We study the magnitude homology of geodesic metric spaces of curvature , especially spaces. We will show that the magnitude homology of such a metric space vanishes for small and all . Consequently, we can compute magnitude homology in small length gradings for spheres , the Euclidean spaces , the hyperbolic spaces and real projective spaces with the standard metric. We also show that the existence of a closed geodesic in a metric space guarantees the nontriviality of magnitude homology.
"Magnitude homology of geodesic metric spaces with an upper curvature bound." Algebr. Geom. Topol. 21 (2) 647 - 664, 2021. https://doi.org/10.2140/agt.2021.21.647