differential geometry w notes from teacher_Part_67

differential geometry w notes from teacher_Part_67 - ϕ •...

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5.4. DEGREE OF A MAP 133 5.4.3 Brouwer Degree Let M and V be two closed (compact without boundary) oriented n -dimensional manifolds. Let ϕ : M V be a smooth map. Let ω Λ n V be an n -form on V . Suppose that Z V ω , 0 . Then we can normalize it so that Z V ω = 1 . Then we can consider the quantity R M ϕ * ω R V ω , which can also be written as R ϕ ( M ) ω R V ω . This quantity counts how many times the image of M wraps around V . Corollary 5.4.1 Let M and V be n-dimensional manifolds and ϕ : M V be a smooth map. Let α and β be n-forms on V such that R V α , 0 and R V β , 0 . Then R M ϕ * α R V α = R M ϕ * β R V β . Proof : 1. Let ω = α R V α - β R V β . di geom.tex; April 12, 2006; 17:59; p. 133

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134 CHAPTER 5. LIE DERIVATIVE 2. We have Z V ω = 0 . 3. Therefore, the form ω is exact. 4. Hence, the form ϕ * ω is exact. 5. Thus, Z M ϕ * ω = 0 . ± The quantity deg( ϕ ) = R M ϕ * ω R V ϕ does not depend on the choice of the form ω but only on the map ϕ . It is called the Brouwer degree of the map
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Unformatted text preview: ϕ . • Example. • In the case of one-dimensional manifolds the degree of the map ϕ : M → S 1 is called the winding number . • Picture. • Theorem 5.4.2 Let V and M be n-dimensional compact oriented man-ifolds without boundary. Let ϕ : M → V be a smooth map. Let y ∈ V be a regular value of ϕ so that the di ﬀ erential ϕ * : T x M → T y V at any point x ∈ ϕ-1 ( y ) is bijective (isomorphism). Then deg( ϕ ) = X x ∈ ϕ-1 ( y ) sign ( ϕ ( x )) , where sign ( ϕ ( x )) = sign (det( ϕ * )) . Proof : 1. (I). Claim: the preimage ϕ-1 ( y ) of a regular value is a ﬁnite set, that is, ϕ-1 ( y ) = { x i ∈ M | ϕ ( x i ) = y , i = 1 , 2 , . . . , N } . di ﬀ geom.tex; April 12, 2006; 17:59; p. 134...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_67 - ϕ •...

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