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For example, for differential geometry, the top-level code is 53, and the second-level codes are: A for classical differential geometry; B for local differential geometry; C for global differential geometry; D for symplectic geometry and contact geometry; In addition, the special second-level code "-" is used for specific kinds of materials.
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.
The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric ...
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point.
See also multivariable calculus, list of multivariable calculus topics. Manifold. Differentiable manifold; Smooth manifold; Banach manifold; Fréchet manifold; Tensor analysis. Tangent vector
In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.
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.
In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory.