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.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
