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Appendix a Differential Forms and Operators on Manifolds Appendix A Differential Forms and Operators on Manifolds In this appendix, we introduce differential forms on manifolds so that the integration on manifolds can be treated in a more general framework. The operators on manifolds can be also discussed using differential forms. The material in this appendix can be used as a supplement of Chapter 2. A.1 Differential Forms on Manifolds Let M be a k-manifold and u =[u1, ··· ,uk] be a coordinate system on M.A differential form of degree 1 in terms of the coordinates u has the expression k i α = aidu =(du)a, i=1 1 k 1 where du =[du , ··· , du ]anda =[a1, ··· ,ak] ∈ C . The differential form is independent of the chosen coordinates. For instance, assume that v = [v1, ··· ,vk] is another coordinate system and α in terms of the coordinates v has the expression n i α = bidv =(dv)b, i=1 i n 1 ∂v ∂v 1 ··· ∈ where b =[b , ,bk] C .Write = j .By ∂u ∂u i,j=1 T ∂v dv =du , ∂u we have T ∂v dvb =du b, ∂u 340 Appendix A Differential Forms and Operators on Manifolds and by T ∂v a = b, ∂u we have (du)a =(dv)b. From differential forms of degree 1, we inductively define exterior differential forms of higher degree using wedge product, which obeys usual associate law, distributive law, as well as the following laws: dui ∧ dui =0, dui ∧ duj = −duj ∧ dui, a ∧ dui =dui ∧ a = adui, where a ∈ R is a scalar. It is clear that on a k-dimensional manifold, all forms of degree greater than k vanish, and a form of degree k in terms of the coordinates u has the expression 1 k 1 ω = a12···kdu ∧···∧du ,a12···k ∈ C . (A.1) To move differential forms from one manifold to another, we introduce the inverse image (also called pullback) of a function f.LetM1 and M2 be two manifolds, with the coordinate systems u and v respectively. Let μ ∈ r C (r 1) be a mapping from M2 into M1, μ(y)=x, y ∈ M2.Letg be the parameterization of M2: y = g(v). We have u =(μ ◦ g)(v). r Hence, u is a C function of v.Letf be a function on M1. Then there is a function h on M2 such that h(y)=f(μ(y)) y ∈ M2. We denote the function h by μ∗f and call it the inverse image of f (by μ). Clearly, the inverse image has the following properties: 1. μ∗(f + g)=μ∗f + μ∗g; ∗ ∗ ∗ 1 2. μ (fg)=(, μ f)(μ g), f ∈ C ; i 3. if df = i aidu ,then ∗ ∗ ∗ i d(μ f)= (μ ai)d(μ u ). i We now define the inverse image of a differential form as follows. Definition A..1. Let ω be a differential form of degree j: ω = fdun1 ∧···∧dunj . Then the inverse image of the differential form ω (by μ) is defined by μ∗ω =(μ∗f)d (μ∗un1 ) ∧···∧d(μ∗unj ) . A.2 Integral over Manifold 341 The properties of inverse image for functions therefore also hold for differen- tial forms. Besides, by Definition A.1, the following properties hold. supp(μ∗ω) ⊂ μ(supp(ω)) and ∗ ∗ ∗ ∗ ∗ μ dω =d(μ ω),μ(ω1 ∧ ω2)=μ ω1 ∧ μ ω2. A.2 Integral over Manifold Let M be a compact k-manifold having a single coordinate system (M,u) and ω be a k-form on M given in (A.1). Let D = u(M) ⊂ Rk. Then the integral of ω over M is defined by 1 k ω = a12···kdu ∧···∧du . M D We often simply denote ω by ω if no confusion arises. Notice that if M the support of a12···k is a subset of D,thenwehave 1 k 1 k a12···kdu ∧···∧du = a12···kdu ∧···∧du . D Rk To extend the definition of the integral over a manifold, which does not have a global coordinate system, we introduce the partition of unity and the orientation of a manifold. Definition A..2. Assume that {Ui} is an open covering of a manifold M. A countable set of functions {φj} is called a partition of unity of M with respect to the open covering {Ui} if it satisfies the following conditions: ∞ 1. φj ∈ C has, compact support contained in one of the Ui; 2. φj 0and j φj =1; 3. each p ∈ M is covered by only a finite number of supp(φj ). It is known that each manifold has a partition of unity. We shall use partition of unity to study orientation of manifolds. Orien- tation is an important topological property of a manifold. Some manifolds are oriented, others