Abstract
We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space (X,d_X). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold Ssubset {{mathbb {R}}}^m, m>n needs to be constructed to approximate a point cloud in {{mathbb {R}}}^m. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov–Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in {{mathbb {R}}}^m and interpolated to a smooth submanifold.
Highlights
Introduction and the Main Results1.1 Geometrization of Whitney’s Extension ProblemIn this paper, we develop a geometric version of Whitney’s extension problem
We develop a geometric version of Whitney’s extension problem
When dealing with inverse problems, it is assumed that the data set X comes from some unknown Riemannian manifold M, and some a priori bounds on the geometry of this manifold are given
Summary
We develop a geometric version of Whitney’s extension problem. Let f : K → R be a function defined on a given (arbitrary) set K ⊂ Rn, and let m ≥ 1 be a given integer. Remark 1 The assumption that X is δ-intrinsic in Theorem 1 is not crucial Without this assumption, the following more technical variant of the theorem holds: If a metric space X is δ-close to Rn at scale r , where δ/r is bounded above by a constant depending on n, there exists a complete (possibly not connected). Theorem 1 has the following corollary that concerns neighborhoods of smooth manifolds and the class of metric spaces that satisfy a weak δ-flatness condition in the scale of injectivity radius and a strong δ-flatness condition in a small-scale r. We only mention the fact that finite-dimensional boundaryless Alexandrov spaces with twosided curvature bounds are Riemannian manifolds with C1,α metrics ([71], see [8, Theorem 14.1]).
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