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The covariant derivative of a function ... The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
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]
The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated index-free methods to represent tensors. In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector ...
The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant derivative [3]
When the torsion tensor is zero, so that [,] =, we may use this fact to write Riemann curvature tensor as [2] (,) =,,. Similarly, one may also obtain the second covariant derivative of a function f as
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 covariant derivatives (also called "tangential derivatives") of Tullio Levi-Civita and Gregorio Ricci-Curbastro provide a means of differentiating smooth tangential vector fields. Given a tangential vector field X and a tangent vector Y to S at p , the covariant derivative ∇ Y X is a certain tangent vector to S at p .
The exterior derivative of a totally antisymmetric type (0, s) tensor field with components A α 1 ⋅⋅⋅α s (also called a differential form) is a derivative that is covariant under basis transformations. It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold.