Reconstruction and interpolation of manifolds II: Inverse problems for Riemannian manifolds with partial distance data

Abstract

We consider how a closed Riemannian manifold and its metric tensor can be approximately reconstructed from local distance measurements. In the part 1 of the paper, we considered the construction of a smooth manifold in the case when one is given the noisy distances d(x, y) = d(x, y) + εx,y for all points x, y ∈ X, where X is a δ-dense subset of M and |εx,y| < δ. In this paper we consider a similar problem with partial data, that is, the approximate construction of the manifold (M, g) when we are given d(x, y) for x ∈ X and y ∈ U ∩X, where U is an open subset of M . As an application, we consider the inverse problem of determining the manifold (M, g) with non-negative Ricci curvature from noisy observations of the heat kernel G(y, z, t) with separated observation points y ∈ U and source points z ∈ M \ U on the time interval 0 < t < 1.