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Differential Geometry of Curves and Surfaces [Tapp] - mklabgr - 09-03-2026 Differential Geometry of Curves and Surfaces Author: Kristopher Tapp Publication date: September 2016 Publisher: Springer Cham Series:Undergraduate Texts in Mathematics Kristopher Tapp’s Differential Geometry of Curves and Surfaces is an undergraduate-level introduction to classical differential geometry that aims to combine rigorous mathematics with geometric intuition and real-world applications. Beginning with the differential geometry of curves, the book develops fundamental ideas such as curvature, torsion, evolutes, involutes and cycloids before progressing to parametrized surfaces, tangent spaces, the first and second fundamental forms, Gaussian curvature and related concepts. Its prerequisites are relatively modest, and proofs contain many intermediate steps, making it suitable both for a first course and as preparation for graduate mathematics or mathematical physics.  The later chapters develop some of the central ideas connecting local and global geometry. Geodesics are studied as the natural analogue of straight lines on curved surfaces, together with parallel transport and related geometric phenomena. The book culminates in the Gauss–Bonnet theorem, which establishes a remarkable relationship between curvature and topology. In its classical form, for a compact oriented surface MM, ∫MK dA=2πχ(M),\int_M K\,dA = 2\pi\chi(M), where KK is the Gaussian curvature and χ(M)\chi(M) is the Euler characteristic. Thus a quantity obtained by measuring curvature locally across the surface determines a global topological invariant—one of the fundamental insights of differential geometry. The book's six principal chapters progress through Curves → Additional Topics in Curves → Surfaces → Curvature → Geodesics → Gauss–Bonnet.  A major strength is Tapp's emphasis on visualization and applications. The text uses extensive color illustrations and examples involving cartography, conformal and area-preserving maps, Huygens' work on pendulum clocks, cycloids, optics, gears, Green's theorem and the planimeter, Clairaut's theorem as a conservation law, Foucault's pendulum, and parallel transport. These applications are not substitutes for rigorous proofs; rather, they motivate abstract concepts and make their geometric meaning clearer. This combination makes the book particularly attractive for students encountering differential geometry for the first time.  Key takeaways Level: Mainly advanced undergraduate, but useful preparation for graduate differential geometry. Core topics: curves, surfaces, curvature, geodesics, parallel transport and the Gauss–Bonnet theorem. Approach: unusually visual and application-oriented while retaining mathematical rigor. Best feature: it shows how local differential quantities such as curvature lead to global geometric and topological conclusions. Recommended for: mathematics students, teachers, and physics students wanting a relatively accessible bridge from multivariable calculus and linear algebra to modern geometry. BOOK