从Berwald空间到Riemann空间的射影变换
|
摘要
给定一个n维紧致无边的微分流形M,已证明:如果tr_FRic≤s_F,那么从Berwald空间(M,F)到Riemann空间(M,F)的任何逐点C-射影变换均是平凡的,并且F关于F是平行的。这里,tr_FRic表示F的Ricci曲率张量Ric关于F的迹,s_F:=tr_FRic是F的数量曲率。特别地:如果tr_FRic≤s_F,那么从Riemann空间(M,F)到另一个Riemann空间(M,F)的任何射影变换都是平凡的。
Given a compact and boundaryless n-dimensional differentiable manifold M,we showed that any pointwise C-projective changes from a Berwald space( M,F) to a Riemann space( M,F) is trivial if tr_FRic≤s_F,where tr_FRic denotes the trace of the Ricci curvature Ric of F with respect to F and s_F: = tr_FRic is the scalar curvature of F. In particular,we showed that any projective change from a Riemann space( M,F) to another Riemann space( M,F) is trivial if tr_FRic≤s_F.
Given a compact and boundaryless n-dimensional differentiable manifold M,we showed that any pointwise C-projective changes from a Berwald space( M,F) to a Riemann space( M,F) is trivial if tr_FRic≤s_F,where tr_FRic denotes the trace of the Ricci curvature Ric of F with respect to F and s_F: = tr_FRic is the scalar curvature of F. In particular,we showed that any projective change from a Riemann space( M,F) to another Riemann space( M,F) is trivial if tr_FRic≤s_F.
引文
[1]BAO D,CHERN S S,SHEN Z.An introduction to Riemann-Finsler geometry[M].Springer Science&Business Media,2012.
[2]CHEN X,SHEN Z.A comparison theorem on the Ricci curvature in projective geometry[J].Annals of Global Analysis and Geometry,2003,23(2):141-155.
[3]SHEN Z.Differential geometry of sprays and Finsler spaces[M].[S.l.]:Kluwer Academic Publishers,2001.
[4]MATSUMOTO M.Projective changes of Finsler metrics and projectively flat Finsler spaces[J].Tensor,NS,1980,34:303-315.
[5]SHEN Z.On projectively related Einstein metrics in Riemann-Finsler geometry[J].Mathematische Annalen,2001,320(4):625-647.
[6]FUKUI M,YAMADA T.On projcctive mappings in Finsler geometry[J].Tensor,NS,1981,31:216-222.
[7]ANTONELLI P L,INGARDEN R S,MATSUMOTO M.The theory of sprays and Finsler spaces with applications in physics and biology[M].[S.l.]:Springer Science&Business Media,2013.
[8]SHEN Z.Lectures on Finsler Geometry[M].[S.l.]:World Scientific Co.,Singapore,2001.
[2]CHEN X,SHEN Z.A comparison theorem on the Ricci curvature in projective geometry[J].Annals of Global Analysis and Geometry,2003,23(2):141-155.
[3]SHEN Z.Differential geometry of sprays and Finsler spaces[M].[S.l.]:Kluwer Academic Publishers,2001.
[4]MATSUMOTO M.Projective changes of Finsler metrics and projectively flat Finsler spaces[J].Tensor,NS,1980,34:303-315.
[5]SHEN Z.On projectively related Einstein metrics in Riemann-Finsler geometry[J].Mathematische Annalen,2001,320(4):625-647.
[6]FUKUI M,YAMADA T.On projcctive mappings in Finsler geometry[J].Tensor,NS,1981,31:216-222.
[7]ANTONELLI P L,INGARDEN R S,MATSUMOTO M.The theory of sprays and Finsler spaces with applications in physics and biology[M].[S.l.]:Springer Science&Business Media,2013.
[8]SHEN Z.Lectures on Finsler Geometry[M].[S.l.]:World Scientific Co.,Singapore,2001.