h3-4 - Math 481 - 3. Checking the manifold definition 1....

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Unformatted text preview: Math 481 - 3. Checking the manifold definition 1. Example: An atlas on P1. Recall that P1 is the space of all lines in R2 through (0,0). The map 4513 is defined on What set U5 C P1? 18 (02356;) a coordinate patch? Are (Ula¢1);(U21¢2) an atlas? o Is every line through3(0,0) Contained in some Ui? 7' IS 951071 0 U2) open ill R2? $2071 0 U2)? c What is 452-0 gfifl? a1 0 <51? ' 2. Similarly, on Worksheet 1, for P2 we have it}! "‘_ _ _‘ ‘ "¢1(L)=(ulau%)a . = (11%: Mg): 0 3. Example: Consider k numbered rods of length 1, in a closed chain with hinged joints, in the plane. What is the configuration spaCe if k = 3, a) assuming the first rod is fined, b) no constraints assumed? ’2 Math 481 4. Orientability 1. A manifold is orientable if it has an atlas such that whenetrer UiflU} 75 (D, then DMZ-043171) has positive determinant. a) Here, a = (1153,... ,uy) : Ui —> an and me,- o tail) is the Jacobian an; 6113 53: est 1 J an? an" which we also write as b) on U; n Uj, 9151- o g5? and gt,- 0 gti‘l are inverse maps, so, by the Multivariable Chain Rule, their Jacobians are inverse matrices. Hence, their determinants are inverse numbers (remember for n X n matrices A, B, det(AB) = det(A) det(B)).- Thus1 these determinants are nonzero. . G) Since det D(¢,~ o it?) 79 0, it has the same sign at all the points in a connected component of U2- n U j. d) Note: switching any pair of 95,; coordinates switches two rows of D(¢io¢j_1). Switching any pair-of qu coordinates switches two columns of D(¢¢- 0 $171). Either switch changes the sign of the determinant. 2. A 2-dimensional manifold is orientabie if and only if it does not contain a Mobius band. For example, P2 is not orientable. 3. a) Suppose M is an orientable manifold and A = {(Ua, 46a 3 (n3, . . . , is an orient- aCUrt, - - - m3) 1 > 0 on U0, flUg). For any other compatible coordinate patch 6(Ufi, . . . , 1 n (U,¢ = (9:1, . . . ,33”)) with U connected, the sign of det is the same for all 3(ua,...,ug) U0, with U n Ua 72 6. Reason: by the Multivariable Chain Rule, we have ing atlas (det 3(371,...,m”) _ 6(331,...,:I:”) 3(u;,...,n2 3(ué,...,u3) F 6(u;,...,ug) 8(ufi,...,ug) and dot is multiplicative. 1)) Conclusion: If a manifold M has a finite atlas of connected coordinate patches, there is a finite recursive procedure for deciding if M is orientable: Start with the first patch and try to alter all those that intersect it to get the determinant of the Jacobians to be positive. If this cannot be done, M is not orientable. If it can, repeat starting with the second coordinate patch. Reference: flankel, pp. 82-85. ...
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h3-4 - Math 481 - 3. Checking the manifold definition 1....

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