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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 ...
Download as PDF; Printable version; ... In mathematics, Cartan formula can mean: one in ... and are Lie derivative, exterior derivative, and interior ...
The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a ...
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 ...
Download as PDF; Printable version; ... through Cartan's magic formula for a given section as = +. It describes ... Lie derivative properties
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
This generalizes to the Lie derivative of any tensor field along the flow generated by . The Lie bracket is an R - bilinear operation and turns the set of all smooth vector fields on the manifold M {\displaystyle M} into an (infinite-dimensional) Lie algebra .
Cartan's magic formula for Lie derivatives can be used to give a short proof of the Poincaré lemma. The formula states that the Lie derivative along a vector field ξ {\displaystyle \xi } is given as: [ 12 ]