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Anubhav Nanavaty AN INTRODUCTION TO DIFFERENTIAL FORMS, STOKES' THEOREM AND GAUSS-BONNET THEOREM ANUBHAV NANAVATY Abstract. This paper serves as a brief introduction to differential geome- try. It first discusses the language necessary for the proof and applications of a powerful generalization of the fundamental theorem of calculus, known as n 3 Stokes' Theorem in R . Further, geometry in R will be discussed to present Chern's proof of the Poincar´e-HopfIndex Theorem and Gauss-Bonnet The- 3 orem in R , both of which relate topological properties of a manifold to its geometric properties. Only a working knowledge of multivariable calculus is needed to understand this paper. All other concepts are introduced and dis- cussed. Contents 1. Introduction 1 2. Differential Forms and Manifolds 2 2.1. Differentiable Manifolds 2 2.2. Differentiable Forms 4 2.3. Exterior Derivatives 6 2.4. Integration of Forms 8 3. Stokes' Theorem 10 3.1. Applications 13 4. Riemannian Manifolds and Geometry in R3 14 4.1. Cartan's Structure Equations in Rn 15 4.2. Curvature in R3 18 5. The Gauss-Bonnet Theorem 20 6. Conclusion 26 Acknowledgments 26 References 26 1. Introduction We first introduce the concept of a manifold, which leads to a discussion of differential forms, the exterior derivative and pull-back map. We then discuss integration of forms in Rn in order to state and prove Stokes' Theorem in Rn. A few applications of Stokes' Theorem are also stated and proved, such as Brouwer's fixed point theorem. In order to discuss Chern's proof of the Gauss-Bonnet Theorem in R3, we slightly shift gears to discuss geometry in R3. We introduce the concept of a Riemannian Manifold and develop Elie Cartan's Structure Equations in Rn to define Gaussian Curvature in R3. The Poincar´e-Hopf Index Theorem is first stated 1 and proved, and the concept of the Euler number is introduced in order to end with a proof of the Gauss-Bonnet Theorem in R3. A few important implications of the theorem are then mentioned. Most of the definitions, along with proofs of the propositions and theorems have been adapted from Do Carmo's Differential Forms and Applications [1], along with Pressley's Elementary Differential Geometry [2]. 2. Differential Forms and Manifolds We begin with the concept of a differentiable manifold. A generic theme in differential geometry is that we associate seemingly `unknown' objects, such as manifolds, with `known' objects, such as Rn, so that we can study the local behavior of the object using concepts such as differential forms. 2.1. Differentiable Manifolds. Definition 2.1. A differentiable manifold is a set M along with a set of injective n maps fα : Uα ! M, such that Uα ⊂ R are open, and: S (1) M = α fα(Uα) −1 −1 (2) For all α; β such that fα(Uα)\fβ(Uβ) = W 6= Ø, it must be that fα (W ); fβ (W ) n −1 −1 are both open in R . Further, fα ◦ fβ and fβ ◦ fα are differentiable. (3) f(fα;Uα)g, called the set of charts or coordinate systems, is maximal in regards to both (1) (2) In other words, an n dimensional manifold, denoted M n is a set that locally `looks' like Rn. The second condition makes sure that if two coordinate systems overlap, then points that are `close together' in one system are mapped to points that are also `close together' in the other system. The last condition serves the purpose of inducing a topology on M n, meaning that open sets on M can now be defined using open sets in Rn. From this definition, it is clear that Rn is a mani- fold (as we can just map the topology on Rn to itself), but there are examples of manifolds whose global properties are drastically different from Rn. For the sake of simplicity, we only look at the cases where the manifold satisfies Haussdorf's axiom and has a countable basis1. We will now describe the concept of the tangent space of a manifold. In the manifold Rn, so to find a tangent vector to a curve n α : I = [a; b] ! R at point p 2 [a; b] we calculate the derivative α0(p), which is in fact the tangent vector. However, it is not obvious how to do such a thing on an arbitrary manifold. Note the following definition. Definition 2.2. Take p 2 M n, and let α : I ! M n be smooth such that α(0) = 2 p 2 M, . and consider D = f' : M ! R j ' is linearg. We define the tangent 0 0 d vector to α at p as a map α (0) : D ! R defined by α (0)(') = dt (' ◦ α)jt=0. 