Taking g i j = δ δ i j the Kronecker delta, i.e. the scalar dot product in Euclidean space, one gets the O ( n ) non-linear sigma model. That is, write ϕ ϕ = u ^ ^ to be the unit vector in R n , so that u ^ ^ ⋅ ⋅ u ^ ^ = 1 , with ⋅ ⋅ the ordinary Euclidean dot product. Then u ^ ^ ∈ ∈ S n − − 1 the ( n − − 1 ) - sphere, the isometries of which are the rotation group O ( n ) . The Lagrangian can then be written as