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This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
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 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 ...
In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve.
Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory.
Non-Euclidean geometry; Elliptic geometry. Spherical geometry; Sphere-world; Angle excess; hyperbolic geometry. hyperbolic space; hyperboloid model; Poincaré disc model; Poincaré half-plane model; Poincaré metric; Angle of parallelism