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Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 137-141) and index.

12. An introduction to affine geometry -- 12.1. Basic definitions -- 12.2. Surfaces with recurrent Ricci tensor -- 12.3. The affine quasi-Einstein equation -- 12.4. The classification of locally homogeneous affine surfaces with torsion -- 12.5. Analytic structure for homogeneous affine surfaces

13. The geometry of type A models -- 13.1. Type A : foundational results and basic examples -- 13.2. Type A : distinguished geometries -- 13.3. Type A : parameterization -- 13.4. Type A : moduli spaces

14. The geometry of type B models -- 14.1. Type B : distinguished geometries -- 14.2. Type B : affine killing vector fields -- 14.3. Symmetric spaces

15. Applications of affine surface theory -- 15.1. Preliminary matters -- 15.2. Signature (2, 2) VSI manifolds -- 15.3. Signature (2, 2) bach flat manifolds.

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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 ax + b group is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type A surfaces. These are the left-invariant affine geometries on R2. Associating to each Type A surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue [mu] = -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 B surfaces; these are the left-invariant affine geometries on the ax + b group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere S2. 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.

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