On Exponential Metrics With Isotropic S-Curvature And Finsler Metrics Which Are Projectively Equivalent To A Riemann Metric

In this thesis, we study some properties of S-curvature of the exponential metric in the form F =αe^ks (where is a Riemannian metric,β= b_i ( x ) yi is a non-zero 1-form, k is a constant and k≠0). We prove that this kind of (α,β)- metrics are of isotropic S-curvature if and only if they are of isotropic mean Berwald curvature . In this case, S-curvature vanishes, i.e. S=0, and they are weakly Berwald metrics. We also discuss the conditions under which the exponential metrics are locally Minkowskian. Further, we study conformally related Finsler metrics. Finally, we give some properties of the Finsler metrics which are projectively related to a Riemann metricα. Besides, using Maple programme, we figure out the geodesic coefficients and the mean Cartan torsion of exponential metric.