# Differential geometry- Local isometry

• Jan 7th 2010, 07:25 AM
lat87
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)

i have so far done...

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

with #=1
• Jan 8th 2010, 01:22 AM
Laurent
Quote:

Originally Posted by lat87
: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)

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...