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valid for any vector fields X and Y and any tensor field T.. Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation ...
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]
covariant derivative. exterior covariant derivative; Levi-Civita connection; parallel transport. Development (differential geometry) connection form; Cartan connection. affine connection; conformal connection; projective connection; method of moving frames; Cartan's equivalence method; Vierbein, tetrad; Cartan connection applications; Einstein ...
There are many ways to understand the gauge covariant derivative. The approach taken in this article is based on the historically traditional notation used in many physics textbooks. [1] [2] [3] Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.
One can also define the covariant derivative by the following geometric approach, which does not make use of Christoffel symbols or local parametrizations. [39] [40] [41] Let X be a vector field on S, viewed as a function S → ℝ 3. Given any curve c : (a, b) → S, one may consider the composition X ∘ c : (a, b) → ℝ 3.
In fact in the above expression, one can replace the covariant derivative with any torsion free connection ~ or locally, with the coordinate dependent derivative , showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.
The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on either a metric tensor or a connection. The Lie derivative of a type (r, s) tensor field T along (the flow of) a contravariant vector field X ρ may be expressed using a coordinate basis as [20]
The covariant derivative is such a map for k = 0. The exterior covariant derivatives extends this map to general k. There are several equivalent ways to define this object: [3] Suppose that a vector-valued differential 2-form is regarded as assigning to each p a multilinear map s p: T p M × T p M → E p which is completely