LectureNotes1G - Math 528 1 Geometry and Topology II Fall 2005 PSU Lecture Notes 1 1 Topological Manifolds The basic objects of study in this class

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 528 Jan 11, 2005 1 Geometry and Topology II Fall 2005, PSU Lecture Notes 1 1 Topological Manifolds The basic objects of study in this class are manifolds. Roughly speaking, these are objects which locally resemble a Euclidean space. In this section we develop the formal definition of manifolds and construct many examples. 1.1 The Euclidean space By R we shall always mean the set of real numbers, which is a well ordered field with the following property. Completeness Axiom . Every nonempty subset of R which is bounded above has a least upper bound . The set of all n-tuples of real numbers R n := { ( p 1 , . . . , p n ) | p i ∈ R } is called the Euclidean n-space. So we have p ∈ R n ⇐⇒ p = ( p 1 , . . . , p n ) , p i ∈ R . Let p and q be a pair of points (or vectors) in R n . We define p + q := ( p 1 + q 1 , . . . , p n + q n ). Further, for any scalar r ∈ R , we define rp := ( rp 1 , . . . , rp n ). It is easy to show that the operations of addition and scalar multiplication that we have defined turn R n into a vector space over the field of real numbers. Next we define the standard inner product on R n by p, q = p 1 q 1 + . . . + p n q n . Note that the mapping · , · : R n × R n → R is linear in each variable and is sym- metric. The standard inner product induces a norm on R n defined by p := p, p 1 / 2 . If p ∈ R , we usually write | p | instead of p . Theorem 1.1.1. (The Cauchy-Schwartz inequality) For all p and q in R n | p, q | p q . The equality holds if and only if p = λq for some λ ∈ R . 1 Last revised: October 2, 2006 1 Proof I. If p = λq it is clear that equality holds. Otherwise, let f ( λ ) := p- λq, p- λq . Then f ( λ ) > 0. Further, note that f ( λ ) may be written as a quadratic equation in λ : f ( λ ) = p 2- 2 λ p, q + λ 2 q 2 . Hence its discriminant must be negative: 4 p, q 2- 4 p 2 q 2 < which completes the proof. Proof II. Again suppose that p = λq . Then p, q = p q p p , q q . Thus it suffices to prove that for all unit vectors p and q we have | p, q | ≤ 1 , and equality holds if and only if p = ± q . This may be proved by using the method of lagrangne multipliers to find the maximum of the function x, y subject to the constraints x = 1 and y = 1. More explicitly we need to find the critical points of f ( x, y, λ 1 , λ 2 ) := x, y + λ 1 ( x 2- 1) + λ 2 ( y 2- 1) = n i =1 ( x i y i + λ 1 x 2 i + λ 2 y 2 i )- λ 1- λ 2 . At a critical point we must have 0 = ∂f/∂x i = y i + 2 λ 1 x i , which yields that y = ± x . The standard Euclidean distance in R n is given by dist( p, q ) := p- q . The proof of the following fact is left as an exercise. Corollary 1.1.2. (The triangle inequality) For all p , q , r in R n ....
View Full Document

This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

Page1 / 10

LectureNotes1G - Math 528 1 Geometry and Topology II Fall 2005 PSU Lecture Notes 1 1 Topological Manifolds The basic objects of study in this class

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online