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Differentiable Manifolds, Tensors, and Curvature
Differentiable Manifolds, Tensors, and Curvature
foundations +67 more results +38 more equations +28 more depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) depends_on (declared) notation 13.1: not:mfd-conventions 13.1 notation definition 13.2: Linear coordinate transformation 13.2 Linear coordinat… definition 13.3: Orthogonal coordinate system 13.3 Orthogonal coord… definition 13.4: Orthogonal coordinate transformation 13.4 Orthogonal coord… definition 13.5: Tensor under orthogonal transformations 13.5 Tensor under ort… definition 13.8: Curve 13.8 Curve definition 13.9: Tangent vector to a curve 13.9 Tangent vector t… definition 13.10: Admissible change of parameter 13.10 Admissible chang… definition 13.11: Arc length 13.11 Arc length definition 13.12: Unit tangent vector 13.12 Unit tangent vec… definition 13.13: Normal curvature vector of a curve 13.13 Normal curvature… definition 13.14: Principal unit normal vector 13.14 Principal unit n… definition 13.15: Unit binormal vector 13.15 Unit binormal ve… proposition 13.6: Levi-Civita identities in three dimensions 13.6 Levi-Civita iden… theorem 13.16: Fundamental theorem of the theory of curves 13.16 Fundamental theo… theorem 13.30: Theorema Egregium; Gauss, 1827 13.30 Theorema Egregiu… corollary 13.31: Bending invariance 13.31 Bending invarian… proposition 13.38: Geodesics as stationary curves of the arc length 13.38 Geodesics as sta… proposition 13.40: Parallel transport is an isometry of the tangent
plane 13.40 Parallel transpo… theorem 13.41: Holonomy equals the enclosed curvature; local
Gauss–Bonnet 13.41 Holonomy equals… proposition 13.46: Compatibility of atlases 13.46 Compatibility of… proposition 13.56: Product manifold 13.56 Product manifold theorem 13.59: Regular value theorem 13.59 Regular value th… theorem 13.62: Regular value theorem in codimension k 13.62 Regular value th… theorem 13.63: Constant rank theorem 13.63 Constant rank th… corollary 13.64: The image of a constant-rank map, locally 13.64 The image of a c… equation 13.1: eq:mfd-eta eq. (13.1) equation 13.13: eq:mfd-condorto eq. (13.13) equation 13.36: eq:mfd-dsdt eq. (13.36) equation 13.38: eq:mfd-tang eq. (13.38) equation 13.43: eq:mfd-freset1 eq. (13.43) equation 13.46: eq:mfd-freset3 eq. (13.46) equation 13.47: eq:mfd-freset2 eq. (13.47) equation 13.67: eq:mfd-L eq. (13.67) equation 13.83: eq:mfd-ecsgauss eq. (13.83) equation 13.87: eq:mfd-sc11 eq. (13.87) equation 13.88: eq:mfd-sc22 eq. (13.88) equation 13.92: eq:mfd-ecscoda1 eq. (13.92) equation 13.95: eq:mfd-riemann-surface eq. (13.95) The chain of Differentiable Manifolds, Tensors, and Curvature: 39 of 172 objects, read left to right from what the chapter assumes to what tests it. Solid lines are declared logical edges, dashed ones were inferred from the structure of the source. Click any node to open its own chain.