| Popis: |
The definition of a manifold given in Chapter 8 assumes that the underlying set M is already known. However, there are situations where we only have some indirect information about the overlap of the domains Ui of the local charts defining our manifold M in terms of the transition functions $$\displaystyle \varphi _i^j = \varphi _{j i}{\colon } \varphi _i(U_i\cap U_j)\rightarrow \varphi _j(U_i\cap U_j), $$ but where M itself is not known. For example, this situation happens when trying to construct a surface approximating a 3D-mesh. If we let Ωij = φi(Ui ∩ Uj) and Ωji = φj(Ui ∩ Uj), then φji can be viewed as a “gluing map” $$\displaystyle \varphi _{j i}{\colon } \Omega _{i j}\rightarrow \Omega _{j i} $$ between two open subsets of Ωi and Ωj, respectively. |
