Martingale optimal transport has attracted much attention due to its application in pricing and hedging in mathematical finance. The essential notion which makes martingale optimal transport different from optimal transport is peacock. A peacock is a sequence of measures which satisfies convex order property. In this paper we study peacock geodesics in Wasserstain space, and we hope this paper can

There has been a recent and ongoing flurry/blizzard of papers making use of the notion of L∞-algebra in relation to field theory. Here we call attention to just some of the more recent developments, but only in the sense of an antipasto: a small dish with a smattering variety of little foods to stimulate the appetite for the main course, for someone else to serve. Various authors belong to different

We establish some comparison theorems on Finsler manifolds with curvature quartic decay. As their applications, we obtain some optimal compact theorems, volume growth and Mckean type estimate for the first Dirichlet eigenvalue for such manifolds. Although we present the results for Finsler manifolds, they are all new results for Riemannian manifolds.

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector

In this paper we construct a family of complete immersed minimal surfaces in R3 of genus one with two embedded catenoid-type ends, one Enneper-type end and total Gauss curvature −16π. The proof of the existence of this one-parameter family of minimal surfaces, was obtained using the Weierstrass representation, the theory of elliptic functions and explicitly solving the period problem.

The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety N with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki [48] for K-contact manifolds to the Sasaki case. This construction

The article presents a new method for constructing exact solutions of non-evolutionary partial differential equations with two independent variables. The method is applied to the linear classical equations of mathematical physics: the Helmholtz equation and the variable type equation. The constructed method goes back to the theory of finite-dimensional dynamics proposed for evolutionary differential

Let F=αϕ(s),s=βα, be an invariant (α,β)-metric on a reductive homogeneous manifold G/H of dimension n≥3 such that ϕ(s) comes from a real analytic function. In this paper, we first obtain the necessary and sufficient conditions for F to be a weakly Berwald metric, and then prove that F has weakly isotropic S-curvature if and only if F has vanishing S-curvature. Based these results, we prove that S=0

In this note we complete a study of globally homogeneous Riemannian quotients Γ\(M,ds2) in positive curvature. Specifically, M is a homogeneous space G/H that admits a G–invariant Riemannian metric of strictly positive sectional curvature, and ds2 is a G–invariant Riemannian metric on M, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that Γ\(M,ds2) is (globally)

Holonomy algebras of Weyl connections in Lorentzian signature are classified. In particular, examples of Weyl connections with all possible holonomy algebras are constructed.

In this paper, we study Hamilton inequality for unbounded Laplacians on weighted graphs satisfying CDE′(−K,∞) for some K≥0. This is a generalization of the main result in [8] for bounded Laplacians. For this purpose, we assume a non-degenerated measure and make some modifications to the solutions of the heat equation. As an application of Hamilton inequality, we derive a type of Harnack inequality

Light carries energy and therefore is source of a gravitational field. In the paper it is proven, in particular, that a laser beam is source of non-linear gravitational waves corresponding, from a quantum point of view, to spin-1 massless particles. This fact suggests both a possible solution to the old problem on the lack of gravitational attraction between two laser beams moving parallel and a new

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface Sm,s with s ends and exactly m of them with infinite genus, such that m,s∈N and 1

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular

The present paper investigates a natural generalization of the duality between Riemannian symmetric pairs of compact type and those of non-compact type à la É. Cartan. The main result of this paper is to construct an explicit description of a one-to-one correspondence between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads, which is called

In this paper, we investigate p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincaré inequality, and we get a vanishing type theorem, which shows that if we put some assumptions on the bound of the Weitzenböck curvature operator and the first eigenvalue of the Laplacian operator, then there is no nontrivial p-harmonic ℓ-form (2≤ℓ≤n−2) with finite Lp norm on M, which generalizes the previous

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation

We give a necessary and sufficient condition for a set of left invariant metrics on a compact Heisenberg manifold to be relatively compact in the corresponding moduli space.

