Magnitude homology of geodesic metric spaces with an upper curvature bound

In this article, we study the magnitude homology of geodesic metric spaces of curvature \(\leq \kappa\), especially \({\rm CAT}(\kappa)\) spaces. We will show that the magnitude homology \(MH^{l}_{n}(X)\) of such a meric space \(X\) vanishes for small \(l\) and all \(n > 0\). Conseqently, we can compute a total \(\mathbb{Z}\)-degree magnitude homology for small \(l\) for the shperes \(\mathbb{S}^{n}\), the Euclid spaces \(\mathbb{E}^{n}\), the hyperbolic spaces \(\mathbb{H}^{n}\), and real projective spaces \(\mathbb{RP}^{n}\) with the standard metric. We also show that an existence of closed geodesic in a metric space guarantees the non-triviality of magnitude homology.