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I'm reading the book Geometrical methods of mathematial physics by Brian Schutz. In chapter 3, on Lie groups, he states and proves that the vector fields on a manifold over which a particular tensor is invariant (i.e. has 0 Lie derivative over) form a Lie algebra. And associated with every Lie algebra is a Lie group.

Does this have any implications on the manifold somehow "containing" a Lie group in it (as some sort of submanifold maybe?), since the exponentiation of the Lie algebra gives you the identity component of the Lie group it's associated with? This seems not right since, for example, the Lie group of symmetries of a manifold may be of higher dimension than the manifold itself (e.g. Minkowski spacetime has dimension 4, while its group of symmetries, the Poincare group is of dimension 10).

Nevertheless, it seems that there has to be some sort of association going on? Or am I just taking crazy pills?

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# Manifolds with symmetries

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