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
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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
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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.
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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 R3 to be the point of Application.
Definition: (Tangent Space)
3 3 3 The set T pR :={v p | v 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 Tp R , 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.
Now we define 1-form on R 3 to be an element of the dual space of R 3 i.e.
3 * (R ) = is R-linear} Definition :
3 A 1-Form is R-linear that sends to every tangent vector v of R a real number . Remark :
3 1) Let p R be a point of application, then the 1-form p is any element of the dual space 3 * ϕ (T pR ) .
3 3 3 * 2) Let F: TpR TqR be a map, then ∀ ϕp (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.
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Definition :
3 Let f: R R be a differentiable map, then the differential is a 1-Form.
Example :
n th 1) Let xi:R R be the i projection map, then the differential of xi at any 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 Tp 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 ∎
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Corollary 2 :
Let f be differentiable map on R 3, then
df= + +
Proof :
3 By definition of the differential of f, we have for all V p Tp R ,