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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 ...
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.
A Riemannian connection that is torsion-free (i.e., for which the torsion tensor vanishes: T α βγ = 0) is a Levi-Civita connection. The Γ α βγ for a Levi-Civita connection in a coordinate basis are called Christoffel symbols of the second kind.
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.
where is the Kronecker delta or identity matrix. Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia. Possible metrics on real space are indexed by signature (,).
As an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3), =, where is the same tensor as , because =, with Kronecker δ acting here like an identity matrix.