Ivancevic_Applied-Diff-Geom

194 Applied Differential Geometry: A Modern Introductionfollows: Let ϕ : M → N be a C k −map, X ∈ X k−1 (M) and Y ∈ X k−1 (N),k ≥ 1. If X ∼ ϕ Y , thenL X (ϕ ∗ f) = ϕ ∗ L Y ffor all f ∈ C k (N, R), i.e., the following diagram commutes:C k ϕ ∗(N, R) ✲ C k (M, R)L Y❄❄C k (N, R) ✲ C k (M, R)ϕ ∗The Lie derivative map L X : C k (M, R) → C k−1 (M, R) is a derivation,i.e., for two functions f, g ∈ C k (M, R) the Leibniz rule is satisfiedL XL X (fg) = gL X f + fL X g;Also, Lie derivative of a constant function is zero, L X (const) = 0.The connection between the Lie derivative L X f of a function f ∈C k (M, R) and the flow F t of a vector–field X ∈ X k−1 (M) is given as:ddt (F ∗ t f) = F ∗ t (L X f) .3.7.2 Lie Derivative of Vector FieldsIf X, Y ∈ X k (M), k ≥ 1 are two vector–fields on M, then[L X , L Y ] = L X ◦ L Y − L Y ◦ L Xis a derivation map from C k+1 (M, R) to C k−1 (M, R). Then there is a uniquevector–field, [X, Y ] ∈ X k (M) of X and Y such that L [X,Y ] = [L X , L Y ] and[X, Y ](f) = X (Y (f)) − Y (X(f)) holds for all functions f ∈ C k (M, R).This vector–field is also denoted L X Y and is called the Lie derivative ofY with respect to X, or the Lie bracket of X and Y . In a local chart(U, φ) at a point m ∈ M with coordinates (x 1 , ..., x n ), for X| U = X i ∂ x iand Y | U = Y i ∂ x i we have[X i ∂ x i, Y j ] (∂ x j = Xi ( ∂ x iY j) − Y i ( ∂ x iX j)) ∂ x j ,