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The covariant derivative is a generalization of the directional derivative from vector calculus.As with the directional derivative, the covariant derivative is a rule, , which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. [7]
Here is the inverse matrix to the metric tensor . In other words, = and thus = = = is the dimension of ... the covariant derivative of the metric vanishes, ...
The metric tensor is an example of a tensor field. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor.
D X is a metric connection in the normal bundle. There are thus a pair of connections: ∇, defined on the tangent bundle of M; and D, defined on the normal bundle of M. These combine to form a connection on any tensor product of copies of TM and T ⊥ M. In particular, they defined the covariant derivative of :
When the manifold is equipped with a metric, covariant and contravariant indices become very closely related to one another. Contravariant indices can be turned into covariant indices by contracting with the metric tensor. The reverse is possible by contracting with the (matrix) inverse of the metric tensor.
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. [1] This is equivalent to: A connection for which the covariant derivatives of the metric on E ...
The covariant derivative can in turn be recovered from parallel transport. [25] In fact ∇ X Y {\displaystyle \nabla _{X}Y} can be calculated at a point p , by taking a curve c through p with tangent X , using parallel transport to view the restriction of Y to c as a function in the tangent space at p and then taking the derivative.
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.