1View [1] for a statement of Haussdorf's axiom and the definition of a countable basis. 2Note that for any α such that p 2 α(I), we can reparametrize α such that α(0) = p 2 The definition of the tangent vector is intuitively the same as the one we had for Rn. In order to why this is the case, set M = Rn and take a curve α : I ! Rn n such that α(t) = (α1(t); : : : ; αn(t)). For any linear f : R ! R, we can calculate the tangent vector at α(0) in the following way using the chain rule: d X @f(α(0)) f(α(t))j = α0 (0) dt t=0 @x i i i h X @ i = α0 (0) f(α(0)) i @x i i d dα As desired, dt f(α(t)) is dependent on dt . Definition 2.3. For M n, the tangent space at a point p 2 M n is denoted as 0 n TpM = fα (0) j α : I ! M ; α(0) = pg. In other words, the tangent space at a point is the set of all tangent vectors at that point. This next proposition allows for us to better understand the tangent space at p. n n Proposition 2.4. For p 2 M , dim(TpM ) = n. Further, if p 2 fα(Uα), then @ n @ span(f j j 2 [n]g) = TpM . The set f g is the set of partial derivative @xj @xj n @ @ operations of with respect to fα such that ' := (' ◦ fα) R @xj @xj n Proof. If p 2 fα(Uα) such that Uα ⊂ R and p = f(0;:::; 0), then for a curve n −1 α : I ! M with α(0) = p we have that f (α(0)) = (x1(t); : : : ; xn(t)). For 3 −1 simplicity's sake , we conflate f ◦ α with α, so that α(t) = (x1(t); : : : ; xn(t)); x 2 n R . This allows us to state that ' ◦ α(t) = '(x1(t); : : : ; xn(t)) for all ' : M ! R. We can now deduce that d (2.5) a0(0)' = (' ◦ α)j dt t=0 d (2.6) = '(x (t); : : : ; x (t))j dt 1 n t=0 n X @ 0 0 = xi(0) We can now write a (0) as:(2.7) @x 0 i=0 i n 0 X 0 @ (2.8) a (0) = xi(0) @x 0 i=0 i Since there are n terms in the summation, it follows that the tangent space is n-dimensional and f @ g is a basis for the space. @xi n n n n Proposition 2.9. If M = R , then for p 2 R , Tp(R ) = R . It is now useful to discuss the concept of orientability. Definition 2.10. A differentiable manifold is orientable if there exists a differen- tiable structure f(fα;Uα)g such that for all α; β where fα(Uα) \ fβ(Uβ) = W 6= Ø, −1 the determinant of the differential map (known as the jacobian J) of fβ ◦ fα is positive. 3This is a standard practice in differential geometry 3 −1 In other words, let F = fβ ◦fα : Uα ! Uβ. It follows, by multivariable calculus, that for p 2 Uα, the differentiable map, or jacobian, is defined by DF (p): Tp(Uα) ! Tf(p)(Uα). If det(dF ) > 0 for all p, this intuitively means that the tangent spaces of the two charts cannot have opposite orientations, and so there is a fixed orientation at each p 2 M. 2.2. Differentiable Forms. Now, we can slightly shift gears to discuss forms on manifolds. Definition 2.11. The dual space at p is denoted as the set of linear functions ∗ (TpM) = f' : TpM ! Rg. Further we define k ^ ∗ ∗ k (TpM) = f' : [(TpM) ] ! Rg 4 as the set of all real linear functions that take k elements of TpM and are k-linear and alternate5 The definitions of a 1-form and 0-form follow. Definition 2.12. An exterior 1-form is a function ! which maps p 2 M n to ∗ !(p) 2 (TpM) . In other words, the 1-form assigns to each point p a real linear function on M n. We can find a basis for these forms in the following manner: we know that lo- n cally, p is parametrized by some Uα ⊂ R such that p 2 fα(Uα). Consequently, n n we can assign to p coordinates of R by the functions xi : R ! R such that −1 −1 xi(fα (p)) = (fα )i(p). In other words, we project the i'th component of p ac- −1 n cording to the coordinate chart Uα. For convenience sake, we conflate fα (p) 2 R with p 2 M, because we assume that we are working under a single chart Uα, so n locally, f(Uα) can be conflated with R .
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