We apply the technique based on twisted extensions of symmetry algebras to construct new nonlinear four-dimensional differential coverings for the hyper-CR equation of Einstein–Weyl structures and for the associated integrable hierarchy. We expose related multi-component three-dimensional covering. By the symmetry reduction of the hyper-CR equation of Einstein–Weyl structures we derive nonlinear three-dimensional

Starting with a non-abelian gerbe represented by a non-abelian differential cocycle, with values in a given crossed-module, this paper explicitly calculates a formula for the derivative of the associated surface holonomy of squares mapped into the base manifold; with spheres later considered as a special case. While the definitions in this paper used for gerbes, their connections, and the induced holonomy

Let (M,g) be an n-dimensional Riemannian manifold with a quasi-Einstein metric g, i.e., g satisfies the following equationRic+Hessf−1τdf⊗df=λg for constants τ>0 and λ, here Ric denotes the Ricci curvature tensor. In this paper, we derive a sharp lower bound estimate for the scalar curvature R of (M,g) when τ>1,λ=0 and f≤0. A more specific estimate can be derived if f satisfies another asymptotic condition

We study Lorentzian manifolds (M,g) of dimension n≥4, equipped with a maximally twisting shearfree null vector field p, for which the leaf space S=M/{exp⁡tp} is a smooth manifold. If n=2k, the quotient S=M/{exp⁡tp} is naturally equipped with a subconformal structure of contact type and, in the most interesting cases, it is a regular Sasaki manifold projecting onto a quantisable Kähler manifold of real

In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M.

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently

Stimulated by S. Ohta and W. Wylie, we establish some compactness theorems for complete Riemannian manifolds via m-Bakry–Émery and m-modified Ricci curvatures with negative m. Our results may be considered as generalizations of the classical compactness theorems via Ricci curvature due to S.B. Myers, W. Ambrose, G.J. Galloway, and J. Cheeger, M. Gromov, and M. Taylor, and relax some previous compactness

A notion of almost contact metric statistical structure is introduced and thereby contact metric and K-contact statistical structures are defined. Furthermore a necessary and sufficient condition for a contact metric statistical manifold to admit K-contact statistical structure is given. Finally, the condition for an odd-dimensional statistical manifold to have K-contact statistical structure is expressed

In this paper we show that Cartan geometries can be studied via transitive Lie groupoids endowed with a special kind of vector-valued multiplicative 1-forms. This viewpoint leads us to a more general notion, that of Cartan bundle, which encompasses both Cartan geometries and G-structures.

This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds are a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result

We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let f:Mcn→Qc˜n+p be an isometric immersion of a Riemannian manifold with constant sectional curvature c into a space form of curvature c˜, and free of weak-umbilic points if c>c˜. We show that the substantial codimension of f is p=n−1 if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank n−1

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.

In this paper, we study hypersurfaces of the homogeneous NK (nearly Kähler) manifold S3×S3. As the main results, we first show that the homogeneous NK S3×S3 admits neither locally conformally flat hypersurfaces nor Einstein Hopf hypersurfaces. Then, we establish a Simons type integral inequality for compact minimal hypersurfaces of the homogeneous NK S3×S3 and, as its direct consequence, we obtain

Assuming suitable constraints on the warping function ρ and on the mean curvature vector field, we apply the Omori-Yau's generalized maximum principle in order to study the behavior of a support function naturally attached to a complete n-dimensional submanifold Σn immersed in a Killing warped product Mn+p×ρR whose base Mn+p has sectional curvature bounded from below. In the case that Σn is compact

This paper is concerned with the existence and compactness problems of perturbed Dirac-harmonic maps into flat tori. We consider two types of asymptotically linear perturbations, non-resonance and resonance cases. For the non-resonance case, we show that the set of solutions in a given free homotopy class is compact and there exist at least (n+1)-distinct solutions in each free homotopy class, where

It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of Rn. In the 3-dimensional case we prove a similar result for a large class of causally simple contractible subsets of an arbitrary globally hyperbolic spacetime applying methods from the theory of contact-convex

A new method (by Kersten, Krasil'shchik and Verbovetsky), based on the theory of differential coverings, allows to relate a system of PDEs with a differential operator in such a way that the operator maps conserved quantities into symmetries of the system of PDEs. When applied to a quasilinear first-order system of PDEs and a Dubrovin–Novikov homogeneous Hamiltonian operator the method yields conditions

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs

In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution

We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify sl(2,C) based automorphic Lie algebras corresponding to all finite

In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally, we give information on how the Page space is related to some other metrics on the same underlying smooth manifold.

