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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.
The Riemannian connection or Levi-Civita connection [9] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in ...
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 ...
β are the same in any basis and form an invariant tensor of type (1, 1), i.e. the identity of the tangent bundle over the identity mapping of the base manifold, and so its trace is an invariant. [21] Its trace is the dimensionality of the space; for example, in four-dimensional spacetime,
2.2.5 Twice-contracted second Bianchi identity. 2.2.6 Ricci identity. 2.2.7 Remarks. ... Because the Levi-Civita connection is metric-compatible, ... is a one-form then
The reciprocal basis (b 1, b 2, b 3) is defined by the relation [4]: 28–29 = where δ i j is the Kronecker delta. The vector v can also be expressed in terms of the reciprocal basis: v = v k b k {\displaystyle \mathbf {v} =v_{k}~\mathbf {b} ^{k}} The components v k are the covariant components of the vector v {\displaystyle \mathbf {v} } .
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.
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 ...