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Unformatted text preview: Math 481
1. Examples of.ManifoIds 1. Examples of manifolds: a) .5'1 or $2 = conﬁguration Space of a pendulum in the plane or 3space.
b) T2 = 81 x 6'1 2 conﬁguration space of at double pendulum in the plane (superposition
allowed). (3) P1 or P2 2 conﬁguration space of a rod with ﬁxed centerpoint [ends not distin
guished) in the plane or 3space. d) Note: Conﬁguration space of a. rod with left endpoint ﬁxed OR right endpoint ﬁxed
is NOT a. manifold:  e) S”={EEE”+I:]:EI=1}
f) T”='Slx...x31(ntimes).
g) P"=S”/{§:'=—f}.  ‘. Question: Can one distinguish between 1"1 Hand SI? Between P2 and 32?
2. Models of manifolds: c) Klein Bottle: 3.  i
Read: Frankel, pp. 1, 1116. j
\ Math 481
2. Deﬁnition of Differentiable Manifold 1. a) TheKlein Bottle, K2. , h) K2 is not homeomorphic to the Space on the right since the self—intersection should
not be there. In fact, K 2 can not be embedded in R3. It can in R4: how? 2. a) In the text, §1.2a, on Topology: Spaces are homeomorphtc ( = topologically equivalent)
if there is a oneto—one map f of M onto N such that f and f ‘1 are continuous. b) In this course, our spaces are manifolds. These are locally homeomorphic to an open
set in R" (see below). 0) Examples of glued spaces that are/are not manifolds? 3. a.) In the text, §1.2 b 85 c, on Deﬁnition of an n—dimensional manifold M : A coordinate
patch is a. map og : U —> IR” (where U C M ) which is a oneto—one map onto an open
set in IR”. b) An atlas is a _collection qu;i'vhere M = UK]. , and any two patches are compatible
(see 4. (3) below) when they overlap. 7 4. 3.) Example: An atlas on 31. Take (U m 31 — (0,1).¢U)T(V * 31  (0, 1), 915v)
Where 463, gtv satisfy _ ¢U(Pl _ SC
2 _ l * y
9151/09) _ 3‘3
2 _ 1+ y
43:2
¢U(P)¢v(P) = 1_ y2 = 4 (see ﬁgure, over). b) Is (U, 9%} a coordinate patch on 81? Similarly, is (V, W)?
c) Do they form an atlas on 31?
o Is U U V == 5'1? _
o Are (150 and qby compatible, i.e. are both ¢U(Un V) and gin/(U n V) open in R? Are
¢U 0 ¢;1 and qsv o 92551 smooth? ' Read: Frankel, pp. 17—20. ...
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 Fall '08
 kapovich
 Math, Topology, Manifold, configuration space, Differentiable manifold, Frankel, oneto—one map

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