precisely we address the question whether the classical functional inequalities (Sobolev, Poincaré) hold in this setting, and as a by-product of our methods we obtain an optimality result concerning the B−constant of the Sobolev inequality.

The class of simple separable KK-contractible (KK-equivalent to {0}) C*-algebra s which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebra s A⊗W is classifiable, where A is a simple separable C*-algebra with finite nuclear dimension and W is the simple inductive limit of Razak algebras with unique trace, which is bounded (see Razak

The study follows the conjecture of Lawson and Simon (1973) and proves the main theorems. If the warping function of a compact warped product Lagrangian submanifold Σ3 isometrically immersed into the nearly Kaehler 6-sphere, satisfies several extrinsic conditions, then there are no stable integral currents in Σ3 and homology groups are vanished in Σ3. As applications, we show that Σ3 is homotopic to

In this paper, we introduce and define hyper-dual split vectors by using hyper-dual numbers. We define three subsets; two of them are subsets of Lorentzian unit hyper-dual sphere and the other is a subset of hyperbolic unit hyper-dual sphere. We show that to each element of these subsets corresponds two intersecting perpendicular lines in Minkowski space. We also introduce and define hyper-dual split

Based on the Chen–Möller–Sauvaget formula, we apply the theory of integrable systems to derive three equations for the generating series of the Masur–Veech volumes VolQg,n associated with the principal strata of the moduli spaces of quadratic differentials, and propose refinements of the conjectural formulas given in Delecroix et al. (0000) and Aggarwal et al. (0000) for the large genus asymptotics

Given a complete hypersurface in the Euclidean space, we assume that the second fundamental form is bounded and the mean curvature is bounded from below by a positive constant. We prove that if the Lp-norm of the trace-free second fundamental form for some p≥2 is finite, then the hypersurface is compact.

The problem of equivalency for linear differential operators of the first order is discussed.

We introduce the notion of εη-Einstein ε-contact metric three-manifold, which includes as particular cases η-Einstein Riemannian and Lorentzian (para) contact metric three-manifolds, but which in addition allows for the Reeb vector field to be null. We prove that the product of an εη-Einstein Lorentzian ε-contact metric three-manifold with an εη-Einstein Riemannian contact metric three-manifold carries

Ignatowski (1910) showed that assumptions about light are not necessary to obtain Lorentzian kinematics as one of only few possibilities. We give a much simplified proof of his result as formulated by Gorini (1971) for n+1-dimensional space–time.

Let C be a polarized nodal curve of compact type. In this paper we study coherent systems (E,V) on C given by a depth one sheaf E having rank r on each irreducible component of C and a subspace V⊂H0(E) of dimension k. Moduli spaces of stable coherent systems have been introduced by King and Newstead (1995) and depend on a real parameter α. We show that when k≥r, these moduli spaces coincide for α big

Lagrangian mean curvature flow has been studied extensively recently and many important results have been obtained. In this paper, we attempt to consider the hyperbolic version of Lagrangian mean curvature flow, which is a nonlinear wave equation for the potential function u(x) of a graph submanifold (x,Du(x)) in R2n. In particular, here we are interested in the one dimensional case. Assume the initial

In [9] author proved that 4-dimensional Lorentzian concircular structure [in brief (CS)4] space–time coincides with generalized Robertson Walker [in brief GRW] space–time. Consequently, to study perfect fluid (CS)4-spacetime obeying Einstein equation under various curvature restrictions on energy–momentum tensor like semi-symmetric, Ricci semi-symmetric etc. are equivalent to study that of perfect

A variational formulation for nonequilibrium thermodynamics was recently proposed in Gay-Balmaz and Yoshimura (2017a, 2017b) for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systematic way to derive, from a unified perspective

In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including systems with degenerate singularities which are invariant under a torus action.

The KdV equation is one of the most famous PDE which has a vast field of applications in fluid dynamics. The scope of our research is the Lie group analysis of time-fractional KdV-type equation including a perturbation term. Thus, the Lie group theory is extended to both cases simultaneously. Geometric vector fields of symmetries and one-dimensional optimal system are found in order to reduce the equation