Differential geometry- Local isometry

:confused: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**