Differential Geometry

A concise and rigorous introduction to differential geometry, taking the reader from curves in three-space to the comparison theorems of Riemannian geometry in a single self-contained volume.

Differential Geometry: A Rigorous Introduction is a graduate-level textbook that treats both the classical and modern phases of the subject, in that order, with full proofs. Part I develops the geometry of curves and surfaces in R³: parametrization, curvature and torsion, the Frenet-Serret formulas, regular surfaces, the first and second fundamental forms, the Gauss map, and Gaussian curvature, culminating in the Theorema Egregium and the local and global Gauss-Bonnet theorems. Part II recasts these ideas in coordinate-free language: smooth manifolds, the tangent bundle, vector fields and flows, the rank theorem and Whitney embedding, the cotangent bundle and differential forms, and Stokes' theorem on oriented manifolds with boundary. Part III develops Riemannian geometry proper: 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, so Proposition 1.5 and Theorem 1.8 both live in Chapter 1.

What is inside

• 14 main chapters and 4 appendices across 170 pages, organized into three parts that read in sequence
• 20 instructional figures with precise captions: the Frenet frame, the osculating circle, the Gauss map, elliptic/parabolic/hyperbolic points, geodesic triangles and angle excess, atlas and transition maps, flow boxes, pullback of forms, the exponential map, Jacobi fields and geodesic spread, and a three-panel comparison of positive, zero, and negative curvature
• 68 exercises graded from routine to substantial, at the end of Chapters 2-5, 8, 12, 13, and at the end of each appendix
• Full solutions to every main-chapter exercise and to odd-numbered appendix exercises, collected in a dedicated Solutions section at the back of the book
• Four self-contained appendices covering point-set topology, multilinear algebra and bilinear forms, 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 in mathematics beginning a formal course in differential geometry or Riemannian geometry
• Advanced undergraduates preparing for graduate study, wanting 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
• Anyone returning to the subject who wants a compact, 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 are aids to geometric intuition; the proofs do not depend on them.