On Some Properties Of The Cartan Type One Form Of A Finsler Manifold

In the paper “Adiabatic Limit and Connections in Finsler Geometry”, Prof. Fengand Dr. Li identify the well-known Chern connection in Finsler geometry with the Bottconnection in foliation theory by using a natural foliation structure in the projectivesphere tangent bundle SM of a Finsler manifold M, and then they considered thedifference H of the Cartan connection and the Bott connection. By taking the trace ofH, they got a global1-form on SM. This1-form is formally similar to thefamous Cartan1-form I but has the different geometric feature. We call it theCartan type1-form.This paper mainly study some properties of the Cartan type1-form. Firstly, wecomputed the exterior differentiations of the Cartan type1-form and the Cartan1-form, and prove that if the base manifold M is a Berwald manifold, then d0, andM is Riemann manifold if and only if dI0. Secondly, we discuss the relationsbetween the Cartan type1-form and the Shen’s distortion. We prove that the verticaldifferentiations of the Shen’s distortion is equal to. Thirdly, by using the Cartan type1-form, we define a natural Randers manifold structure on SM and prove that thisRanders manifold is Langsberg if M is Riemannian. At last, we prove that the dualvector field of the Cartan type1-form (resp. the dual vector field of the Cartan1-form)is conformal if and only if it is Killing and further, if and only if it is Riemannian.