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Home , Differential form

1 Differential Forms

A differential form is a of the base-forms ω ∈ Ωk(X),

ω = f(x)dxi1 ∧ dxi2 ∧ · · · ∧ dxik

Where the ij are chosen in increasing order, 1 ≤ i1 < i2 < ··· < ik ≤ n. Although not technically the definition, there is no detail lost in defining the value on the standard for n, namely ∂ , ∂ ,..., ∂ , to be R ∂x1 ∂x2 ∂xn  ∂ ∂ ∂  dxi1 ∧ dxi2 ∧ · · · ∧ dxik , ,..., ≡ +1 ∂xi1 ∂xi2 ∂xik That is, we get +1 if all the indices match. From this definition, all other values can be computed. For intuition we stick to n dimensional forms. Thus we only have one possible form to work with,

dx1 ∧ · · · ∧ dxn

∂ . If we are given other random vectors ξ1, . . . ξn, we must express them in the basis, forming the transition ∂xi

M ∂ →ξi ∂xi Thus to compute dx1 ∧ · · · ∧ dxn (ξ1, . . . , ξn) we simply expand each ξj in the standard basis, then use the alternating-ness of the form and recognize what pops out is the

   ∂ ∂    dx ∧ · · · ∧ dx (ξ , . . . , ξ ) = det M ∂ dx ∧ · · · ∧ dx ,..., = det M ∂ (+1) 1 n 1 n →ξi 1 n →ξi ∂xi ∂x1 ∂xn ∂xi When we deal with , we may talk about “top-forms” on them, i.e. maximal dimensional forms. So if M is n-dimensional, we have just one of n forms, say it is labeled Ω. Really the only way to work with it is to push it n −1∗ −1 down to R , so suppose that we have two charts (Uα, ϕα), (Uβ, ϕβ). Then we know that ϕ Ω := Ω ◦ ϕ∗ , which will be somewhat demonstrated shortly. Then for the two charts, since Ω must push-down to an n-form, we konw that

−1∗ ϕα Ω = λα(x)dx1 ∧ dx2 ∧ · · · ∧ dxn

−1∗ ϕβ Ω = λβ(x)dy1 ∧ dy2 ∧ · · · ∧ dyn Where the different letters are signifying the different choice of coordinates. What we are interested in is the comparison of −1 n n λα, λβ. This is done the following way. Form the transition map Φ := ϕβ ◦ ϕα , which is R → R ,

Φ(x1, . . . , xn) = (y1(x1, . . . , xn), . . . , yn(x1, . . . , xn)) then use it so push-back λβ(x)dy1 ∧ dy2 ∧ · · · ∧ dyn into “x-coordinates” (really this means into ϕα(Uα)). That is, consider

∗ Φ (λβ(x)dy1 ∧ dy2 ∧ · · · ∧ dyn) = λβ(Φ(x))dy1 ∧ dy2 ∧ · · · ∧ dyn(Φ∗·, Φ∗ ··· ,..., Φ∗·)

D ∂ E Where Φ∗ is the map from the “x-coordinate” (whose basis is ) to the “y-coordinate” tangent space ∂xi D E (whose basis is ∂ ), and we will be shown the “definition” of here now. Given some x-basis vector ∂ , we have that ∂yi ∂xi   ∂ ∂y1 ∂ ∂y2 ∂ ∂yn ∂ Φ∗ = + + ··· + ∂xi ∂xi ∂y1 ∂xi ∂xi ∂xn ∂yn

In our consideration above, we really need only to check the value of this push back on the standard basis ∂ . We ∂xi compute:     ∗ ∂ ∂ ∂ ∂ ∂ Φ (λβ(x)dy1 ∧ dy2 ∧ · · · ∧ dyn) ,..., = λβ(Φ(x))dy1 ∧ dy2 ∧ · · · ∧ dyn Φ∗ , Φ∗ ,..., Φ∗ (1) ∂x1 ∂xn ∂x1 ∂x2 ∂xn

∂ Here I must resign from detail. Use the definition of Φ∗( ) above and expand each in the equation above. Then use ∂xi multilinearity of the differential form. Then recognize we can swap entries for (−1) by anti-linearity. Then recognize that     ∂ ∂ ∂ y1, . . . , yn λβ(Φ(x))dy1 ∧ dy2 ∧ · · · ∧ dyn Φ∗ , Φ∗ ,..., Φ∗ = λβ() det Jac( ) ∂x1 ∂x2 ∂xn x1, . . . , xn

1 This being done though, we remember that ∂ ∂ λα(x)dx1 ∧ dx2 ∧ · · · ∧ dxn( ,..., ) = λα() ∂x1 ∂xn Since these forms are uniquely determined by their values on this input we must have   y1, . . . , yn λα = λβ() det Jac( ) x1, . . . , xn We now see why this applies to Rn.

