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— Andrew Pressley: Elementary Differential Geometry, p. 183 Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle moves along a geodesic under no forces other than those that keep it on the surface.
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The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics.
Differential geometry of curves for a full treatment of curves embedded in a Euclidean space of arbitrary dimension; Dioptre, a measurement of curvature used in optics; Evolute, the locus of the centers of curvature of a given curve; Fundamental theorem of curves; Gauss–Bonnet theorem for an elementary application of curvature
Calibrated geometry; Cartan connection; Cartan's equivalence method; Catalan's minimal surface; Caustic (mathematics) Cayley's ruled cubic surface; Center of curvature; Chentsov's theorem; Chern–Simons form; Chern–Weil homomorphism; Chern's conjecture (affine geometry) Chern's conjecture for hypersurfaces in spheres; Clairaut's relation ...
Tangent developable of a curve with zero torsion. The tangent developable is a developable surface; that is, it is a surface with zero Gaussian curvature.It is one of three fundamental types of developable surface; the other two are the generalized cones (the surface traced out by a one-dimensional family of lines through a fixed point), and the cylinders (surfaces traced out by a one ...