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Orientations as Equivalence Class of Nowhere vanishing n-forms
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On a connected Manifold M, there is a bike tien between orientations and equivalence classes of smooth nowhere vanishing n forms on M. Where [X1, ..., Xn] is associated to w such that w(X1,...,Xn) is everywhere positive. (w and w' are equivalent if w = f w' for a positive function f). Explanation: - the space of n-covectors is of dim 1, so at each p, w = f(p)w'. As f is continuous and M is connected, f(M) is connected. As f is nowhere vanishing, f(M) lies in punctured line, which has two connected components. So f is either always positive or always negative. - we will see later why there is an w that is no where vanishing - we will see later why w(X1, .., Xn) has Constant sign on M. - well defn: if w and w' are equivalent then they differ by a positive function so w(X1,..,Xn) and w'(X1,...,Xn) have the same sign. - well defn: if Yi another element active, at each p, then w(X1, ....,Xn) = det A w(Y1, ..., Yn) for transition matrix A but det A > 0. Call w an orientation form on M. Orientation on 0 -dim manifolds are functions that assign to each point 1 or -1.
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