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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 ...
The trace or tensor contraction, considered as a mapping. The map , representing scalar multiplication as a sum of outer products. The generalized Kronecker delta or multi-index Kronecker delta of order is a type tensor that is completely antisymmetric in its upper indices, and also in its lower indices. Two definitions that differ by a factor ...
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D. Given a smooth curve γ on (M, g) and a vector field V along γ its derivative is defined by. Formally, D is the pullback connection γ*∇ on the pullback bundle γ*TM. In particular, is a vector field along the curve γ itself.
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds.
The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex Hermitian matrices means that we can express any Hermitian matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
This can be simplified by performing a contraction on the Levi-Civita symbols, = =, where is the Kronecker delta function (= when and = when =) and is the generalized Kronecker delta function. We can reason out this identity by recognizing that the index k {\displaystyle k} will be summed out leaving only i {\displaystyle i} and j ...
The Kronecker delta is like the identity matrix when multiplied and contracted: ... For a Levi-Civita connection this tensor is defined to be zero, which for a ...
A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita ...