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In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γ i jk are zero. The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900). [7]
A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.
The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field , the matrix () is skew-symmetric; equivalently, it is an element of the Lie algebra (). This can be seen as follows.
the connection is torsion-free, i.e., T ∇ is zero, so that ∇ X Y − ∇ Y X = [X, Y]; parallel transport is an isometry, i.e., the inner products (defined using g) between tangent vectors are preserved. This connection is called the Levi-Civita connection. The term "symmetric" is often used instead of torsion-free for the first property.
Conversely a manifold with such a connection is locally symmetric (i.e., its universal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case.
Symmetric and exterior power connections [ edit ] Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power, S k E , Λ k E ⊂ E ⊗ k {\displaystyle S^{k}E,\Lambda ^{k}E\subset E^{\otimes k}} , the definition of the tensor product connection applies in a straightforward manner to ...
Symmetric and antisymmetric relations. By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").
Torsion-free affine connection, an affine connection whose torsion tensor vanishes; Torsion-free metric connection or Levi-Civita connection, a unique symmetric connection on the tangent bundle of a manifold compatible with the metric