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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 4520 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 DMZ043171)
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 pairof qu coordinates switches two columns of D(¢¢ 0 $171). Either switch changes
the sign of the determinant. 2. A 2dimensional 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. 8285. ...
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This note was uploaded on 04/23/2010 for the course MATH Math 481 taught by Professor Kapovich during the Fall '08 term at University of Illinois at Urbana–Champaign.
 Fall '08
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 Math

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