Book IV continues the discussion begun in the first three volumes. Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces {which} are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group ℝ² is Abelian and the �������� + ���� group\index{ax+b group} is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type ���� surfaces. These are the left-invariant affine geometries on ℝ². Associating to each Type ���� surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue ����=-1$ turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type ���� surfaces; these are the left-invariant affine geometries on the �������� + ���� group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere ����². The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.
Esteban is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He received his Ph.D. in 2011 under the direction of E. Garcia-Rio and R. Vazquez-Lorenzo. His research specialty is Riemannian and pseudo-Riemannian geometry. He has published more than 20 research articles and books.Eduardo is a Professor of Mathematics at the University of Santiago (Spain). He is a member of the editorial board of Differential Geometry and its Applications and The Journal of Geometric Analysis and leads the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He received his Ph.D. in 1992 from the University of Santiago under the direction of A. Bonome and L. Torron. His research specialty is Differential Geometry. He has published more than 120 research articles and booksPeter is a Professor of Mathematics and amember of the Institute of Theoretical Science at the University of Oregon. He is a fellow of the American Mathematical Society and is a member of the editorial board of Results in Mathematics, Differential Geometry and its Applications, and The Journal of Geometric Analysis. He received his Ph.D. in 1972 from Harvard University under the direction of L. Nirenberg. His research specialties are Differential Geometry, Elliptic Partial Differential Equations, and Algebraic topology. He has published more than 260 research articles and books.JeongHyeong is a Professor of Mathematics at Sungkyunkwan University and is an associate member of he KIAS Korea).She received her Ph.D. in 1990 from Kanazawa University in Japan under the direction of H. Kitahara. Her research specialties are spectral geometry of Riemannian submersion and geometric structures on manifolds like eta-Einstein manifolds and H-contact manifolds. She organized the geometry section of AMC 2013 (The Asian Mathematical Conference 2013), the ICM 2014 satellite conference on Geometric analysis, and geometric structures on manifolds (2016). She has published more than 81 research articles and books.Ramon is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He is a member of the Spanish Research Network on Relativity and Gravitation. He received his Ph.D. in 1997 from the University of Santiago de Compostela under the direction of E. Garcia-Rio. His research focuses mainly on Differential Geometry with special emphasis on the study of the curvature and the algebraic properties of curvature operators in the Lorentzian and in the higher signature settings. He has published more than 50 research articles and books.
