"Calculus and Mechanics on Two-Point Homogeneous Riemannian Spaces" is a specialized book that deals with the mathematical foundations and applications in geometry. It explores the concepts of constant curvature and their connections to other areas of mathematics, such as integrable partial differential equations and Hamiltonian systems. The book highlights the role of geodesic flows on compact surfaces with constant negative curvature and their significance for ergodic theory. Additionally, the hyperbolic space H(R) is treated as the space of velocities in special relativity, emphasizing the relevance of the topic in modern physics. The comprehensive analysis and theoretical foundations make this work a valuable resource for students and professionals engaged with the mathematical aspects of Riemannian geometry and its applications in various sciences.