This book provides an introduction to the basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas. This new edition includes new chapters, sections, examples, and exercises.
From the reviews: "There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books." --EMS NEWSLETTER
I: GENERAL DIFFERENTIAL THEORY. 1: Differential Calculus. 2: Manifolds. 3: Vector Bundles. 4: Vector Fields and Differential Equations. 5: Operations on Vector Fields and Differential Forms. 6: The Theorem of Frobenius. II: METRICS, COVARIANT DERIVATIVES AND RIEMANNIAN GEOMETRY. 7: Metrics. 8: Covariant Derivatives and Geodesics. 9: Curvature. 10: Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle. 11: Curvature and the Variation Formula. 12: An Example of Seminegative Curvature. 13: Automorphisms and Symmetries. III: VOLUME FORMS AND INTEGRATION. 15: Volume Forms. 16: Integration of Differential Forms. 17: Stokes' Theorem. 18: Applications of Stokes' Theorem. Appendix: The Spectral Theorem.
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Softcover reprint of the original 1st ed. 1999