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