Elements of Differential GeometryThis 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. |
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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 assume basis calculus called Chapter circle of latitude compute concept convex coordinate chart coordinate patch curve a(s da/dt defined Definition derivative differentiable function differential equations differential geometry Dupin indicatrix Euclidean Example 1.7 FIGURE formula Frenet-Serret apparatus Gauss-Bonnet Gauss's Gaussian curvature geodesic coordinate patch given gives global helix Hence hyperbolic inner product integral intrinsic inverse isometry Lemma Let a(s linear connection linear transformation matrix metric space normal curvature Note one-to-one open set orthogonal orthonormal osculating plane parallel translation perpendicular plane curve Problem 3.1 Proof Proposition Prove radius regular curve reparametrization Riemannian manifold Riemannian metric Section simple surface space curve sphere straight line surface in R3 surface of revolution tangent plane tangent spherical image tangent vector tangent vector field tensor theory tion torsion total curvature unit speed curve unit vector vector field vector space zero