一类Randers度量(英文)

研究了一类特殊的Randers度量,找到了这类度量与黎曼度量α逐点射影等价的几个方程.

Projectively flat arctangent Finsler metric

In this work, we study a class of special Finsler metrics F called arctangent Finsler metric, which is a special （α, β）-metric, where a is a Riemannian metric and β is a 1-form, We obtain a sufficient and necessary condition that F is locally projectively fiat if and only if α and β satisfy two special equations. Furthermore we give the non-trivial solutions for F to be locally projectively fiat. Moreover, we prove that such projectively fiat Finsler metrics with constant flag curvature must be locally Minkowskian.

Projectively flat Asanov Finsler metric

In this work, we study the Asanov Finsler metric F=α（β^2/α^2＋gβ/α＋1）^1/2exp{（G/2）arctan[β/（hα）＋G/2]}, where α=（αijy^iy^i）^1/2 is a Riemannian metric and β=by^i is a 1-fom, g∈（-2,2）, h=（1-g^2/4）^1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and ,Sis parallel with respect to α. Moreover, we proved that the Douglas tensor of Asanov Finsler metric vanishes if and only if β is parallel with respect to α.

On some projectively flat polynomial (α,β)-metrics

In this paper, we consider some polynomial （a,fl）-metrics, and discuss the sufficient and necessary conditions for a Finsler metric in the form F=α＋α1β＋α2β^2/α＋α4β^4/α^3 to be projectively flat, where ai 0=1,2,4） are constants with a1≠0, a is a Riemannian metric and β is a 1-form. By analyzing the geodesic coefficients and the divisibility of certain polynomials, we obtain that there are only five projectively flat cases for metrics of this type. This gives a classification for such kind of Finsler metrics.

Khler Finsler Metrics Are Actually Strongly Khler

In this paper, the Kahler conditions of the Chern-Finsler connection in complex Finsler geometry are studied, and it is proved that Kahler Finsler metrics are actually strongly Kahler.

Einstein Finsler metrics and Killing vector fields on Riemannian manifolds

We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S^3 with Ric = 2F^2, Ric = 0 and Ric =-2F^2, respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.

Laplacian on Complex Finsler Manifolds

In this paper, the Laplacian on the holomorphic tangent bundle T 1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M, F ). Utilizing the initiated "Bochner technique", a vanishing theorem for vector fields on the holomorphic tangent bundle T 1,0 M is obtained.

Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on(S^n, F) with an irreversible Finsler metric F are irrationally elliptic,there exist either exactly 2 [(n+1)/2] or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler(S^3, F) if any prime closed geodesic has non-zero Morse index.

Affinely equivalent Khler-Finsler metrics on a complex manifold

The purpose of the present paper is to investigate afflnely equivalent Kahler-Finsler metrics on a complex manifold. We give two facts （1） Projectively equivalent Kahler-Finsler metrics must be affinely equivalent; （2） a Khhler-Finsler metric is a Kaihler-Berwald metric if and only if it is affinely equivalent to a Kahler metric. Furthermore, we give a formula to describe the affine equivalence of two weakly Kaihler-Finsler metrics.

Convergence of Finslerian metrics under Ricci flow

In this work,we study the convergence of evolving Finslerian metrics first in a general flow and next under Finslerian Ricci flow.More intuitively it is proved that a family of Finslerian metrics g（t）which are solutions to the Finslerian Ricci flow converges in C~∞ to a smooth limit Finslerian metric as t approaches the finite time T.As a consequence of this result one can show that in a compact Finsler manifold the curvature tensor along the Ricci flow blows up in a short time.

Conformal vector fields on some Finsler manifolds

We study conformal vector fields on a Finsler manifold whose metric is defined by a Riemannian metric, a 1-form and its norm. We find PDEs characterizing conformal vector fields. Then we obtain the explicit expressions of conformal vector fields for certain spherically symmetric metrics on R^n.

The Schwarz-Pick lemma for planar harmonic mappings

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.

On Unitary Invariant Weakly Complex Berwald Metrics with Vanishing Holomorphic Curvature and Closed Geodesics

In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-valued smooth positive function of w ∈ R,and z is in a unitary invariant domain M C^n. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f, we prove that all the real geodesics of F =√[rf(s- t)] on every Euclidean sphere S^(2n-1) M are great circles.