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WannabeNewton

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WannabeNewton

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Fredrik

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It seems to me that you have understood this correctly, so I think the only thing you have overlooked is that you're supposed to conclude that k≥m and k≤m together imply that k=m. (If

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WannabeNewton

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Hey Fredrik, thanks for the reply. I'm a little more sure now. Going back to the example on [itex]S^{2}[/itex], we have the three killing vector fields [itex]R = \partial _{\phi }, S = cos\phi \partial _{\theta} - cot\theta sin\phi \partial _{\phi }, T = -sin\phi \partial _{\theta} - cot\theta cos\phi \partial _{\phi }[/itex] and these three vector fields satisfy [itex]L _{V(a)}V_{(b)} = [V_{(a)}, V_{(b)}] = \alpha ^{c}V_{(c)}[/itex] where c runs through all k. Here we have k = 3, and a submanifold of dimension 2 but [itex]S^{2}[/itex] is still an integral submanifold of the set. I used a different text as reference this time and it made it clearer: we choose a set of k vector fields from the original manifold M such that the k vector fields span a subbundle of TM where the subbundle is of dimension m <= k. We don't assume they are all linearly independent here so it is ok if the number of vector fields in the set is greater than the dimension of the subbundle. According to the text, at each point on the submanifold we pick k = m vector fields from the set that are linearly independent so that they form a basis for each tangent space at each point and all the members of the set can be expressed as a linear combination of them so that the members of the set form a lie algebra. I guess the other text forgot to mention that. Thanks Fredrik.

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