The discussion centers on the application of Frobenius's theorem to smooth frame fields on n-dimensional smooth manifolds. It establishes that for a smooth frame field defined by n linearly independent smooth vector fields, the Lie bracket [X,Y] of any two frame field vectors X and Y lies in the span of those vectors at each point, confirming that the Frobenius condition for complete integrability is satisfied. The conversation also explores the relationship between Frobenius's theorem and the Straightening theorem, emphasizing that while the former provides conditions for integrability of subbundles, the latter relates to defining local coordinates along flows of vector fields.
PREREQUISITESThis discussion is beneficial for mathematicians, particularly those specializing in differential geometry, as well as students and researchers interested in the applications of Frobenius's theorem and its connection to other geometric theorems.
It is of course a Lie group. It's a two-dimensional smooth manifold. A typical element looks likecianfa72 said:Sorry, but the above Borel subgroup of ##\operatorname{SL}(2,\mathbb R)## is not itself a Lie group ?