Differential Geometry: A Rigorous Introduction

Differential Geometry: A Rigorous Introduction is a concise, single-volume graduate textbook covering both the classical and modern phases of the subject, with every theorem stated precisely and proved in full.

Part I: Curves and Surfaces in Three-Space. Parametrized curves, arc length, the Frenet–Serret formulas, regular surfaces, the first and second fundamental forms, the Gauss map, Gaussian curvature, the Theorema Egregium, and the local and global Gauss–Bonnet theorems.

Part II: Smooth Manifolds. Charts and atlases, smooth maps, tangent spaces, vector fields and flows, the rank theorem and Whitney embedding, the cotangent bundle, differential forms, and Stokes' theorem on oriented manifolds with boundary.

Part III: Riemannian Geometry. Riemannian metrics and the Levi-Civita connection, geodesics and the exponential map, the Hopf–Rinow theorem, the Riemann curvature tensor and sectional curvature, Jacobi fields, and the comparison theorems of Bonnet–Myers and Cartan–Hadamard.

The sign conventions follow Lee and do Carmo: the sectional curvature of the round unit sphere is +1, and the shape operator of the outward-oriented sphere is minus the identity. Definitions, theorems, propositions, lemmas, corollaries, examples, and remarks share a single numbering stream within each chapter.

Inside the book:

• 14 main chapters and 4 appendices across 200 pages, organized into three parts that read in sequence
• 20 original figures illustrating the Frenet frame, the Gauss map, classification of surface points, geodesic triangles, the exponential map, Jacobi fields, and curvature comparison
• 110 exercises graded from routine computation to substantial technique, at the end of every chapter and every appendix
• Worked solutions to all odd-numbered exercises, collected in a dedicated section at the back of the book — even-numbered exercises are left unsolved, so the book can be used as a main text for a course that assigns homework
• Four self-contained appendices on point-set topology, multilinear algebra, ordinary differential equations, and Sard's theorem, provided for reference so the main chapters stay focused

Prerequisites. Fluency in linear algebra, real analysis on Rⁿ (continuity, differentiation, the implicit and inverse function theorems), and the basic language of point-set topology. Familiarity with ordinary differential equations is useful; the results actually needed are reviewed in Appendix C. No prior exposure to differential geometry, smooth manifolds, or tensor analysis is assumed.

Who this book is for:

• First-year graduate students beginning a formal course in differential or Riemannian geometry
• Advanced undergraduates preparing for graduate study who want the classical and modern pictures in one volume
• Physicists, engineers, and computer scientists who need a mathematically complete reference for curvature, geodesics, and the manifold apparatus
• Instructors seeking a compact main text for a two-semester sequence, with half the exercise bank available to assign as unsolved homework
• Anyone returning to the subject who wants a theorem-by-theorem treatment with every proof present and every sign convention fixed

Differential Geometry: A Rigorous Introduction is a textbook, not an informal tour. Every theorem is stated precisely, proved in full, and placed in the logical architecture of the subject. The figures support intuition; the proofs do not depend on them.