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As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor. As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector. [5] Under a general coordinate change, the components of the permutation tensor are multiplied by the Jacobian of the transformation matrix ...
The Kronecker delta has the so-called sifting property that for : = =. and if the integers are viewed as a measure space, endowed with the counting measure, then this property coincides with the defining property of the Dirac delta function () = (), and in fact Dirac's delta was named after the Kronecker delta because of this analogous property ...
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
This is the Levi-Civita connection on the tangent bundle TM of M. [2] [3] A local frame on the tangent bundle is an ordered list of vector fields e = (e i | i = 1, 2, ..., n), where n = dim M, defined on an open subset of M that are linearly independent at every point of their domain. The Christoffel symbols define the Levi-Civita connection by
In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.
where δ ij is the Kronecker delta, and ε ijk is the Levi-Civita symbol. It is not as obvious how to determine the rotational operator compared to space and time translations. We may consider a special case (rotations about the x , y , or z -axis) then infer the general result, or use the general rotation matrix directly and tensor index ...
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form.
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.