Invariant Randers Metrics On Flag Manifolds And Cohomogeneity One Manifolds

Being a part of algebra and manifold learning,Lie group,Lie algebra and representation theory manifest their potential in both mathematics and physics.Generalized flag manifold is considered as an important branch of Lie algebra and representation theory.The corresponding theory is widely applied in quantum mechanics.Generalized flag manifold is an adjoint orbit of an element in a Lie algebra.In 1934,Ehresman took the lead in studying flag manifolds by digging into the properties of homogeneous space.Einstein metric are the solutions of the vacuum Einstein field equations.A globally defined Einstein metric is lacking.So researching Einstein metrics on flag manifolds will make a contribution to the final solution.We begin with studying generalized flag manifolds as homogeneous spaces.Since the existence of the invariant fields on homogeneous space is already confirmed,we can know invariant fields on flag manifolds do exist.It follows that Ricci curvature should have some kind of specific form.And then we directly obtain the Einstein metric because the Ricci curvature is proportional to it.Next we come to Randers matric,which can be described by navigation data.We construct the Finsler structure through navigation data in order to find out the conditions which the Randers metric should satisfy.Combine the conditions of being Einstein metric and Randers metric,we can construct the invariant Einstein-Randers metrics.Second,we pay attention to the isometry groups of cohomogeneity one manifold.With this we are capable of studying Randers metric with specific curvatures on this kind of manifold and the proof of their existence and nonexistence are given.If the group action,which is effective,of the cohomogeneity one manifold is a semisimple Lie group with its maximal compact subgroup being nontrivial and its center is an infinity group,then there is no invariant Randers metrics with negative flag curvature and isotropy S-curvature on it.Third,we choose a certain full flag manifold to calculate the invariant Einstein metric on it.Starting with its root space decomposition,we get the equations of the coefficients from the formula of Ricci tensor on homogeneous manifolds.By solving it,we find out that there is only one invariant Einstein metric on this full flag manifold.