When is the Carath\'eodory metric a K\"ahler metric?
When is the Carath\'eodory metric a K\"ahler metric?
We prove that if the Carath\'eodory metric on a strictly pseudoconvex domain with a smooth boundary is a K\"{a}hler metric, then the domain is biholomorphic to a ball. Additionally, we highlight the relationships between the Lu constant and the holomorphic Bergman curvature.