are not. For example, in R3 the M¨obius strip is a non- oriented 2-dimensional manifold, while the sphere S2 is an oriented one. To define the orientation of a manifold, we introduce the orientation of a vector space. Definition A..3. Let E = {e1, ··· , en} and B = {b1, ··· , bn} be two bases of the space Rn.LetA be a matrix, which represents the linear transformation from B to E such that ei = Abi,i=1, 2, ··· ,n. 342 Appendix A Differential Forms and Operators on Manifolds We say that E and B have the same orientation if det(A) > 0, and that they have the opposite orientations if det(A) < 0. Therefore, for each n ∈ Z+, Rn has exactly two orientations, in which the orientation determined by the canonical o.n. basis is called the right-hand orientation, and the other is called the left-hand orientation. The closed k-dimensional half-space is defined by k k H = {[x1, ··· ,xk] ∈ R : xk 0}. (A.2) Definition A..4. A connected k-dimensional manifold M (k 1) is said to be oriented if, for each point in M, there is a neighborhood W ⊂ M and a k k diffeomorphism h from W to R or H such that for each x ∈ W ,dhx maps the given orientation of the tangent space TxM to the right-hand orientation of Rk. By the definition, if M is a connected oriented manifold, there is a differential { } ∈ ∩ structure (Wi,hi) such that at each p hi(Wi) hj(Wj ), the linear trans- ◦ −1 Rk → Rk ◦ −1 formation d(hi hj )p : , satisfies det d(hi hj )p > 0foreach p. We give two examples below to demonstrate the concept of orientation. Example A..1. Let U =[−π, π] × (−1, 1). M¨obius strip M2 is given by the parametric equation g : U → R3: u u u T g(u, v)= 2+v sin cos u, 2+v sin cos u, v cos . (A.3) 2 2 2 However, g is not a diffeomorphism from U to g(U) ⊂ R3 since it is not one-to-one (on the line u = π). To get an atlas on M2, we define u u u T g˜(u, v)= −2+v cos cos u, −2+v cos cos u, v sin , 2 2 2 (A.4) and set U o =(−π, π) × (−1, 1). Then {(g(U o),g−1), (˜g(U o), g˜−1)} is a differ- entiable structure on M2 with g(U o) ∪ g˜(U o)=M2. Note that the following lines T L1 = {[2, 0,t] : −1 <t<1}, T L2 = {[t, 0, 0] ; −3 <t<−1}, do not reside on g(U o) ∩ g˜(U o). We compute the determinant det(d(˜g−1 ◦ g)) on different arcs in U o: −1 1(u, v) ∈ (0, π) × (−1, 1) det(d(˜g ◦ g)) = . −1(u, v) ∈ (−π, 0) × (−1, 1) The opposite signs show that the M¨obius strip M2 is not oriented. Example A..2. The spherical surface S2 defined in Example 2.1 is oriented. 2 Recall that {(W1,h1), (W2,h2)} is an atlas on S . The transformation from A.2 Integral over Manifold 343 h2 =(˜u, v˜)toh1 =(u, v) is given in (2.25). The Jacobian of the transforma- tion is negative everywhere, for we have 1 det(df(˜ ˜))=− < 0. u,v (˜u2 +˜v2)2 2 Hence, S is oriented. Since det(df(˜u,v˜)) < 0, in order to obtain the same orientation over both W1 and W2, we can replace h2 by −h2. The definition of integrable functions on a manifold M needs the concept of zero-measure set on manifold. To avoid being involved in the measure theory, we define the integration over a manifold for continuous functions only. Definition A..5. Let M be a connected and oriented k-dimensional man- ifold, and Φ = {φj}j∈Γ be a partition of unity of M so that the support of each φj ∈ Φ is contained in the domain of a coordinate system. Let ω be a continuous differential form of degree k on M. The integral of ω over M is defined as ω = φj ω, j∈Γ provided the series converges. 1 If M is compact, then a partition of unity of M is a finite set. Hence, ω over a compacted manifold always exists. It is clear that the integral is independent of the chosen partition. As an application, we introduce the volume formula of a manifold (M,g), where g is a parameterization of M. We define the volume differentiation of (M,g)bythek-form dV = |Gg| du, where Gg is the Riemannian metric generated by g. Then the volume of M is V (M)= 1. M The integral on an oriented manifold has the following properties. Theorem A..1. Assume that M is k-dimensional oriented manifold.
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