The Risk Transmission Mechanism of Global Stock Markets from the Perspective of Entropy-Riemann Geometry: Theoretical Construction and Empirical Analysis

Abstract This article innovatively combines entropy theory and Riemannian geometry to construct a new framework for in-depth analysis of the global stock market’s complexity and risk-transmission mechanism. By abstracting the stock markets of various countries as submanifolds on Riemannian manifolds and introducing entropy functions to quantify market uncertainty and complexity, this paper successfully integrates entropy information with Riemannian metrics, allowing for the accurate calculation of geometric distances between different markets. Through the Ricci curvature and the Laplace–Beltrami operator, the market risks’ aggregation and diffusion are analyzed in-depth. Research has found that emerging markets like Argentina exhibit lower entropy values and higher independence, while developed economies such as the United States and the United Kingdom show higher entropy values and complexity, thus revealing these markets’ core positions and efficient risk–transmission pathways in the global risk network. Additionally, through the heterogeneity analysis of volatility and returns, this article reveals significant differences in risk cyclicality and market volatility among different countries, thereby forming multi-level roles and core-periphery structures in the global financial network. This entropy-Riemann-geometry-based analysis method not only provides a powerful tool for understanding the global stock market’s dynamics but also offers investors a refined investment strategy based on geometric distance and risk aggregation, demonstrating its cutting-edge nature and practicality in financial research.