The course "Math 740: Foundations of Differential Geometry" is a foundational graduate-level course that delves into the concepts of connection and curvature on manifolds, with a focus on Riemannian and Lorentzian metrics. Key topics include tensor computations, the exponential map, and various interpretations of curvature. The course aims to provide students with detailed knowledge of these concepts and their applications, such as understanding the precession of Mercury's orbit through relativistic effects in the Schwarzschild model of spacetime. The course format includes two 75-minute lectures per week, graded problem sets, and a written final exam. Recommended resources include "Semi-Riemannian Geometry" by B. O'Neill and "Introduction to Smooth Manifolds" (2nd ed.) by J.M. Lee.The course "Math 740: Foundations of Differential Geometry" is a foundational graduate-level course that delves into the concepts of connection and curvature on manifolds, with a focus on Riemannian and Lorentzian metrics. Key topics include tensor computations, the exponential map, and various interpretations of curvature. The course aims to provide students with detailed knowledge of these concepts and their applications, such as understanding the precession of Mercury's orbit through relativistic effects in the Schwarzschild model of spacetime. The course format includes two 75-minute lectures per week, graded problem sets, and a written final exam. Recommended resources include "Semi-Riemannian Geometry" by B. O'Neill and "Introduction to Smooth Manifolds" (2nd ed.) by J.M. Lee.