We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator $\Db$ on $Δ\otimesΛ^1$. We establish various obstructions to the existence of topological reductions to PSU(3). For compact manifolds, we also give sufficient conditions for topological PSU(3)-structures that can be lifted to topological SU(3)-structures. We also construct the first known compact example of an integrable non-symmetric PSU(3)-structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic Kähler structures which are defined by an $Sp(1)\cdot Sp(2)$-invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for $\Db$ and that these equivalent conditions produce constraints on the Ricci tensor.
Here, it is introduced a concept of convolution metric in Finslerian Geometry. This convolution metric is a kind of function obtained by a given mathematical operation between two Finslerian metrics. Some basic properties of the Finslerian convolution metrics are studied. Then it is characterized Finslerian convolution metrics which are of type Riemannian, Minkowskian as well as Randers. Furthermore, some examples of the Finslerian convolutions are given.
Given a parabolic geometry, it is sometimes possible to find special metrics characterised by some invariant conditions. In conformal geometry, for example, one asks for an Einstein metric in the conformal class. Einstein metrics have the special property that their geodesics are distinguished, as unparameterised curves, in the sense of parabolic geometry. This property characterises the Einstein metrics. In this article we initiate a study of corresponding phenomena for other parabolic geometries, in particular for the hypersurface CR and contact Legendrean cases.
We review results about $G_2$-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.
In this paper, first we derive an explicit formula for the flag curvature of a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce it for naturally reductive homogeneous Finsler space with the above mentioned metrics.
In this paper, first we prove the existence of invariant vector field on a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce an explicit formula for the the $S$-curvature of homogeneous Finsler space with these metrics. Using this formula, we further derive the formula for mean Berwald curvature of the homogeneous Finsler space with the above mentioned metrics.
We describe the Special Kähler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the Kähler potential. This extends to the case of a singular spectral curve and we show that this defines the Special Kähler structure on certain natural integrable subsystems. Examples include the extreme case where the metric is flat.
The notion of a Douglas space of second kind of a Finsler space with $(α, β)$-metric was introduced by I. Y. Lee [9]. Since then, so many geometers have studied this topic e. g., [14]. In this paper, we prove that a Douglas space of second kind with special $% (α, β)$-metric $α+εβ+ k \frac{β^2}{α}$ is conformally transformed to a Douglas space of second kind. Further, we obtain some results which prove that a Douglas space of second kind with certain $(α, β)$-metrics such as Randers metric, Kropina metric, first approximate Matsumoto metric and Finsler space with square metric is conformally t…
Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dime…
In this paper, we study geodesics and geodesic vectors for homogeneous exponential Finsler space and homogeneous infinite series Finsler space. Further, we find necessary and sufficient condition for a non-zero vector in these homogeneous spaces to be a geodesic vector.
In this paper, we study left invariant conic Finsler metrics on the 2-dimensional non-Abelian Lie group $G$ with nowhere vanishing spray vector fields, and classify those satisfying the constant curvature condition, the Landsberg condition or the Berwald condition respectively. We prove that any left invariant conic Landsberg metric on $G$ must be Berwald. This discovery enable us to propose a homogeneous conic Landsberg Conjecture, which guesses that every homogeneous conic Landsberg metric is Berwald, and prove the 2-dimensional case for it.
We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms, connection 1-forms and equivariant $3\times3$ positive-definite symmetric matrix-valued functions on principal $G$-bundles over 3-manifolds. As applications, we describe regular parts of $G_2$-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of $G=\mathrm{T}^3$ and $\math…
Local Models for Special Kähler Metric Singularities Along the Discriminant Locus of the $\mathrm{SL}_2(\mathbb{C})$ Hitchin Base
College of Information Science and Technology, Jinan University（暨南大学信息科学技术学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Oeljeklaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension $K$ of $Q$ and a subgroup of units $U$. We characterize the existence of pluriclosed metrics (also known as strongly K\" ahler with torsion (SKT) metrics) on any OT manifold $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number $b_3$ and the Dolbeault cohomology group $H^{2,1}_{\overline{\partial}}$. Combined with the main result in [D20], these numerical conditions render explicit examples of…
We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $n_1$-dimensional and $n_2$-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $(n_1-1)$-dimensional group of transformations to the case of positive constant Ricci curvature ($λ>0$).
Igarashi introduce the concept of $(α, β)$-metric in Cartan space $\ell^{n}$ analogously to one in Finsler space and obtained the basic important geometric properties and also investigate the special class of the space with $(α, β)$-metric in $\ell^{n}$ in terms of $ 'invariants' $. In the present paper we determine the $ 'invariants' $ in two different cases of deformed infinite series metric which characterize the special classes of Cartan spaces $\ell^{n}$.
There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and the universal covering group $\widetilde{\mathrm{E}_0}(2)$ of the connected component of the Euclidean group. For each $G$ among these Lie groups, we give explicitly the list of all Lorentzian left invariant metrics on $G$, up to un automorphism of $G$. Moreover, for any Lorentz…
We obtain necessary conditions for the existence of special Kähler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we determine the whole moduli space of special Kähler structures with fixed singularities. The tool we develop for this aim is a correspondence between special Kähler structures and pairs consisting of a cubic differential and a hyperbolic metric.
School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China（中国科学院吴文俊数学重点实验室，中国科学技术大学）；School of Mathematical Sciences, Luoyang Normal University（洛阳师范学院数学科学学院）
Hamel functions of a spray play an important role in the study of the projective metrizability of the concerned spray, and Funk functions are special Hamel functions. A Finsler metric is a special Hamel function of the spray induced by the metric itself and a Funk metric is a special Funk function of a Minkowski spray. In this paper, we study sprays on a Hamel or Funk function model. Firstly, we give some basic properties of a Hamel or Funk function of a spray and some curvature properties of a Hamel or Funk function in projective relations. We use the Funk metric to construct a family of spra…
