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I Chart coordinate maps of topological manifolds

  1. Mar 1, 2017 #1
    Hello every one .
    first of all consider the 2-dim. topological manifold case
    My Question : is there any difference between
    $$f \times g : R \times R \to R \times R$$
    $$(x,y) \to (f(x),g(y))$$
    and $$F : R^2 \to R^2$$
    $$(x,y) \to (f(x,y),g(x,y))$$
    Consider two topological manifolds the torus ##T^2## and the euclidean plane ##R^2## such that both of them for any point (p) that lies on each manifold can be represented by an ordered pair ##p=(a,b)##
    Since ##R^2 = R \times R##
    and ##T^2 = S^1 \times S^1##
    and consider chart maps ##X(p)## for the two manifolds separately such that
    ##X : U \to D##
    for ##M=R^2 , U \subset R^2 and D \subset R^2## same for ##T^2##
    for the case of ##R^2##
    $$M=R^2={ (a,b) \in R^2 }$$
    choose ##U \subset R^2## and ##D \subset R^2##
    and say that $$X : U \to D$$
    $$(a,b) \to (f(a,b),g(a,b))$$
    but for the torus ##T^2## we consider the notion of PRODUCT MANIFOLD
    $$M=T^2={(a,b) \in T^2 : T^2 = S^1 \times S^1}$$
    choose ##U \subset S^1 , D \subset R##
    such that $$X : U_1 \to D$$
    $$a \to X(a)$$ same for the other circle with ##Y## chart map
    now consider the Cartesian product of the two chart maps $$X \times Y$$
    $$X \times Y : U_1 \times U_2 \to D \times D$$
    $$(a,b) \to (f(a),g(b))$$
    for the manifold being the euclidean plane ##(R^2)##we used the chart map such that each coordinate is a function of the point which is function of two variables ##(a,b)## and for the manifold being the torus ##(T^2)## we used the chart map such that each coordinate is function of each point which is function of single variable .
    The question is , can we use the method or the notion of PRODUCT MANIFOLD for the Euclidean Plane (Since ##R^2 = R \times R## ) same as it was used for the Torus ##T^2## ?
    Example for some chart maps for both the manifolds
    ##M=R^2##
    $$(a,b) \to (f(a,b),g(a,b))$$
    $$(a,b) \to (\sqrt {a^2 + b^2} , \tan^{-1}(\frac a b))$$
    for ##U=\{(a.b) \in R^2 : a \gt 0 \}## and ##D=\{(f(a,b),g(a,b)) \in R^2 :f(a,b) \gt 0 ,0 \gt g(a,b) \lt \pi \}##
    and for ##M=T^2## if we consider the configuration space of the double pendulum being the Torus ##T^2## such that $$(a,b) \mapsto (f(a),g(b))$$
    $$(a,b) \mapsto (\sin(a) , \sin(b))$$
    for ##U=\{(a,b) \in T^2 : 0 \gt a \lt \frac{\pi }{2} , 0 \gt b \lt \frac{\pi }{2} \}##



    Thanks.

    <Moderation note: fixed some minor LaTex errors to improve readability>
     
    Last edited by a moderator: Mar 3, 2017
  2. jcsd
  3. Mar 6, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Mar 6, 2017 #3
    It's ok , are there any replies for this post ?
     
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