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In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. [1] The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric , allowing distances to be measured on that surface.
Christoffel symbols satisfy the symmetry relations = or, respectively, =, the second of which is equivalent to the torsion-freeness of the Levi-Civita connection. The contracting relations on the Christoffel symbols are given by
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 Christoffel symbol of the second kind. The quantity R i j k m {\displaystyle R_{ijk}^{m}} represents the mixed components of the Riemann-Christoffel curvature tensor . The general compatibility problem
Following standard practice, [1] one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing (,). They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second ...
This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle.
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. -ref- See, for instance, (Spivak 1999) and (Choquet-Bruhat & DeWitt-Morette 1977)-/ref- The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be ...
The torsion-free spin connection is given by = + = , where are the Christoffel symbols. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold ...