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h1-2 - Math 481 1 Examples of.ManifoIds 1 Examples of...

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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 3-space. 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 3-space. 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, 11-16. j \ Math 481 2. Definition of Differentiable Manifold 1. a) The-Klein 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 one-to—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 one-to—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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h1-2 - Math 481 1 Examples of.ManifoIds 1 Examples of...

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