1.1 Application to Rn Suppose that U ⊂ Rn is open. We DEFINE the of the differential form Z Z f(x)dx1 ∧ · · · ∧ dxn ≡ f(x)dx1dx2 ··· dxn U U n Now suppose that V ⊂ R is another with coordinates v1, . . . , vn. Then we form the transition map

F : V → U, F (v1, . . . , vn) = (x1(v1, . . . , vn), . . . , xn(v1, . . . , vn)) Then considering these open sets to be manifolds, we have that Z Z Z   ∗ x1, . . . , xn f(x)dx1 ∧ · · · ∧ dxn = f(F (v))F (dx1 ∧ · · · ∧ dxn) = f(F (v)) det Jac( ) dv1 ∧ · · · ∧ dvn U V V v1, . . . , vn 2 Orientation and Riemannian

Now we lay groundwork for volume elements that will be key for our Haar Measures.

1. (Orientation Equivalence) Two bases of any (ei)i, (fj)j are said to be of same orientation if the transition matrix between them has positive , that is, if

det (Me→f ) > 0 Note that we haven’t declared which bases are “positively” oriented, but merely established two equivalence classes of bases. Typically however we will immediately after declare “positive” to be the one of same orientation as the standard Rn basis, or its push-up to the .

2. (Orientation via charts) Given charts (Uα, ϕα), each chart determines an orientation on the tangent spaces on the manifold, just those Tp(M), p ∈ Uα, via declaring that the positively oriented bases at each point are those with same orientation as ϕ−1( ∂ ), j = 1, . . . , n (note this is done at every different p, but since our chart is C∞, these orientations ∗ ∂xj are in some sense “smooth.” −1 n n Two charts are said to be “coherently oriented” if the transition map ϕβ ◦ ϕα : R → R induces an orientation- −1 −1 preserving map of tangent spaces, that is, if det(Jac(ϕβϕα )) > 0, since Jac(ϕβϕα ) is the map that maps n n T ( ) → T −1 ( ) x Rα ϕβ ϕα (x) Rβ The α, β signifying different choices of coordinates. Thus, the natural way to define an orientable manifold is to say that it is orientable if it may be covered by coherently- oriented charts. 3. (Orientation via a differential form) An alternative definition of orientation is to say that there is a non-vanishing, top- dimensional differential form Ω on M, that induces a pointwise form Ωp at every p ∈ M. The reason this determines an orientation is that since given two bases of a tangent space Tp(M), say [e1p, . . . , enp], [f1p, . . . , fnp], we have, using the same arguments as above, that

Ω(f1, . . . , fn) = det(Me→f )Ω(e1, . . . , en) Thus Ω splits the bases of the tangent spaces on whether or not their Ω evaluations have the same sign, giving an orientation. Again we haven’t declared which bases are (+), (−), but we will soon. This definition is equivalent to the chart definition since if you have the charts above, coherently oriented, each one can induce a differential form on the tangent spaces (pushing up the standard dx1 ∧ · · · ∧ dxn), and then use a partition of unity subordinate to the charts to “stitch together” the individual forms.

2 4. (Riemannian Metric) A Riemannian Metric is nothing more than inner product on all the tangent spaces of the points of the manifold, i.e. Φ assigns to each point p an inner-product Φp an inner-product on Tp(M). Since given a neighborhood of p and a chart (U, ϕ) this chart induces a standard tangent-space basis E1p,...,Enp on the nearby tangent spaces, we have at each point u ∈ U, that Φu is determined by

Φu(Eiu,Eju) ≡ gij(u)

∞ The metric is said to be C if for fixed i, j the function gij is a smooth function of u (of course only on the neighborhood of p).

Theorem 1. If M is smooth manifold together with a Riemannian metric and an orientation (i.e. a choice of “(+)” bases, then using the orientation given we may create a differential form Ω such that Ω(F1p,...,Fnp) = +1 on every positively oriented orthonormal basis of the respective tangent spaces Tp(M), p ∈ M. As a consequence of this declaration, we discover that the value on the chart basis Eip, i = 1, . . . , n above is given q Ω(E1p,...,Enp) = det(gij(p)) where gij(p) is referring to the full matrix of entries for the inner product Φp.

Corallary 2. If we have an oriented , then there exists the Ω above. This Ω has push-down to Rn given by −1∗ q ϕ Ω = det(gij(p))dx1 ∧ · · · ∧ dxn

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