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Differential Forms and Connections

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Differential Forms and Connections Introduction to Differential Forms and Connections Done by: Abdulla Eid Supervised by : Prof. Steven Bradlow Fall 2008, University of Illinois at Urbana Champign Dec 07, 2008 1 Abstract In this project, the main aim is to understand the main concepts of differential forms and connection in elementary setting of differential geometry. This is an extra work load for a differential Geometry course that I am taking under Prof. Steven Bradlow at the University of Illinois at Urabana-Champaign. In this course we have to demonstrate the ability to read and understand a topic from the book that is not covered in the lecture, we have to write a short report that shows our understanding and how we can present the topic that we studied by ourself. The content of my report will be about the basics of differential forms and connections, the report will be 3 chapters, first chapter is about differential form, second about connection and the final chapter is about Connections, especially the Levi Civita connection and the fundamental theorem of Riemannian Geometry in dimension 2) . Main part of the material will be from the text in the course which “Elementary Differential Geometry” by O’Neil. Many of the exercise will be provided as an example to show my understanding to the material and how to present them. My thanks go to Prof. Bradlow for choosing the material and give me the time and the direction to develop this small report 2 Table of Contents Introduction to Differential Forms and Connections Chapter 1: Differential Forms § 1.1: 1-Forms (Section 1.5) § 1.2: Differential Form (section 1.6) § 1.3: Differential Form in a surface (4.4) Chapter 2: Connections § 2.1: Covariant Derivative (section 2.5, section 7.3) § 2.2: Connection Forms (Section 2.7) § 2.3: The Structural Equation § 2.4: The Fundamental Equation a surface in R 3 § 2.5: Form Computation Chapter 3: Curvature § 3.1: Geometric Surfaces . § 3.2: Gaussian Curvature . § 3.3 : Levi Civita Connection and The Fundamental Theorem of Riemannian Geometry for dimension 2. 3 Chapter 1 : Differential Forms § 1.1 1-Forms : 3 We start this section by defining 1-Form on the set of all tangent vectors of R . So fix a point p Ǩ R3 to be the point of Application. Definition: (Tangent Space) 3 3 3 The set T pR :={v p | v Ǩ R } is called the tangent space of R at p . Remark : 3 1- Tp R is an R-vector space with the following addition and scalar multiplication: 3 Vp,w p Ǩ Tp R , Ǩ R, i) V p+w p:=(v+w) p ii) Vp=( v) p 3 3 2- Tp R is R-isomorphic as an R-vector space to R via the natural map that assign p to the 3 origin point of R i.e. ͧ ͧ : ͎+͌ ǫ ͪ+ Ŵ ͪ Ǩ ͌ Now we define 1-form on R 3 to be an element of the dual space of R 3 i.e. 3 * (R ) = ʜ: ͌ͧ Ŵ ͌ | is R-linear} Definition : 3 A 1-Form ʜ: ͌ͧ Ŵ ͌ | is R-linear that sends to every tangent vector v of R a real number ʚͪʛ. Remark : 3 1) Let p R be a point of application, then the 1-form p is any element of the dual space 3 * Ǩ ϕ (T pR ) . 