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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 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 ...
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} .
Download as PDF; Printable version ... We define the Lie derivative: ... ( Hodge dual of constant function 1 is the volume form) Co-differential operator 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
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
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 with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra.