Riemannian geometry on the diffeomorphism group of the circle

Abstract

The topological group $\mathcal{D}^k(\mathbb{S})$ of diffeomorphisms of the unit circle $\mathbb{S}$ of Sobolev class H ^k, for k large enough, is a Banach manifold modeled on the Hilbert space $H^k(\mathbb{S})$. In this paper we show that the H ¹ right-invariant metric obtained by right-translation of the H ¹ inner product on $T_{\rm id}\mathcal{D}^k(\mathbb{S})\simeq H^k(\mathbb{S})$ defines a smooth Riemannian metric on $\mathcal{D}^k(\mathbb{S})$, and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H ¹ right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.