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읽기 경로 — Pressley

고전 매개변수 흐름의 장별 경로

읽음 0/0 갱신 2026-06-26

How 원전 소개 — Pressley organizes elementary differential geometry. Its method is classical and parametric: everything is computed through explicit parametrizations $\gamma(t)$ and $\sigma(u,v)$, building from concrete formulas toward global theorems. Compare the very different reading path oneill, which front-loads frame fields and differential forms.

The three arcs

Arc I — Curves (Ch. 1–3)

  1. Ch. 1 매개화된 곡선, 호장과 재매개화 — curves as maps; the unit-speed (regular) reparametrization that simplifies everything.
  2. Ch. 2 곡률과 비틀림 — $\kappa$ (deviation from a line), signed curvature for plane curves, $\tau$ (deviation from a plane), the Frenet–Serret frame, and the fundamental theorem: $\kappa,\tau$ determine a space curve up to rigid motion.
  3. Ch. 3 곡선의 대역적 성질 — the leap from local to global: Hopf's Umlaufsatz, isoperimetric inequality, four vertex theorem.

Arc II — Surfaces and their fundamental forms (Ch. 4–8)

  1. Ch. 4–5 곡면과 곡면 조각 — surfaces as locally-$\mathbb{R}^2$ sets; patches, atlases, tangent planes, orientability; quadrics, ruled and revolution surfaces.
  2. Ch. 6 제1기본형식 — the intrinsic metric: lengths, angles, areas, and the maps it governs (isometry, conformal, equiareal; spherical geometry).
  3. Ch. 7 제2기본형식 — the extrinsic form via the Gauss/Weingarten map; normal vs. geodesic curvature; 평행이동.
  4. Ch. 8 가우스곡률과 평균곡률, 주곡률 — $K=\det\mathcal{W}$, $H=\tfrac12\operatorname{trace}\mathcal{W}$; eigenvalues and Euler's theorem.

Arc III — Deep structure and capstones (Ch. 9–13)

  1. Ch. 9 측지선 — straightest curves; the geodesic equations (intrinsic).
  2. Ch. 10 빼어난 정리 — Gauss/Codazzi–Mainardi equations; $K$ is intrinsic; rigidity of surfaces.
  3. Ch. 11 hyperbolic geometry — constant negative curvature; the half-plane, disc, and Klein models.
  4. Ch. 12 극소곡면 — $H=0$; Plateau's problem; holomorphic representation.
  5. Ch. 13 가우스–보네 정리 — the capstone: $\iint_S K\,dA = 2\pi\chi(S)$, geometry meets topology.

The throughline

Pressley repeatedly pairs a local invariant with a global/rigidity theorem:

Object Local invariant Global / uniqueness result
Plane curve signed curvature $\kappa_s$ determined up to isometry (§2.2)
Space curve $\kappa,\tau$ determined up to isometry (§2.3)
Surface first + second fundamental forms determined up to isometry (§10.1)
Surface Gaussian curvature $K$ intrinsic (Egregium §10.2); $\iint K = 2\pi\chi$ (§13.4)

The arc bends from "how do we measure a curve/surface?" to "what is preserved, and what does the total curvature know about the global shape?" The endpoint, Gauss–Bonnet, is where local differential data ($K$) determines a topological invariant ($\chi$).

See also