definition 13.18 Surface

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Supports

• depends_on definition 13.21 First fundamental form ¶
  • depends_on definition 13.22 First fundamental coefficients ¶
    • depends_on definition A.476 Disc-type surfaces and the two functionals ¶
      • depends_on definition A.481 Douglas' boundary functional ¶
        • depends_on definition A.486 The normalised admissible class ¶
        • depends_on lemma A.484 Lower semicontinuity of the Douglas functional ¶
        • depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
        • depends_on theorem A.482 Douglas' identity ¶
      • depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
        • depends_on lemma A.487 The normalisation costs nothing ¶
        • depends_on lemma A.485 Conformal automorphisms of the disc ¶
      • depends_on lemma A.489 Courant–Lebesgue ¶
        • depends_on theorem A.490 Equicontinuity of the normalised class ¶
      • depends_on proposition A.478 $D\ge A$, with equality exactly for conformal maps ¶
    • depends_on definition 13.29 Gaussian curvature ¶
      • depends_on theorem 13.30 Theorema Egregium; Gauss, 1827 ¶
        • depends_on corollary 13.31 Bending invariance ¶
    • depends_on definition 13.23 Inverse metric tensor ¶
      • depends_on definition 13.26 Christoffel symbols ¶
        • depends_on definition 13.37 Geodesic lines ¶
        • depends_on definition 13.27 Riemann symbols ¶
  • depends_on proposition 13.40 Parallel transport is an isometry of the tangent plane ¶
    • depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
      • depends_on example 13.42 The sphere, the solid angle, and the pole ¶
• depends_on definition 13.19 Parameter curves ¶
  • depends_on definition 13.22 First fundamental coefficients ¶ ↺
  • depends_on definition 13.20 Unit normal vector of a surface ¶
    • depends_on definition 13.28 Normal curvature vector of a surface ¶
      • depends_on definition 13.29 Gaussian curvature ¶ ↺
      • depends_on definition 13.36 Geodesic curvature vector ¶
        • depends_on definition 13.37 Geodesic lines ¶ ↺
    • depends_on definition 13.25 Second fundamental coefficients ¶
      • depends_on definition 13.29 Gaussian curvature ¶ ↺
      • depends_on definition 13.27 Riemann symbols ¶ ↺
      • depends_on proposition A.493 A harmonic conformal map is a minimal surface ¶
    • depends_on definition 13.24 Second fundamental form ¶
      • depends_on definition 13.25 Second fundamental coefficients ¶ ↺