摘要
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).
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）.
基金
supported by the Program for New Century Excellent Talents in Fujian Province and Natural Science Foundation of China (Grant Nos. 10971170,10601040)
