The Stiffness Of The Finsler Manifold And The Geodesic Vector On The Lie Group

Two parts are included in this thesis.Firstly, we study geometric properties of the Finsler manifolds with strictly negative flag curvature as well as constant S-curvature and Finsler manifold with strictly negative scalar curvature. We obtain some results if some non-Riemann quantities(for example: the mean Cartan tortion, Matsumoto-tortion and so on) on the Finsler manifold satisfy certain growth condition. Main re-sults are as following:Theorem 3.1.1 Suppose that (M, F) is an n-dimensional complete Finsler manifold with flag curvature K ? - ?(? is a fixed positive constant), and has almost constant S-curvature. If the mean Cartan tortion I grows sub-exponen tially a rate of (?), then F is Riemannian.Theorem 3.2.1 Suppose that (M, F) is an n-dimensional (n> 3) complete Finsler manifold with scalar curvature K ?-?(? is a fixed positive constan-t). If the Matsumoto-tortion My grows sub-exponentially a rate of (?), then F is Randers metric. Specially, if it is a Finsler manifold with scalar curvature K ?-?, then it must be Randers metric.The geodesic vector has important meaning for studying geodesics of the Finsler metric on a Lie group. Thus, the second part of this thesis contributes to the geodesic vector of a 3-dimensional Lie group with left invariant Kropina and Matsumoto metric. We choose a suitable basis, then geodesic vectors are described by a set of equations the component of the geodesic vector satisfies under this properly choosen basis. We have the following result:Theorem 4.1.1 Let F be a left invariant Kropina metric defined by a Riemann metric a and vector field X = ?e1(0 < ? < 1) on a connected u-nimodular Lie group G of dimension n ? 3, i.e. F(x,y) =ax(y,y)/ax(X,y). Then y=y1e1+y2e2+y3e3?g is a geodesic vector if and only if it satisfies the follow-ing equations:(?2-?3)y2y3 = 0,2(?3-?1)y12y3+?1y3|y|2=0,2(?1-?2)y12y2-?1y2|y|2=0.Here |y|2=y12+y22+y32, {?i} is the structure constant of Lie group G with a properly choosen basis {ei}. Specially, if ?1??2 =?3 ?0, y?g is a geodesic vector if and only if y?Span{e1}.