In this paper, we present the classification of 2 and 3-dimensional Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric.

Analytic curves are classified w.r.t. their symmetry under a given regular and separately analytic Lie group action G×M→M on an analytic manifold. We show that a non-constant analytic curve γ:D→M is either free or exponential – i.e., up to analytic reparametrization of the form t↦exp⁡(t⋅g→)⋅x. The vector g→∈g is additionally proven to be unique up to (non-zero scalation and) addition of elements in

A field of endomorphisms R is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of R called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures

Based on our deep understanding on the essential relationships between the Zermelo navigation problems and the geometries of indicatrix on Finsler manifolds, we study navigation problems on conic Kropina manifolds. For a conic Kropina metric F(x,y) and a vector field V with F(x,−Vx)≤1 on an n-dimensional manifold M, let F˜=F˜(x,y) be the solution of the navigation problem with navigation data (F,V)

Finite-dimensional coverings from systems of differential equations are investigated. Multivector fields defining fibers of coverings and their Schouten bracket are used. A method for constructing coverings is given and demonstrated by the example of the Laplace equation.

We recall the notion of a differential operator over a map (in linear and non-linear settings) and consider its versions such as formal ħ-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and operators arising as quantization of symplectic micromorphisms.

The existence problem for attractors of foliations with transverse linear connection is investigated. In general foliations with transverse linear connection do not admit attractors. Sufficient conditions are found for the existence of a global attractor that is a minimal set. An application to transversely similar pseudo-Riemannian foliations is obtained. The global structure of transversely similar

In this paper, the well-known log-Minkowski inequality is extended to the Orlicz space. We first propose and establish an Orlicz logarithmic Minkowski inequality by introducing two new concepts of mixed volume measure and Orlicz mixed volume measure, and using the Orlicz Minkowski inequality for the mixed volumes. The Orlicz logarithmic Minkowski inequality in special case yields the Stancu's logarithmic

In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is a Hadamard manifold, i.e. that it is geodesically complete and has everywhere negative sectional curvature. An important consequence for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.

We propose a geometrical approach to the mechanics of continuous media equipped with inner structures and give the basic (Navier–Stokes, mass conservation and energy conservation) equations of their motion.

In this paper, we compute the sectional curvature of the group whose Euler-Arnold equation is the quasi-geostrophic (QG) equation in geophysics and oceanography, or the Hasegawa-Mima equation in plasma physics: this group is a central extension of the quantomorphism group Dq(M). We consider the case where the underlying manifold M is rotationally symmetric, and the fluid flows with a radial stream

This paper aims to define and study a notion of orientability in the Heisenberg sense (H-orientability) for the Heisenberg group Hn. In particular, we define such notion for H-regular 1-codimensional surfaces. Analysing the behaviour of a Möbius Strip in H1, we find a 1-codimensional H-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, H-orientability

We consider non-degenerate graph immersions into affine space An+1 whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs (J,γ), where J is an n-dimensional real Jordan algebra and γ is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine

In this paper, we study the eigenvalue of p-Laplacian on finite graphs. Under generalized curvature dimensional condition, we obtain a lower bound of the first nonzero eigenvalue of p-Laplacian. Moreover, a upper bound of the largest p-Laplacian eigenvalue is derived.

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σk-curvature in the interior and constant Hk-curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type

In this paper, we provide a necessary and sufficient condition for a set in PSL(2,R) or in T1H2 to be a fundamental domain for a given Fuchsian group via its respective fundamental domain in the hyperbolic plane H2.

Let M be a real hypersurface in complex projective space. The almost contact metric structure on M allows us to consider, for any nonnull real number k, the corresponding k-th generalized Tanaka-Webster connection on M and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi operator Rξ

We classify and explicitly describe maximal antipodal sets of some compact classical symmetric spaces and those of their quotient spaces by making use of suitable embeddings of these symmetric spaces into compact classical Lie groups. We give the cardinalities of maximal antipodal sets and we determine the maximum of the cardinalities and maximal antipodal sets whose cardinalities attain the maximum