Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 179(1) α ∧ ∗β = 〈α, β〉 µ = β ∧ ∗α;(2) ∗1 = µ, ∗µ = (−1) Ind(g) ;(3) ∗ ∗ α = (−1) Ind(g) (−1) k(n−k) α;(4) 〈α, β〉 = (−1) Ind(g) 〈∗α, ∗β〉, where Ind(g) is the index of the metric g.3.6.2.3 Exterior Differential SystemsHere we give an informal introduction to exterior differential systems (EDS,for short), which are expressions involving differential forms related to anymanifold M. Later, when we fully develop the necessary differential geometricalas well as variational machinery (see (5.8) below), we will give amore precise definition of EDS.Central in the language of EDS is the notion of coframing, which is a realfinite–dimensional smooth manifold M with a given global cobasis and coordinates,but without requirement for a proper topological and differentialstructures. For example, M = R 3 is a coframing with cobasis {dx, dy, dz}and coordinates {x, y, z}. In addition to the cobasis and coordinates, acoframing can be given structure equations (3.10.2.4) and restrictions. Forexample, M = R 2 \{0} is a coframing with cobasis {e 1 , e 2 }, a single coordinate{r}, structure equations {dr = e 1 , de 1 = 0, de 2 = e 1 ∧ e 2 /r} andrestrictions {r ≠ 0}.A system S on M in EDS terminology is a list of expressions includingdifferential forms (e.g., S = {dz − ydx}).Now, a simple EDS is a triple (S, Ω, M), where S is a system on M,and Ω is an independence condition: either a decomposable k−form or asystem of k−forms on M. An EDS is a list of simple EDS objects wherethe various coframings are all disjoint.An integral element of an exterior system (S, Ω, M) is a subspace P ⊂T m M of the tangent space at some point m ∈ M such that all forms in Svanish when evaluated on vectors from P . Alternatively, an integral elementP ⊂ T m M can be represented by its annihilator P ⊥ ⊂ TmM, ∗ comprisingthose 1−forms at m which annul every vector in P . For example, withM = R 3 = {(x, y, z)}, S = {dx ∧ dz} and Ω = {dx, dz}, the integralelement P = {∂ x + ∂ z , ∂ y } is equally determined by its annihilator P ⊥ ={dz − dx}. Again, for S = {dz − ydx} and Ω = {dx}, the integral elementP = {∂ x + y∂ z } can be specified as {dy}.