Lie bracket of tangent vectors on embedded vs induced metric

[ruwn]

Im sorry to bother you, but I am trying to understand one thing about embedding. Consider you have sphere embedded in the R^3, so you have a flat metrik. Otherwise you could describe the same sphere without embedding but with an induced metric.

My problem is to make clear that the Lie-Brackets of two tangentvectors in R^3 on the sphere are equal to the equivalent tangentvectors according to the induced metric.

( [X,Y]=[X',Y'] with g(X,Y)=g_induced(X',Y'))

thanks

by the way i think intuitionally it works...

 

[StatusX]

The Lie bracket is defined independently of the metric, so this should work. The only thing you need to check is that the Lie bracket of two vector fields tangent to the sphere is another vector field tangent to the sphere, but this follows for any submanifold from Frobenius' integrability condition.