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Math Help - Differential geometry- Local isometry

  1. #1
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    Differential geometry- Local isometry

    i have the defintion of a local isometry that


    i have used [ as a mod sign.

    [dfp(v)]=[v]

    Question;

    Show that a smooth map f: M ----> M' is a local isometry if and only if

    dfp(v) . dfp(w) =v . w (p E M, v,w E TpM)


    Hence show that a smooth map f : M----->M' is a local isometry if and only if, for all p E M, there exists a basis { ei } of TpM such that dfp (ei) . dfp(ej) = ei . ej (i,j = 1,...m)




    Please HELPPPPP!


    i have so far done...

    [dfp (v) . dfp(w) ] = # (p) [v] . #(p) [w]=
    # squared (p) [v] .[w]

    with #=1
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  2. #2
    MHF Contributor

    Joined
    Aug 2008
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    Paris, France
    Posts
    1,174
    Quote Originally Posted by lat87 View Post
    i have the defintion of a local isometry that


    i have used [ as a mod sign.

    [dfp(v)]=[v]

    Question;

    Show that a smooth map f: M ----> M' is a local isometry if and only if

    dfp(v) . dfp(w) =v . w (p E M, v,w E TpM)


    Hence show that a smooth map f : M----->M' is a local isometry if and only if, for all p E M, there exists a basis { ei } of TpM such that dfp (ei) . dfp(ej) = ei . ej (i,j = 1,...m)




    Please HELPPPPP!


    i have so far done...

    [dfp (v) . dfp(w) ] = # (p) [v] . #(p) [w]=
    # squared (p) [v] .[w]

    with #=1
    (I don't understand your notation with the #...)

    For the first question, note that \|df_p(v)+df_p(w)\|^2=\|df_p(v+w)\|^2=\|v+w\|^2 and expand these squares...
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