Differentiable Manifolds/Diffeomorphisms and related vector fields

Vector fields, covector fields, the tensor algebra and tensor fields Lie algebras and the vector field Lie bracket Diffeomorphisms We shall now define the notions of homeomorphisms and diffeomorphisms for mappings between manifolds. 0 100% #FFFFFF Definition 6.1: Let be a manifold of class , where as usual , and let be a manifold of class , where also . A function is called a homeomorphism iff it is bijective both itself and are continuous 0 100% #FFFFFF Definition 6.2: Let be a manifold of class , where , and let be a manifold of class , where also . Let . A function is called a diffeomorphism of class iff it is bijective both itself and are differentiable of class The rank of the differential 0 100% #FFFFFF Definition 6.3: Let , be a -dimensional manifold of class , be a -dimensional manifold of class , be open, and differentiable of class . The rank of is defined as . The dimension of is well-defined since is a linear function, which is why it's image is a vector space; further, it is a vector subspace of , which is a -dimensional vector space, which is why it has finite dimension. 0 100% #FFFFFF Theorem 6.4: Let be a -dimensional manifold of class and be a -dimensional manifold of class . Related vector fields 0 100% #FFFFFF Definition 6.3: Let be a manifold of class , where , let be a manifold of class , where also , and let be differentiable of class , where also . We call and -related iff 0 100% #FFFFFF Theorem 6.4: Let be a manifold of class , where , let be a manifold of class , where also , let be a diffeomorphism of class , where also , and let . Then is the unique vector field such that and are -related. Proof: 1. We show that and are -related. Let be arbitrary. Then we have: 2. We show that there are no other vector fields besides which are -related to . Let be also contained in such that and are -related. We show that , thereby excluding the possibility of a different to -related vector field. Indeed, for every we have: Due to the bijectivity of , there exists a unique such that , and we have . Therefore, and since was required to be -related to : 0 100% #FFFFFF Theorem 6.5: Let be a manifold of class , where , let be a manifold of class , where also , let be a diffeomorphism of class , where also . If is differentiable of class where and , then the unique -related vector field is differentiable of class . Proof: Let . Inserting a few definitions from chapter 2, we obtain , and therefore Since is differentiable of class and , is also differentiable of class . Further, the function is differentiable of class , because is differentiable of class . Due to lemma 2.17, it follows that also is differentiable of class , and therefore, due to the definition of differentiability of vector fields, so is . Vector fields, covector fields, the tensor algebra and tensor fields Lie algebras and the vector field Lie bracket