Editing Curvature (section)

==Generalizations==
[[File:Parallel Transport.svg|thumb|Moving a vector along a curve from A → N → B → A produces another vector. The inability to return to the initial vector is measured by the holonomy of the surface. In a space with no curvature, the angle α is 0 degrees, and in a space with curvature, the angle α is greater than 0 degrees. The more space is curved, the greater the magnitude of the angle α. ]]
The mathematical notion of ''curvature'' is also defined in much more general contexts.<ref>{{citation |last1=Kobayashi |first1=Shōshichi |author-link1=Shoshichi Kobayashi |title=Foundations of Differential Geometry |title-link=Foundations of Differential Geometry |last2=Nomizu |first2=Katsumi |author-link2=Katsumi Nomizu |publisher=Interscience |year=1963 |isbn=978-0-470-49647-3 |place=New York |chapter=2–3}}</ref> Many of these generalizations emphasize different aspects of the curvature as it is understood in lower dimensions.

One such generalization is kinematic. The curvature of a curve can naturally be considered as a kinematic quantity, representing the force felt by a certain observer moving along the curve; analogously, curvature in higher dimensions can be regarded as a kind of [[tidal force]] (this is one way of thinking of the [[sectional curvature]]). This generalization of curvature depends on how nearby test particles diverge or converge when they are allowed to move freely in the space; see [[Jacobi field]].

Another broad generalization of curvature comes from the study of [[parallel transport]] on a surface. For instance, if a vector is moved around a loop on the surface of a sphere keeping parallel throughout the motion, then the final position of the vector may not be the same as the initial position of the vector. This phenomenon is known as [[holonomy]].<ref>{{citation |last1=Henderson |first1=David W. |author-link1=David W. Henderson |title=Experiencing Geometry: Euclidean and Non-Euclidean with History |last2=Taimin̦a |first2=Daina |author-link2=Daina Taimiņa |publisher=Pearson Prentice Hall |year=2005 |isbn=978-0-13-143748-7 |edition=3rd |place=Upper Saddle River, NJ |pages=98–99 |doi=10.3792/euclid/9781429799850 |doi-access=free }}</ref> Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see [[curvature form]]. A closely related notion of curvature comes from [[gauge theory]] in physics, where the curvature represents a field and a [[vector potential]] for the field is a quantity that is in general path-dependent: it may change if an observer moves around a loop.

Two more generalizations of curvature are the [[scalar curvature]] and [[Ricci curvature]]. In a curved surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the same radius in flat space. This difference (in a suitable limit) is measured by the scalar curvature. The difference in area of a sector of the disc is measured by the Ricci curvature. Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are particularly important in relativity theory, where they both appear on the side of [[Einstein's field equations]] that represents the geometry of spacetime (the other side of which represents the presence of matter and energy). These generalizations of curvature underlie, for instance, the notion that curvature can be a property of a [[measure (mathematics)|measure]]; see [[curvature of a measure]].

Another generalization of curvature relies on the ability to [[comparison theorem|compare]] a curved space with another space that has ''constant'' curvature. Often this is done with triangles in the spaces. The notion of a triangle makes senses in [[metric space]]s, and this gives rise to [[CAT(k) space|{{math|CAT(''k'')}} spaces]].