An Introduction to Differential Geometry |
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affine connexion angle arbitrary arc length asymptotic lines called Chapter compact surfaces components condition connexion coefficients consider constant contravariant vector coordinate neighbourhood coordinate system corresponding covariant differentiation covariant tensors covariant vector curvature tensor defined definite denote differentiable manifold differential equations differential geometry direction dv² Euclidean space example Əxi follows formula function g₁ Gaussian curvature geodesic arc geodesic curvature given gives Hence identity integrable intrinsic isometric isomorphism linear lines of curvature mapping Math matrix metric tensor n-dimensional null obtain orthogonal parallel field parameter parametric curves principal curvatures proof properties prove r-planes real numbers real-valued relation relative respect Riemannian manifold Riemannian metric Riemannian space satisfy Show suffixes symmetric tangent space tangent vector tensor field tensor of type theorem torsion total curvature vector field vector space zero αβ λι ди მა