# Angles between curves in metric measure spaces

## TL;DR

This paper introduces a new way to define angles between curves in metric measure spaces, linking it with optimal transportation and curvature conditions, and proves the cosine formula in certain curvature-dimension spaces.

## Contribution

It proposes a novel notion of angle in metric spaces that aligns with classical concepts and applies to spaces with Ricci curvature bounds.

## Key findings

- Validates the cosine formula in $RCD^{*}(K,N)$ spaces
- Ensures compatibility with classical angles in Riemannian and Alexandrov spaces
- Connects angle notions with optimal transportation theory

## Abstract

The 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.