Differential Operators Over Manifolds

Abstract

The modern concept of a closed “manifold” allows us to generalize and simultaneously drastically simplify the index problem by eliminating boundary conditions. For example, the homogeneous Laplace equation Δu = 0 on the disk has infinitely many linearly independent solutions, viz. the harmonic functions, while the corresponding Laplace equation on the sphere has a one-dimensional solution space consisting of the constant functions. In this respect the notion of differentiable manifold does not make the mathematics more complicated, but is a genuine first approximation to the “difficult” boundary value problems in Euclidean space ℝⁿ.