Aspects of Differential Geometry IV

Abstract

Book IV continues the discussion begun in the first three volumes. Although it is aimed at firstyear 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 R 2 is Abelian and the ax C 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 R 2 . Associating to each Type A surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue D 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 C b group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere S 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.

Symmetric affine surfaces with torsion

“…The possibilities of Theorem 3 are not exclusive; there are geometries which can be realized both as Type A and Type B structures. However, no Type A or Type B structure is also Type C. We refer to Calviño-Louzao et al [3] for additional information in this regard.…”

“…We compute that 0 = ρ 22;2 = 2(A 12 1 − 1)A 12 1 . There are two subcases: (3). If on the other hand A 11 1 = 2 and A 11 2 = 0, then we can rescale x 2 to obtain Assertion (4) and again we have fixed the gauge as the parameter b plays no role.…”

The curve shortening flow in the metric-affine plane

“…During last decades, many results have appeared in the differential geometry of a manifold with an affine connection ∇ (which is a method for transporting tangent vectors along curves), e.g., collective monographs [2,8]. The difference T = ∇ − ∇ (of ∇ and the Levi-Civita connection ∇ of g), is a (1,2)-tensor, called contorsion tensor.…”

The Curve Shortening Flow in the Metric-Affine Plane

“…In recent decades, many results have appeared in the differential geometry of a manifold with an affine connection ∇ (which is a method for transporting tangent vectors along curves), e.g., collective monographs [6,7]. The difference T = ∇ − ∇ (of ∇ and the Levi-Civita connection ∇ of g), is a (1,2)-tensor, called contorsion tensor.…”