Ivancevic_Applied-Diff-Geom

178 Applied Differential Geometry: A Modern IntroductionFor any k−form α ∈ Ω k (M) and l−form β ∈ Ω l (M), the wedge productis defined fiberwise, i.e., (α ∧ β) m= α x ∧ β m for each point m ∈ M. It isalso associative, i.e., (α ∧ β) ∧ γ = α ∧ (β ∧ γ), and graded commutative,i.e., α∧β = (−1) kl β∧α. These properties are proved in multilinear algebra.So M =⇒ Ω k (M) is a contravariant functor from the category M into thecategory of real graded commutative algebras [Kolar et al. (1993)].Let M be an n−manifold, X ∈ X k (M), and α ∈ Ω k+1 (M). The interiorproduct, or contraction, i X α = X⌋α ∈ Ω k (M) of X and α (with insertionoperator i X ) is defined asi X α(X 1 , ..., X k ) = α(X, X 1 , ..., X k ).Insertion operator i X of a vector–field X ∈ X k (M) is natural withrespect to the pull–back F ∗ of a diffeomorphism F : M → N between twomanifolds, i.e., the following diagram commutes:Ω k (N)F ∗✲ Ω k (M)i X❄Ω k−1 (N)F ∗i F ∗ X❄✲ Ω k−1 (M)Similarly, insertion operator i X of a vector–field X ∈ Y k (M) is naturalwith respect to the push–forward F ∗ of a diffeomorphism F : M → N, i.e.,the following diagram commutes:Ω k (M)F ∗✲ Ω k (N)i Y❄Ω k−1 (M)F ∗i F∗Y❄✲ Ω k−1 (N)In case of Riemannian manifolds there is another exterior operation.Let M be a smooth n−manifold with Riemannian metric g = 〈, 〉 and thecorresponding volume element µ. The Hodge star operator ∗ : Ω k (M) →Ω n−k (M) on M is defined asα ∧ ∗β = 〈α, β〉 µfor α, β ∈ Ω k (M).The Hodge star operator satisfies the following properties for α, β ∈ Ω k (M)[Abraham et al. (1988)]: