A doubt on a problem in Manifolds, Tensor Analysis and Applications - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:13:57Z http://mathoverflow.net/feeds/question/53772 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53772/a-doubt-on-a-problem-in-manifolds-tensor-analysis-and-applications A doubt on a problem in Manifolds, Tensor Analysis and Applications Giuseppe Tortorella 2011-01-30T10:33:13Z 2011-01-30T17:57:14Z <p>Having tried to solve exercise 4.4-7 to have another proof of Frobenius Theorem, I would ask you a question.</p> <p>This is what I have understood: In Step 2 there is to prove, for any tangent subbundle \$E\$ on a manifold \$M,\$ that if the module \$\Gamma(E)\$ of its sections is a Lie subalgebra of \$\Gamma(TM)\$ then it is locally generated by \$k\$ independent commuting vector fields. In Step 1 there is to prove the integrability of \$E\$ when it is locally generated by \$k\$ independent commuting vector fields.</p> <p>Now this is the question: Because, in the step 1, the authors assume additionally that \$\Gamma(E)\$ has to be an abelian Lie subalgebra of \$\Gamma(TM)\$?<br> There is some point that I don't understand? or is it a lapsus calami?</p> <p>Thank you very much for the attention.</p>