Elements of Differential Geometry This text is intended for an advanced undergraduate (having taken linear algebra and multivariable calculus). It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material. For all readers interested in differential geometry. |
Contents
Preliminaries | 1 |
Local Curve Theory | 13 |
Global Theory of Plane Curves | 49 |
Copyright | |
6 other sections not shown
Common terms and phrases
angle arc length basis calculus called Chapter circle of latitude compute concept constant coordinate patch curve a(s da/dt defined DEFINITION differentiable function differential geometry du¹ du² Dupin indicatrix Euclidean Example 1.7 FIGURE formula Frenet-Serret apparatus Gauss Gauss-Bonnet Gaussian curvature geodesic coordinate patch given global helix Hence hyperbolic inner product intrinsic isometry Lemma Let a(s linear connection matrix Note one-to-one open set orthonormal parallel translation parametrized piecewise plane curve Problem Proof PROPOSITION Prove radius regular curve reparametrization Riemannian metric s₁ Section segment simple surface sin² space curve sphere straight line surface in R³ surface of revolution tangent plane tangent spherical image tangent vector Theorem torsion u¹)² u²)² unit speed curve unit vector vector field vector space x(u¹ x₁ zero ди дик