An Introduction to Differential Geometry |
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affine connexion angle arbitrary arc length asymptotic lines axis basis C₁ Chapter compact surface components condition consider contravariant vector coordinate neighbourhood coordinate system corresponding covariant differentiation covariant vector curvature tensor defined definition denote derivative differentiable manifold differential equations differential geometry direction dv² Euclidean space example Əxi follows formula function Gaussian curvature geodesic arc geodesic curvature given gives helicoid helix Hence identity integrable intrinsic isometric isomorphism linear lines of curvature mapping metric tensor n-dimensional obtained orthogonal osculating plane parallel field parameter parametric curves position vector prove r-planes r₁ radius real number relation respect Riemannian manifold Riemannian space satisfy Show sphere suffixes surface of revolution symmetric tangent space tangent vector tensor field tensor of type theorem torsion total curvature transformation unit vector vector field vector space zero ди эт მა