3 3 3 * 2) Let F: TpR TqR be a map, then ∀ ϕpǨ (T pR ) , we have 3 3 * ϕq= ϕp∘F : TqR R is in (T qR ) . The remaining part of this section, we will be concerned on how to convert a real valued function f:R 3R into a differential 1-Form. 4 Definition : 3 Let f: R R be a differentiable map, then the differential ͧ is a ͚͘: ͎+͌ ǫ ͪ+ Ŵ ͕͛ͦ͘ ʚ͚ʛ · ͪ Ǩ ͌ 1-Form. Example : n th 1) Let xi:R R be the i projection map, then the differential of xi at any ǫ ʚͬͥ, ͬͦ, … , ͬ)ʛ Ŵ ͬ$ Ǩ th point v p is dxi(v p)=grad(xi)=(0,0,…,1,…,0)v p=(v p)i=i coordinate of v p. 3 2) Let f 1,f 2,f 3:R R be a map, then we define a 1-Form 3 ψ := f 1dx 1+ f2dx 2+ f 3dx 3 :T pR R by ψ(v) = f 1(p)dx 1(v)+ f 2(p)dx 2(v)+ f 3(p)dx 3(v) = f 1(p)v 1+ f 1(p)v 2+ f 1(p)v 3 Now one may ask, is every 1-Form can be written in the form in (2) above? i.e. given a 1-form ψ, is there f 1,f 2,f 3 such that ψ := f 1dx 1+ f 2dx 2+ f 3dx 3 ? The answer is yes, as we shall see in the following lemma. Lemma 1 : 3 Let be a 1-Form on R , then =f 1dX 1+f 2dX 2+f 3dX 3, where f i= (e i). Proof : 3 Let f i= (e i) (i=1,2,3), then for all v p ǨTp R , we have (f1dx1+f 2dx2+f 3dx3)( мз)= f1 (p)dx1(v p)+f 2 (p)dx2(v p)+f 3 (p)dx3 (v p) = f 1 (p)(v1)+f 2 (p)(v 2)+f 3 (p)(v 3) = (e 1)v 1 + (e 2)v 2+ (e 3)v 3 = (v 1e1+ v 2e2+ v 3e3)= (v p) So = f 1dX 1+f 2dX 2+f 3dX 3 ∎ 5 Corollary 2 : Let f be differentiable map on R 3, then df= i! +i! +i! i3 ͬ͘ i4 ͭ͘ i5 ͮ͘ Proof : 3 By definition of the differential of f, we have for all V p Ǩ Tp R , i! i! i! i! i! i! df(v p)= =( , ) (v 1,v 2,v 3)= v1+ v2+ v3, but v i=dX i(v p) ͕͛ͦ͘ ʚ͚ʛ · ͪ i3 i4 , i5 · i3 i4 i5 i! i! i! So df(v p)= + + (v p) i3 ͬ͘ i4 ͭ͘ i5 ͮ͘ ∎ Using Corollary 2, we reached to the famous Calculus formula we had in Calc III, here is a list of properties about the differential of a map. Proposition 3 : Let f,g:R 3R , h:R R, be differentiable maps, then i)d(f+g)=d(f)+d(g) (ii) d(fg)=fdg+gdf (iii) d(h °f))=h ’(f)df Proof : i) d(f+g)= iʚ!ͮ"ʛ +iʚ!ͮ"ʛ +iʚ!ͮ"ʛ i3 ͬ͘ i4 ͭ͘ i5 ͮ͘ =i! +i! +i! + i" +i" +i" i3 ͬ͘ i4 ͭ͘ i5 ͮ͘ i3 ͬ͘ i4 ͭ͘ i5 ͮ͘ =d(f)+d(g) ii) put x 1:=x, x 2:=y, x 3:=z, then 6 d(fg)= ͧ i! = ͧ i! i" = ͧ i! + ͧ i" =gdf+fdg ∑$Ͱͥ i3 ͬ͘ ∑$Ͱͥʚi3 ͛ ƍ ͚ i3 ʛͬ͘ ∑$Ͱͥ ͛ i3 ͬ͘ ∑$Ͱͥ ͚ i3 ͬ͘ 3 ° iii) h °f:R R, d(h °f)= = ͧ i# ! = = ͧ ′ i! =h’(f)df ---by chain rule. ∑$Ͱͥ i3 ͬ͘ ∑$Ͱͥ ͜ ʚ͚ʛ i3 ͬ͘ ∎ Example : 3 2 2 1- f: R ǫ ʚͬ, ͭ, ͮʛ Ŵ ʚx -1) y + (y +2) z Ǩ ͌ df=(2xdx)y+(x 2-1)dy+2y dy z+(y 2+2) dz = 2xy dx + (x 2-1+2yz) dy + (y2+2) dz So as a result, we have i! = 2xy, i! = x 2-1+2yz, i! = y 2+2 i3 i4 i5 2- (exercise 6,b) 3 yz f: R ǫ ʚͬ, ͭ, ͮʛ Ŵxe Ǩ ͌ df= i! +i! +i! = e yz dx+ xz e yz dy + xye yz dz i3 ͬ͘ i4 ͭ͘ i5 ͮ͘ 7 §1.2- Differential Form : In this section, we will generalize the 1-Form we introduced in section 1, also we might be interesting to work in the n-dimensional Euclidean Space R n instead of just R 3. A- Operation on Differentials : Let dx i,dx j be two differentials defined as in section 1, we are interested to define an operation between them called the multiplication of forms and denoted by the symbol ʗ “Wedge operator”. Definition : Let dx i,dx j be two forms, then one define dx i ʗ dx j := -dxi ʗ dx j (i ≠j) and 0 if i=j. Where dx i ʗ dx j is an expression for the multiplication dx idx j represents a 2-Form and by the same way we can abstractly define the p-form of R n . Definition : (p-Form) A p-Form is an expression contains p dx i (i=1,2,…,n) and such an expression is said to have degree p . Example : Let f: R nR be a smooth map, then 1- A 0-Form is just the expression of f. 2- A 1-form is an expression like fdx 1 or fdx 2 ,… 3- A 2-form is an expression like fdx idx j 4- A 3-Form is an expression like fdx idx jdx k and so on… Remark : n 1- In R , there is only n-Form, because n+1-Form will contain two similar form dx i and by wedge rule, we can shift them to be next to each other and then get a result of zero, so if p > n, then p-Forms are automatically zero. 8 B- Wedge Operator of Forms : n Let , be two 1-Form in R , i.e. = ) i!Ĝ and = ) i"Ĝ , then & & ∑$Ͱͥ ͬ͘$ ∑$Ͱͥ ͬ͘$ i3Ĝ i3Ĝ := ) ) i!Ĝ i"ĝ , So the product of two 1-Form yield 2-form and so in & ʗ ∑$Ͱͥ ∑%Ͱͥ ͬ͘%ͬ͘$ i3Ĝ i3ĝ general the product of p-Form and q-Form is p+q-Form.
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