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Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra.
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 ...
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.
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics . Differential geometry of curves and surfaces
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
Differential geometry stubs (1 C, 115 P) Pages in category "Differential geometry" The following 200 pages are in this category, out of approximately 379 total.
In the language of differential geometry, this derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact , meaning that it is not the derivative of a 0-form (that is, a function): the angle θ {\displaystyle \theta } is not a globally defined smooth function on the entire punctured plane.
Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century. [5] The basis of modern tensor analysis was developed by Bernhard Riemann in a paper from 1861. [6]