Why Does an Isometry Preserve Lie Brackets Between Vector Fields?

[rafax]

I have an isometry between two manifolds M and N. Can someone help me why the following is true

DF([X,Y])=[DF(X),DF(Y)], where X,Y are vector fields on M.

Regards

 

[Fredrik]

Is the isometry called F or DF? Do you mean dF? How is your D defined?