Ivancevic_Applied-Diff-Geom

196 Applied Differential Geometry: A Modern Introduction(3) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 for all X, Y, Z ∈ X k (M) – theJacobi identity;(4) [fX, Y ] = f[X, Y ] − (Y f)X, i.e., L fX (Y ) = f(L X Y ) − (L Y f)X for allX, Y ∈ X k (M) and f ∈ C k (M, R);(5) [X, fY ] = f[X, Y ] + (Xf)Y , i.e., L X (fY ) = f(L X Y ) + (L X f)Y for allX, Y ∈ X k (M) and f ∈ C k (M, R);(6) [L X , L Y ] = L [x,y] for all X, Y ∈ X k (M).The pair (X k (M), [, ]) is the prototype of a Lie algebra [Kolar et al.(1993)]. In more general case of a general linear Lie algebra gl(n), which isthe Lie algebra associated to the Lie group GL(n), Lie bracket is given bya matrix commutator[A, B] = AB − BA,for any two matrices A, B ∈ gl(n).Let ϕ : M → N be a diffeomorphism. Then L X : X k (M) → X k (M) isnatural with respect to push–forward by ϕ. That is, for each f ∈ C k (M, R),L ϕ∗ X(ϕ ∗ Y ) = ϕ ∗ L X Y,i.e., the following diagram commutes:X k (M)ϕ ∗✲ X k (N)L X❄X k (M)ϕ ∗L ϕ∗ X❄✲ X k (N)Also, L X is natural with respect to restrictions. That is, for U open inM and f ∈ C k (M, R),[X|U, Y |U] = [X, Y ]|U,where U : C k (M, R) → C k (U, R) denotes restriction to U, i.e., the followingdiagram commutes [Abraham et al. (1988)]:X k (M)|U✲ X k (U)L X❄X k (M)|UL X|U❄✲ X k (U)