David J. Wraith

Ptolemaic spaces and CAT (0)

We consider Ptolemy’s inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy’s inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

Exotic spheres and curvature

Since their discovery by Milnor in 1956, exotic spheres have provided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres.

Moduli Spaces of Riemannian Metrics

Part I: Positive scalar curvature.- The (moduli) space of all Riemannian metrics.- Clifford algebras and spin.- Dirac operators and index theorems.- Early results on the space of positive scalar curvature metrics.- Kreck-Stolz invariants.- Applications of Kreck-Stolz invariants.- The eta invariant and applications.- The case of dimensions 2 and 3.- The observer moduli space and applications.- Other topological structures.- Negative scalar and Ricci curvature.- Part II: Sectional curvature.- Moduli spaces of compact manifolds with positive or non-negative sectional curvature.- Moduli spaces of compact manifolds with negative and non-positive sectional curvature.- Moduli spaces of non-compact manifolds with non-negative sectional curvature.- Positive pinching and the Klingenberg-Sakai conjecture.

On a theorem of Ambrose

A Riccati inequality involving the Ricci curvature can be used to deduce many interesting results about the geometry and topology of manifolds. In this note we use it to present a short alternative proof to a theorem of Ambrose.

On Ptolemaic metric simplicial complexes

We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy’s inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions.

Spaces of metrics

The aim of this introductory chapter is to present some basic aspects of analysis and topology for the space R(M) of complete Riemannian metrics on a smooth manifold M. We also consider the corresponding moduli space M(M), which is the quotient of R(M) by the action of the diffeomorphism group Diff(M).

Clifford algebras and spin

The aim of this chapter is to develop the concept of a spin group and certain related ideas such as spin structures, spinor representations and spinor bundles. These all play a crucial role in positive scalar curvature geometry, which we will explore in subsequent sections.

Non-negative sectional curvature moduli spaces on open manifolds

In this chapter we will describe results about spaces and moduli spaces of complete Riemannian metrics with non-negative sectional curvature on open manifolds. A new and important tool for understanding these spaces involves employing properties of the so-called ‘souls’ of the metrics, and we start with a discussion of these.

Moduli spaces of Riemannian metrics with negative sectional curvature

In this chapter we shall discuss moduli spaces of negatively curved metrics. Here, in fact, only the case of sectional curvature is of any further interest as a consequence of the work of Lohkamp discussed in §8.5.

A survey of other results

In this chapter we present a diverse selection of results about spaces and moduli spaces of metrics, which stand to some extent outside the themes developed in this book so far. The first section concerns the work of Botvinnik and Gilkey on moduli spaces of positive scalar curvature metrics for spin manifolds in odd dimensions with finite fundamental groups. This topic is the most closely related to subject matter presented earlier in that it involves index theory, and in particular utilizes the eta invariant in a significant way.