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Levi-Civita symbol. In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and ...
Tullio Levi-Civita, ForMemRS [1] (English: / ˈ t ʊ l i oʊ ˈ l ɛ v i ˈ tʃ ɪ v ɪ t ə /, Italian: [ˈtulljo ˈlɛːvi ˈtʃiːvita]; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas.
Levi-Civita connection. In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo-) Riemannian metric and is torsion -free.
The Levi-Civita connection is the torsion-free Riemannian connection on a manifold. It is unique by the fundamental theorem of Riemannian geometry. For every Riemannian connection, one may write a (unique) corresponding Levi-Civita connection. The difference between the two is given by the contorsion tensor.
In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric.
The structure constants are , where is the antisymmetric Levi-Civita symbol. In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors.
Spin connection. In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection ...
Christoffel symbols. In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. [1] The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine ...