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Abstract
AbstractThe goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.
Highlights
The goal of the paper is to study the angle between two curves in the framework of metric spaces
In order to define the angle for geodesics in a more general framework, a crucial observation is that a geodesic can be seen as gradient flow of the distance function, i.e. a geodesic γ ‘represents’ the gradient of −d(γ0, γ1) d(γ1, ·) on each point γt
Inspired by the seminal work of De Giorgi on gradient flows [15], given an arbitrary metric space (X, d) with a geodesic γ : [0, 1] → X and a Lipschitz function f : X → R, we say that γ represents ∇f at time 0, or γ represents the gradient of the function f at the point p = γ(0) if the following inequality holds lim t→0 f
Summary
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. Let (Xj, dj, mj, xj), j ∈ N∪{∞} be a p-mGH converging sequence of pointed metric measure spaces and let fj ∈ L2(Xj, mj), j ∈ N ∪ {∞} be a sequence of functions.
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