A local classification of a class of (α,β)(α,β) metrics with constant flag curvature

We first compute Riemannian curvature and Ricci curvature of (α,β)(α,β) metrics. Then we apply these formulae to discuss a special class (α,β)(α,β) metrics F=α(1+βα)p (|p|⩾1|p|⩾1) which have constant flag curvature. We obtain the sufficient and necessary conditions that F=(α+β)2α have constant flag curvature. Then we prove that such metrics must be locally projectively flat and complete their local classification. Using the same method we find a necessary condition that flag curvature of F=α2α+β is constant and proved that there are no non-trivial Matsumoto metrics. Furthermore, we give a negative answer whether there are non-trivial metrics F=α(1+βα)p (|p|⩾1|p|⩾1) of constant flag curvature when β is closed.