Introduction

Summary

Global analysis has as its primary focus the interplay between the local analysis and the global geometry and topology of a manifold. This is seen classically in the Gauss–Bonnet theorem and its generalizations, which culminate in the Atiyah–Singer Index Theorem [Atiyah and Singer 1968a]. This places constraints on the solutions of elliptic systems of partial differential equations in terms of the Fredholm index of the associated elliptic operator and characteristic differential forms which are related to global topological properties of the manifold.

The Atiyah–Singer Index Theorem has been generalized in several directions, notably by Atiyah and Singer themselves [1971] to an index theorem for families. The typical setting here is given by a family of elliptic operators P = {P_b} on the total space of a fibre bundle F → M → B, where P_b is defined on the Hilbert space L²(p⁻¹(b), dvol(F)). In this case there is an abstract index class ind(P) ∈ K⁰(B). Once the problem is properly formulated it turns out that no further deep analytic information is needed in order to identify the class. These theorems and their equivariant counterparts have been enormously useful in topology, geometry, physics, and in representation theory.

A smooth manifold Mⁿ with an integrable p-dimensional subbundle F of its tangent bundle TM may be partitioned into p-dimensional manifolds called leaves such that the restriction of F to the leaf is just the tangent bundle of the leaf.