The Conjugate And Cut Locus Of Symmetric Spaces And The Homogeneous Randers-Einstein Metrics On The Aloff-Wallach Spaces

This dissertation focuses on the classification of a special kind of connected and simply-connected Riemannian symmetric spaces, the relationships between conjugate loci and cut loci on connected and simply-connected Lorentzian symmetric spaces, and the classification of the homogeneous Randers-Einstein metrics on Aloff-Wallach spaces.Firstly, Based on the results in the articles[49,50], by discussion of the vertex set of Cartan polyhedra, we obtain the relationship between the antipodal set and the cut loci, then we describe the antipodal set for each point on the simply-connected compact irreducible Riemannian symmetric spaces, and then we obtain the classification of con-nected and simply-connected compact irreducible Riemannian symmetric spaces with the property:there is only one antipodal point for each point, then we could find all such connected and simply-connected compact Riemannian symmetric spaces with the property above by the de Rham decomposition of simply-connected manifolds.Next in this article we mainly study the relationships between conjugate loci and cut loci on connected and simply-connected Lorentzian symmetric spaces:According to the classification of connected and simply-connected Lorentzian symmetric spaces, we Firstly discuss the problem on two kinds of important spacetimes(the universal cover spaces of de Sitter spacetimes and Cahen-Wallach spaces), and based on these discussions we consider the problems on the products of these spacetimes and simply-connected Riemannian symmetric spaces, and finally we could give a proof of a conclu-sion similar to the conclusion which R.Critendden gave in1962in the article[14] except for the universal cover spaces of Anti-de Sitter spacetimes and the related products.At the end of the article, based on the conclusion of the articles[17,18,19], using the formulae of Zermelo navigation, we could give a classification of homogeneous Randers-Einstein metrics by finding the Killing fields which are invariant under the known homogeneous (Riemannian) Einstein metrics on Aloff-Wallach spaces.