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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 ...
Conceptually, the Lie bracket [,] is the derivative of along the flow generated by , and is sometimes denoted ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X {\displaystyle X} .
one in differential geometry: = +, where ,, and are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms. See interior product for the detail. It is also called the Cartan homotopy formula or Cartan magic formula .
The Lie derivative is the rate of change of a vector or tensor ... The linear operator which assigns to each function its derivative is an example of a differential ...
The inverse function theorem together with the derivative of the exponential map provides information about the local behavior of exp. Any C k, 0 ≤ k ≤ ∞, ω map f between vector spaces (here first considering matrix Lie groups) has a C k inverse such that f is a C k bijection in an open set around a point x in the domain provided df x is
Here ω(Y) is the g-valued function obtained by duality from pairing the one-form ω with the vector field Y, and X(ω(Y)) is the Lie derivative of this function along X. Similarly Y(ω(X)) is the Lie derivative along Y of the g-valued function ω(X). In particular, if X and Y are left-invariant, then
We define the Lie derivative: () through Cartan's magic formula for a given section () as L X = d ∘ ι X + ι X ∘ d . {\displaystyle {\mathcal {L}}_{X}=d\circ \iota _{X}+\iota _{X}\circ d.} It describes the change of a k {\displaystyle k} -form along a flow ϕ t {\displaystyle \phi _{t}} associated to the section X {\displaystyle X} .
The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence. If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small n × n {\displaystyle n\times n} matrices, then Z {\displaystyle Z} can be computed as the logarithm of e X e Y ...
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