Differential Geometry

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This text serves as a graduate-level introduction to differential geometry for mathematics and physics students, tracing the historical evolution of connection and curvature concepts to elucidate the Chern–Weil theory of characteristic classes on a principal bundle. Key milestones in differential geometry, such as Gauss' Theorema Egregium and the Gauss–Bonnet theorem, are explored. The book includes exercises that challenge the reader's understanding and highlight extensions of the theory. A basic familiarity with manifolds is required, with a deeper knowledge of differential forms necessary after the first chapter. Understanding de Rham cohomology is essential for the final third of the text. Prerequisite material is found in the author's earlier work, which can be mastered in one semester. Appendix A reviews fundamental manifold theory to aid readers. To enhance self-containment, sections on algebraic constructs like the tensor product and exterior power are also included. Differential geometry, rooted in the seventeenth century with Newton and Leibniz, gained prominence in the nineteenth century through Gauss and Riemann's contributions. Today, it is crucial for comprehending physical theories, including Einstein's general relativity, and has applications in various mathematical fields, such as topology and algebraic geometry. Its relevance extends to group theory and probability theory, making it an essential tool for math