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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} .
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 idea of Lie dragging a function along a congruence leads to a definition of the Lie derivative, where the dragged function is compared with the value of the original function at a given point. The Lie derivative can be defined for type ( r , s ) {\displaystyle (r,s)} tensor fields and in this respect can be viewed as a map that sends a type ...
Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis. Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatments of Lie theory often begin with the classical groups.
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
In mathematics, Cartan formula can mean: one in differential geometry: = +, where ,, and are Lie derivative, exterior derivative, and interior product, respectively ...
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D. Given a smooth curve γ on (M, g) and a vector field V along γ its derivative is defined by = ˙ (). Formally, D is the pullback connection γ*∇ on the pullback bundle γ*TM.