ABSTRACT
PR8 The group of isometries of an n-dimensional Euclidean space is isomorphic to affine group Rn x O(n;R).
Information. The space En is defined by one chart with the previous metric element. An isometry f defined by n differentiable functions
I f'( 1 n)y = x ,....x is such that
araj'g (Xl) =__ g,..(yP(Xl )) /I aX' axf
The space being Euclidean, we have: OJ' Of'
0/1 = L ax '
which means that ( ~~) is an orthogonal matrix. Next by writing the diffeomorphisms of En the reader will be able to conclude.
