quot;atlas of class C^p (p &gt;= 0)&quot; on X is a collection
| of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the
| following conditions:
| 
| AT 1.  Each U_i is a subset of X and the U_i cover X.
| 
| AT 2.  Each q_i is a bijection of U_i onto an open subset q_i U_i
|        of some Banach space E_i and for any i, j, [it holds that]
|        q_i (U_i |^| U_j) is open in E_i.
|
| AT 3.  The map
|
|        q_j o q_i^-1  :  q_i (U_i |^| U_j)  --&gt;  q_j (U_i |^| U_j)
|
|        is a C^p-i