In this paper, we consider fractional CR Yamabe type equations of order 2γ, 0<γ

We show that the 2-lobed Delaunay tori are stable as constrained Willmore surfaces in the 3-sphere. The images are made by Nick Schmitt.

We improve a lemma in our former article [4] which describes the group of isometries of a polar of a compact Lie group.

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this

This paper continues the project, begun in [1], of harmonizing Cartan's classical equivalence method and the modern equivariant moving frame in a framework dubbed involutive moving frames. As an attestation of the fruitfulness of our framework, we obtain a new, constructive and intuitive proof of the Lie-Tresse theorem (Fundamental basis theorem) and a first general upper bound on the minimal number

Jung et al. [7] introduced a geometric structure on Sym+(p), the set of p×p symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of Sym+(p), defined by eigenvalue multiplicities, and fibers of the “eigen-composition” map F:M(p):=SO(p)×Diag+(p)→Sym+(p). When M(p) is equipped with a suitable Riemannian metric, the fiber structure leads to

In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in 2n variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of

We obtain upper bounds for the Steklov eigenvalues σk(M) of a smooth, compact, n-dimensional submanifold M of Euclidean space with boundary Σ that involve the intersection indices of M and of Σ. One of our main results is an explicit upper bound in terms of the intersection index of Σ, the volume of Σ and the volume of M as well as dimensional constants. By also taking the injectivity radius of Σ into

The symplectic sum is employed to construct stable generalized complex 6-manifolds. We show that there exist a myriad of stable generalized complex 6-manifolds which have either 0 or negative Euler characteristic and contain a type change locus in which each path-connected component is diffeomorphic to T4.

A global secondary CR invariant is defined as the integral of a pseudo-hermitian invariant which is independent of a choice of pseudo-Einstein contact form. We prove that any global secondary CR invariant on strictly pseudoconvex CR five-manifolds is a linear combination of the total Q′-curvature, the total I′-curvature, and the integral of a local CR invariant.

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian 3-manifold (M¯,g) with (possibly empty) boundary and a smooth flow of isometries ϕt:M¯→M¯ we show that, if M¯ has non-empty boundary

The secant caustic of a planar curve M is the image of the singular set of the secant map of M. We analyze the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of this set. In particular, we investigate in detail some of the geometrical properties of

We deal with solitons of the mean curvature flow. The definition of translating solitons on a light-like direction in Minkowski 3-space is introduced. Firstly, we classify those which are graphical, translation surfaces, obtaining space-like and time-like, entire and not entire, complete and incomplete examples. Among them, all our time-like examples are incomplete. The second family consists of those

There are various statements in the physics literature about the stratification of quantum states, for example into orbits of a unitary group, and about generalized differentiable structures on it. Our aim is to clarify and make precise some of these statements. For A an arbitrary finite-dimensional C*-algebra and U(A) the group of unitary elements of A, we observe that the partition of the state space

We show that the extended principal bundle of a Cartan geometry of type (A(m,R),GL(m,R)), endowed with its extended connection ωˆ, is isomorphic to the principal A(m,R)-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows

We study the problem of prescribing the Ricci curvature in the class of naturally reductive metrics on a compact Lie group. We derive necessary as well as sufficient conditions for the solvability of the equations and provide a series of examples.

We apply the method of linear perturbations to the case of Spin(7)-structures, showing that the only nontrivial perturbations are those determined by a rank one nilpotent matrix. We consider linear perturbations of the Bryant-Salamon metric on the spin bundle over S4 that retain invariance under the action of Sp(2), showing that the metrics obtained in this way are isometric.

On a 2-step nilmanifold with abelian complex structure, there exists an invariant (1,0)-form ρ such that dρ is type-(1,1). It acts on the kernel of ρ by contraction. When this contraction map is non-degenerate, for any given infinitesimal generalized complex deformation Γ1 we construct a solution (Γ,∂→) for the extended Maurer-Cartan equation. It amounts to identifying the obstruction ∂→ for Γ1 to

We show that non-compact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof is to show, via a cohomogeneity-one approach, that non-compact homogeneous spaces admitting an ideal isomorphic to sl2(R) admit no homogeneous Einstein metrics of

In this paper, we give a classification of real hypersurfaces M2n−1 in a nonflat complex space form M˜n(c)(n≧2) whose ⁎-Ricci tensor is D-recurrent. By virtue of this result, we also know a classification of real hypersurfaces in a nonflat complex space form whose ⁎-Ricci tensor is D-parallel.

In this note, we show that the examples of non-Berwaldian Landsberg surfaces with vanishing flag curvature, obtained in [5], are in fact Berwaldian. Consequently, Bryant's claim is still unverified.

We develop two new methods of constructing sequences of manifolds with positive scalar curvature that converge in the Gromov-Hausdorff and Intrinsic Flat sense to limit spaces with “pulled regions”. The examples created rigorously using these methods were announced a few years ago and have influenced the statements of some of Gromov's conjectures concerning sequences of manifolds with positive scalar

In the present paper, we study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group UK+C of the unitization of the compact operators K(H)+C, or equivalently, the quotient UK+C/UD(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As

We study the variational integration problem for Lie algebroids and Finsler manifold in time-dependent fractal dimension. The theory in general is complexified and complex geodesics are obtained accordingly. However, decomplexification or the transition from C→R is possible if the action integral by itself is complexified. Their implications in Finsler manifold were also discussed and several consequences

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