Christoffel Symbol

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Christoffel Symbol


The Christoffel symbol of a quadratic differential form

is a symbol for the abbreviated representation of the expression

The symbol Γk, ij is called the Christoffel symbol of the first kind in contrast to the Christoffel symbol of the second kindChristoffel Symbol which is defined by the relation

where gkt is determined from the equalities

The symbols were introduced by E. Christofl’el in 1869.

References in periodicals archive ?
The non-vanishing independent components of the christoffel symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
The components of the corresponding metric tensor h and the Christoffel symbols on the manifold N will be denoted by [h.
Now, an alternative (although implicit) definition of the Christoffel symbols is contained in the equation that states the vanishing of covariant derivatives of the metric:
This theory has some advantages over the general relativity; the quantities such as christoffel symbols and others become tensors which otherwise in Riemannian geometry they are not.
One straightforwardly goes through the tedious calculation of the Christoffel symbols and the components of the Ricci tensor, finding:
n], n > k this leads to the Ricci tensor R, in components [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are Christoffel symbols.
n]) is a coordinate chart on M, then the Christoffel symbols [[GAMMA].
v[lambda] are the Christoffel symbols of the second kind, and [S.
We recall that the Christoffel symbols of second kind related to a given [THETA](4)-invariant spacetime metric [3] are the components of a [THETA](4)-invariant tensor field and depend on ten functions [B.
as the Christoffel symbols (symmetric in their two lower indices) and