Differentiable Manifolds/Lie algebras and the vector field Lie bracket

Diffeomorphisms and related vector fields Integral curves and Lie derivatives Lie algebras 0 100% #FFFFFF Definition 7.1: Let be a -dimensional real vector space. is called a Lie algebra iff it has a function such that for all and the three rules and (bilinearity) (skew-symmetry) (Jacobi's identity) hold. 0 100% #FFFFFF Definition 7.2: Let with be a Lie algebra. A subset of which is a Lie algebra with the restriction of on that subset is called a Lie subalgebra. The vector field Lie bracket 0 100% #FFFFFF Definition 7.3: Let be a manifold of class . We define the vector field Lie bracket, denoted by , as follows: Theorem 6.4: If are vector fields of class on , then is a vector field of class on (i. e. really maps to ) Proof: 1. We show that for each , . Let and . 1.1 We prove linearity: 1.2 We prove the product rule: 2. We show that is differentiable of class . Let be arbitrary. As are vector fields of class , and are contained in . But since are vector fields of class , and are contained in . But the sum of two differentiable functions is again differentiable (this is what theorem 2.? says), and thus is in , and since was arbitrary, is differentiable of class . Theorem 6.5: If is a manifold, and is the vector field Lie bracket, then and form a Lie algebra together. Proof: 1. First we note that as defined in definition 5.? is a vector space (this was covered by exercise 5.?). 2. Second, we prove that for the vector Lie bracket, the three calculation rules of definition 6.1 are satisfied. Let and . 2.1 We prove bilinearity. For all and , we have and hence, since and were arbitrary, Analogously (see exercise 1), it can be proven that 2.2 We prove skew-symmetry. We have for all and : 2.3 We prove Jacobi's identity. We have for all and : , where the last equality follows from the linearity of and . Diffeomorphisms and related vector fields Integral curves and Lie derivatives