Some Geometric Properties Of A Class Of Index Measure

Finsler geometry including its important exception Riemannian geometry is an important frontier subject in modern mathematics, geometry method devel-oped by finsler geometry is quite useful in exploring the theory of physics,biology mathematics and information geometry.In this thesis, the author studies the ge-ometrical properties of a class of special (a, β)-metric, i.e. exponential met-ric F=a(e1+β/a+1), where a(x,y)=√aij(x)yiyj is a Riemann metric and β(x,y)=bi(x)yi is one-form on a smooth manifold M. The sufficient and nec-essary conditions for it to be locally projectively flat, locally dually flat and for exponential metric to be of isotropic S—curvature have been discussed in this thesis. Main conclusions are as follows:Theorem3.2Let (M, a) be a n-dimensional Riemann manifold and β a1-form on M, F=a(e1+β/a+1) be a Finsler metric on M. Then F is locally projec-tively flat if and only if:(1) a is locally projectively flat, i.e. a is of constant sectional curvature.(2)β is parallel with respect to a, i.e. bi|j=0.Theorem3.3Let (M, a) be a n-dimensional Riemann manifold andβa1-form on M, F=a(e1+β/a+1) be a Finsler metric on M. Then F is locally projec-tively flat if and only if:(1) a is Euclidean metric.(2) βis constant vector.Theorem3.4Let (M, a) be a n-dimensional Riemann manifold and βa1-form on M, F=a(e1+β/a+1) be a Finsler metric on M. if F is locally projectively flat, Then its Flag Curvature is0.Theorem4.1Let (M, a) be a n-dimensional Riemann manifold andβa1- form on M,F=α(e1+β/α+1)be a Finsler metric on M.Then F is locally dually flat if and only if:(1)Gαm=1/3θmα2+2/3θym.(2)roo=2/3θβ-2/3bιθια2.(3)sιO=(4(βθι-θbι))/9. where θ=θkyk is1-form on M,θm=αimθi.Theorem5.1Let(M,α)be a n-dimensional Riemann manifold(n≥3)and β a1-form on M,F=a(e1+β/α+1) be a Finsler metric on M.Then the following conditions are equivalent:(1)F is of isotropic S-curvature.(2)β is a length-fixed one form with α.(3)S=0.(4)F is Berwald metric.(5)β is parallel with respect to α.(6)F and α have the same spay coefficients.(7)F and α are projectively related.