Ivancevic_Applied-Diff-Geom

174 Applied Differential Geometry: A Modern IntroductionSimilarly, second–order ODEs are organized into a category ODE 2 . Asecond–order ODE on M is usually constructed as a vector–field on T M,ξ : T M → T T M, and a morphism of vector–fields (M 1 , ξ 1 ) → (M 2 , ξ 2 ) is amap f : M 1 → M 2 such that the following diagram commutesT T M 1T T f ✲ T T M 2✻✻ξ 1ξ 2T M 1✲ T MT f2Unlike solutions for first–order ODEs, solutions for second–order ODEs arenot in general homomorphisms from R, unless the second–order ODE is aspray [Kock and Reyes (2003)].3.6.2 Differential Forms on Smooth ManifoldsRecall (see section 2.1.4.2 above) that exterior differential forms are a specialkind of antisymmetrical covariant tensors, that formally occur as integrandsunder ordinary integral signs in R 3 . To give a more precise exposition,here we start with 1−forms, which are dual to vector–fields, and afterthat introduce general k−forms.3.6.2.1 1−Forms on MDual to the notion of a C k vector–field X on an n−manifold M is a C kcovector–field, or a C k 1−form α, which is defined as a C k −section of thecotangent bundle T ∗ M, i.e., α : M → T ∗ M is smooth and π ∗ M ◦ X =Id M . We denote the set of all C k 1−forms by Ω 1 (M). A basic exampleof a 1−form is the differential df of a real–valued function f ∈ C k (M, R).With point wise addition and scalar multiplication Ω 1 (M) becomes a vectorspace.In other words, a C k 1−form α on a C k manifold M is a real–valuedfunction on the set of all tangent vectors to M, i.e., α : T M → R with thefollowing properties:(1) α is linear on the tangent space T m M for each m ∈ M;(2) For any C k vector–field X ∈ X k (M), the function f : M → R is C k .Given a 1−form α, for each point m ∈ M the map α(m) : T m M → R isan element of the dual space T ∗ mM. Therefore, the space of 1−forms Ω 1 (M)