In Riemannian geometry, a Jacobi field is a vector field along a geodesic in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.
In Riemannian geometry, a Jacobifield is a vector field along a geodesic γ {\displaystyle \gamma } in a Riemannian manifold describing the difference...
Carl Gustav Jacob Jacobi (/dʒəˈkoʊbi/; German: [jaˈkoːbi]; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions...
theorem is formulated using Jacobifields to measure the variation in geodesics. As the tangential part of a Jacobifield is independent of the geometry...
there exists a non-zero Jacobifield along γ {\displaystyle \gamma } that vanishes at p and q. Recall that any Jacobifield can be written as the derivative...
zeros of the second variation along a geodesic γ arise along Jacobifields. Jacobifields are thus regarded as variations through geodesics. By applying...
geodesic γ {\displaystyle \gamma } are called conjugate if there is a Jacobifield on γ {\displaystyle \gamma } which has a zero at p and q. Convex function...
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other...
fieldJacobi's four-square theorem Jacobi form Jacobi's formula Jacobi group Jacobian ideal Jacobi identity Jacobi integral Jacobi's logarithm Jacobi...
mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an...
diverge or converge when they are allowed to move freely in the space; see Jacobifield. Another broad generalization of curvature comes from the study of parallel...
nor 1 − a is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth...
through data recorded in the Jacobifield, a vector field along the geodesic. One and a quarter centuries after Gauss and Jacobi, Marston Morse gave a more...
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple...
} In fact, by taking the Taylor expansion of the metric applied to a Jacobifield along a radial geodesic in the normal coordinate system, one has g i...
A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating, fluid body of uniform density...
\lambda }(p)} , where the projection operator, or the Fourier transform of Jacobifield operator obtained by applying Peierls braket on Schwinger's variational...
relativity General relativity resources History of general relativity Hamilton–Jacobi–Einstein equation Mathematics of general relativity Numerical relativity...
Lotte Jacobi (August 17, 1896 – May 6, 1990) was a leading American portrait photographer and photojournalist, known for her high-contrast black-and-white...
related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after...
arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel–Jacobi theorem states...
Nicolas Jacobi (born 13 April 1987) is a German professional field hockey player who currently plays as a goalkeeper for Delhi Waveriders in the Hockey...
fundamental form. It continues with geodesics on Riemannian manifolds, Jacobifields, the Morse index, the Rauch comparison theorems, and the Cartan–Hadamard...
Berger–Kazdan comparison theorem Warner comparison theorem for lengths of N-Jacobifields (N being a submanifold of a complete Riemannian manifold) Bishop–Gromov...
Walter Jacobi (January 13, 1918 – August 19, 2009) was a rocket scientist and member of the "von Braun rocket group", at Peenemünde (1939–1945) working...
in the early 19th century, with the use of Jacobi sums and their prime decomposition in cyclotomic fields. Gauss sums over a residue ring of integers...
Global Information