Hey guys, this question is more or less related to the way Frobenius' theorem is presented in my text. Consider an n - manifold M, an m - dimensional submanifold S of M, and a set of k linearly independent vector fields [itex]V^{\mu }_{(a)}[/itex] such that [itex]k \geq m[/itex]. In order for S to be an integral submanifold of the [itex]V^{\mu }_{(a)}[/itex]'s, each vector field in [itex]V^{\mu }_{(a)}[/itex] must be tangent to S everywhere. The text states that, if this condition is met, then at each [itex]p\in S[/itex], each [itex]V^{\mu }_{(a)}(p)[/itex] will be an element of the tangent space [itex]T_{p}(S)[/itex] and since they are linearly independent, they will span the tangent space. The conclusion then follows that since each vector field is tangent to S everywhere, the vector fields themselves will span each tangent space so as to span the tangent bundle to S. I can agree with this if k = m but if k > m I don't see how the vector fields can be linearly independent so as to span the space. Isn't the linearly independent set maximal when it spans a space? If the dimension of each tangent space is m, how can the vector fields span each tangent space when there are k > m linearly independent vector fields? The argument only makes sense to me when k = m. I am not disputing the claims any way by the way, in fact the condition in Frobenius' theorem that in order for a submanifold to be an integral submanifold of a set of vector fields the vector fields must form a lie algebra is easily seen by the set of three killing vector fields, on [itex]S^{2}[/itex], which [itex]S^{2}[/itex] is an integral submanifold of. I am just asking for a clarification because I am obviously overlooking something. Thanks all and sorry in advance if this is the wrong section.(adsbygoogle = window.adsbygoogle || []).push({});

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# Span of a vector space question

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