원전 소개 — Pressley

Summary

Pressley develops the differential geometry of curves and surfaces in $\mathbb{R}^3$ using only calculus, vectors, and linear algebra as prerequisites.¹ Its organizing strategy is classical and parametric: every object is studied through explicit parametrizations $\gamma(t)$ (curves) and $\sigma(u,v)$ (surface patches), and geometric quantities are computed as concrete formulas in those parameters. The book deliberately avoids higher machinery — there is no mention of connections in the modern sense — in favor of "the simplest approach that will yield the desired results."²

The narrative builds in three arcs. Curves (Ch. 1–3): parametrization and arc-length lead to curvature and torsion, packaged in the Frenet–Serret equations, which determine a space curve up to a rigid motion. Surfaces (Ch. 4–8): a surface is something locally like a piece of $\mathbb{R}^2$; its intrinsic measurement comes from the first fundamental form and its bending in space from the second fundamental form / Weingarten map, yielding Gaussian, mean, and principal curvatures. Deeper structure (Ch. 9–13): geodesics, Gauss's Theorema Egregium (Gaussian curvature is intrinsic), and three capstone chapters — hyperbolic geometry, minimal surfaces, and the Gauss–Bonnet theorem linking curvature to topology.³

A distinctive feature is that the second edition treats the tangent plane geometrically, so the first/second fundamental forms and the Weingarten map are genuine geometric objects rather than mere matrices, and it introduces parallel transport to connect geodesics with Gaussian curvature.⁴ The book is exercise-heavy (200+ problems with full solutions), making it well suited to self-study.

Key Takeaways

• A regular curve has a unit-speed (arc-length) reparametrization; arc-length is essentially the only unit-speed parameter.⁵
• Curvature $\kappa$ measures deviation from a straight line; torsion $\tau$ measures deviation from a plane; together they determine a space curve up to a direct isometry of $\mathbb{R}^3$.⁶
• The first fundamental form $E\,du^2 + 2F\,du\,dv + G\,dv^2$ encodes all intrinsic measurements (length, angle, area). The second fundamental form $L\,du^2 + 2M\,du\,dv + N\,dv^2$ encodes extrinsic bending.
• Gaussian curvature $K = \det(\mathcal{W}) = (LN-M^2)/(EG-F^2)$ and mean curvature $H = \tfrac12\operatorname{trace}(\mathcal{W})$ come from the Weingarten map; principal curvatures are its eigenvalues.
• Theorema Egregium: $K$ is intrinsic — preserved by local isometries — even though it is defined via the extrinsic second fundamental form.⁷
• Gauss–Bonnet ties the integral of $K$ over a compact surface to its Euler characteristic, $\iint_S K\,dA = 2\pi\chi(S)$.

Entities & Concepts

• 매개화된 곡선 — curves as maps $\gamma:(\alpha,\beta)\to\mathbb{R}^n$
• 호장과 재매개화 — unit-speed curves, regularity
• 곡률과 비틀림 — Frenet–Serret frame and equations
• 곡선의 대역적 성질 — isoperimetric inequality, four vertex theorem
• 곡면과 곡면 조각 — atlases, smooth surfaces, orientability
• 제1기본형식 — intrinsic metric; isometries, conformal & equiareal maps
• 제2기본형식 — Weingarten/Gauss map, normal curvature
• 가우스곡률과 평균곡률 — $K$ and $H$ from the Weingarten map
• 주곡률 — eigenvalues of the Weingarten map, Euler's theorem
• 측지선 — straightest curves; geodesic equations
• 평행이동 — covariant derivative, holonomy
• 빼어난 정리 — intrinsic nature of $K$; Codazzi–Mainardi equations
• 극소곡면 — $H=0$ surfaces
• hyperbolic geometry — non-Euclidean models
• 가우스–보네 정리 — curvature and topology
• 읽기 경로 — Pressley — chapter-by-chapter route through this book

Relation to Other Wiki Pages

This is one of two primary textbooks in the wiki. It contrasts with 원전 소개 — O'Neill, which reaches much of the same material via Cartan's moving-frames method (differential forms, frame fields, structural equations). Where Pressley computes with explicit parametrizations, O'Neill works with frame fields and connection forms; the two reading paths 읽기 경로 — Pressley and reading path oneill make the difference concrete. Shared concept pages (e.g. 가우스곡률과 평균곡률, 측지선) cite both sources and note where the approaches diverge.