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The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface .
The differential-geometric properties of a parametric curve (such as its length, its Frenet frame, and its generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class itself. The equivalence classes are called C r-curves and are central objects studied in the differential geometry of curves.
Differential geometry is also indispensable in the study of gravitational lensing and black holes. Differential forms are used in the study of electromagnetism. Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
If in addition E = 1, so that H = G 1 ⁄ 2, then the angle φ at the intersection between geodesic (x(t),y(t)) and the line y = constant is given by the equation = ˙ ˙. The derivative of φ is given by a classical derivative formula of Gauss: [60]
Chern class; Pontrjagin class; spin structure; differentiable map. submersion; immersion; Embedding. Whitney embedding theorem; Critical value. Sard's theorem; Saddle point; Morse theory; Lie derivative; Hairy ball theorem; Poincaré–Hopf theorem; Stokes' theorem; De Rham cohomology; Sphere eversion; Frobenius theorem (differential topology ...
Do Carmo's main research interests were Riemannian geometry and the differential geometry of surfaces. [3]In particular, he worked on rigidity and convexity of isometric immersions, [26] [27] stability of hypersurfaces [28] [29] and of minimal surfaces, [30] [31] topology of manifolds, [32] isoperimetric problems, [33] minimal submanifolds of a sphere, [34] [35] and manifolds of constant mean ...
Michael David Spivak [1] (May 25, 1940 – October 1, 2020) [2] [3] was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press.
A solution is to adopt the overdot notation, in which the scope of a vector derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define In this case, if we define