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

A study is made of R6 as a singular quotient of the conical space R+×CP3 with holonomy G2, with respect to an obvious action by U(1) on CP3 with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of SO(3) on R6, which can be used effectively to describe

We construct 1-parameter families of well known solutions to the Yamabe problem defined on maximal flag manifolds to determine bifurcation instants for these families looking at changes of the Morse index of these metrics when the parameter varies on the interval [0,1]. A bifurcation point for such families is an accumulation point of others solutions to the Yamabe problem and a local rigidity point

In this paper, we investigate a duality between Hermitian and almost Kähler structures on the tangent manifold TM induced by pairs of conjugate connections on its base, affine Riemannian manifold M. In the context of information geometry, the classical theory of statistical manifold (which we call S-geometry) prescribes a parametrized family of probability distributions with a Fisher-Rao metric g and

We prove that the structure Jacobi operator Rξ on a real hypersurface M in the complex projective space CP2 satisfies Rξ+κϕ2=0 with κ∈R⁎ (or equivalently, constant Reeb sectional curvature) if and only if either M is of type (A) or M is locally congruent to a non-homogeneous Hopf hypersurface with vanishing Hopf principal curvature.

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover,

We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show S3 does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration

Let M13(c) be an 3-dimensional Lorentz space form and C(M13(c)) denote the conformal transformation group of M13(c). A spacelike surface x:M2→M13(c) is called a conformal homogeneous spacelike surface. If there exists a subgroup G⊂C(M13(c)) such that the orbit G(p)=x(M2),p∈x(M2). In this paper, we classify completely conformal homogeneous spacelike surfaces up to a conformal transformation of M13(c)

We show that every forward complete Finsler manifold of infinite fundamental group and not homotopy-equivalent to S1 has infinitely many geometrically distinct geodesics joining any given pair of points p and q. In the special case in which β1(M;Z)≥1 and M is closed, the number of geometrically distinct geodesics between two points grows at least logarithmically.

Donaldson-Friedman constructed anti-self-dual classes on K3#3CP2‾ using twistor space. We show that some of these conformal classes have almost-Kähler representatives.

A submanifold Mn of a Euclidean space EN is called biharmonic if ΔH→=0, where H→ is the mean curvature vector of Mn. A well known conjecture of B.Y. Chen states that the only biharmonic submanifolds of Euclidean spaces are the minimal ones. Ideal submanifolds were introduced by Chen as those which receive the least possible tension at each point. In this paper we prove that every δ(r)-ideal biharmonic

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature η–invariants for the family of Berger metrics on the odd dimensional spheres.

In this paper, we mainly investigate curvature properties and harmonicity of invariant vector fields on the four-dimensional Oscillator groups endowed with three left-invariant pseudo-Riemannian metrics of signature (2,2). We determine all harmonic vector fields, vector fields which define harmonic maps and the vector fields which are critical points for the energy functional restricted to vector fields

This note studies invariant generators for a certain class of invariant smooth generalized subbundles of exact Courant algebroids, which generalizes existing results for invariant vector subbundles of exact Courant algebroids. As a result, we provide a simple and geometric proof for Theorem 1 in [1].

We investigate the holomorphic differential operators on a Riemann surface M. This is done by endowing M with a projective structure. Let L be a theta characteristic on M. We explicitly describe the jet bundle Jk(E⊗L⊗n), where E is a holomorphic vector bundle over M equipped with a holomorphic connection, for all k and n. This provides a description of global holomorphic differential operators from

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥5. Denote Lg the Paneitz-Branson type operator. In this paper, we show that there exists a nodal solution (solution with changing sign) of the nonlinear Paneitz-Branson type equation Lgv=ϵ|v|N−2v where ϵ<0. At the end, we give a geometric application of the above equation.

In this paper we prove that, under certain conditions, in a quasi-Einstein semi-Riemannian warped product the fiber is necessarily an Einstein manifold. We provide all the quasi-Einstein manifolds when r-Bakry-Emery tensor is null, the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n−1)-dimensional translation group and the fiber is Ricci-flat. As an

We prove that a non-totally vicious n-dimensional compact spacetime (M,g) admitting a parallel lightlike vector field is foliated by compact totally geodesic null hypersurfaces. As a consequence, assuming non-negative Ricci curvature on the leaves then the first Betti number of M is bounded above by n with equality if and only if M is diffeomorphic to the torus.

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and

The spherical Fourier transform on a harmonic Hadamard manifold (Xn,g) of positive volume entropy is studied. If (Xn,g) is of hypergeometric type, namely spherical functions of X are represented by the Gauss hypergeometric functions, the inversion formula, the convolution rule together with the Plancherel theorem are shown by the representation of the spherical functions in terms of the Gauss hypergeometric

We discuss natural transformations in the context of Lie groupoids, and their infinitesimal counterpart. Our main result is an integration procedure that provides smooth natural transformations between Lie groupoid morphisms.

Key to H. C. Wang's quantitative study of Zassenhaus neighbourhoods of non-compact semisimple Lie groups are two constants that depend on the root system of the corresponding Lie algebra. This article extends the list of values for Wang's constants to the exceptional Lie groups and also removes their dependence on dimension. The first application is an improved upper sectional curvature bound for a

In this paper, we study the cross curvature soliton. We study the cross curvature soliton with a warped product structure. On the other hand, we show that the volume entropy is decreasing along the cross curvature flow.

In this paper, we classify n-dimensional (n≥4) gradient shrinking Ricci solitons with bounded Ricci curvature. Precisely, we obtain that such a soliton satisfying Ric+Hessf=λg with 0≤Ric≤λng is a finite quotient of the Gaussian shrinking soliton Rn. In the case of dimension 4, we prove that a radially flat gradient shrinking Ricci soliton with 0≤Ric≤λ2g is a finite quotient of R4 or R2×S2.

Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotient N˜11=(Sp(2)×S4)/S3, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotient N˜11/S3, we construct a transnormal system with transnormal hypersurfaces diffeomorphic

The Fefferman space of a contact Riemannian manifold carries a Lorentzian spin structure canonically. On the Lorentzian spin manifold, we investigate the Dirac operator and the twistor operator closely. In particular, we show that, if the contact Riemannian manifold is integrable, then there exist non-zero global solutions of the twistor equation.

The aim of this paper is to investigate the mean curvature flow soliton solutions on the Heisenberg group H when the initial data is a ruled surface by straight lines. We give a family of those solutions which are generated by Iso0(H) (the isometries of H for which the origin is a fix point). We conclude that the function which describe the motion of these surfaces under MCF, is always a linear affine