Finsler流形的Cartan型1-形式的一些性质

On Some Properties of the Cartan Type One Form of a Finsler Manifold

给出了Cartan型1-形式的外微分与Bott-陈联络的曲率之间的关系,探讨了其与畸变、S曲率之间的关系,证明了Berwald流形的Cartan型1-形式为恰当形式.利用Cartan型1-形式构造了Finsler流形的射影球丛上的一个Randers度量,证明该度量为Landsberg度量的充要条件是底流形为Riemann流形.证明了Cartan型1-形式及Cartan 1-形式的对偶向量场为共型向量场的充要条件是底流形为Riemman流形.

In this paper ,we establish the relationships between the Cartan type one form and the curvature of the Bott-Chern connection ,distortion and S curvature .It is proved that the Cartan type one form is ex-act for a Berwald manifold .Using the Cartan type one form ,we define a natural Randers metric on the projective sphere bundle of a Finsler manifold .We prove that this Randers metric is the sufficient and nec-essary condition of Landsberg if and only if the base manifold is Riemannian .We also prove that the dual of the Cartan type one form is a conformal vector field if and only if the base manifold is Riemannian and give a similar discussion about the Cartan one form .