Ivancevic_Applied-Diff-Geom

192 Applied Differential Geometry: A Modern IntroductionLet M be a compact, oriented Riemannian manifold, E a vector bundlewith a bundle metric 〈·, ·〉 over M,D = d + A : Ω p−1 (Ad E ) → Ω p (Ad E ), with A ∈ Ω 1 (Ad E )– a tensorial and R−linear metric connection on E with curvature F D ∈Ω 2 (Ad E ) (Here by Ω p (Ad E ) we denote the space of those elements ofΩ p (End E ) for which the endomorphism of each fibre is skew symmetric;End E denotes the space of linear endomorphisms of the fibers of E).3.7 Lie Derivatives on Smooth ManifoldsLie derivative is popularly called ‘fisherman’s derivative’. In continuum mechanicsit is called Liouville operator. This is a central differential operatorin modern differential geometry and its physical and control applications.3.7.1 Lie Derivative Operating on FunctionsTo define how vector–fields operate on functions on an m−manifold M, wewill use the Lie derivative. Let f : M → R so T f : T M → T R = R × R.Following [Abraham et al. (1988)] we write T f acting on a vector v ∈ T m Min the formT f · v = (f(m), df(m) · v) .This defines, for each point m ∈ M, the element df(m) ∈ TmM.∗ Thusdf is a section of the cotangent bundle T ∗ M, i.e., a 1−form. The 1−formdf : M → T ∗ M defined this way is called the differential of f. If f is C k ,then df is C k−1 .If φ : U ⊂ M → V ⊂ E is a local chart for M, then the local representativeof f ∈ C k (M, R) is the map f : V → R defined by f = f ◦ φ −1 . Thelocal representative of T f is the tangent map for local manifolds,T f(x, v) = (f(x), Df(x) · v) .Thus the local representative of df is the derivative of the local representativeof f. In particular, if (x 1 , ..., x n ) are local coordinates on M, then thelocal components of df are(df) i = ∂ x if.