Sard's theorem provides such an example. Given a random smooth map between two manifolds (lets say compact and of the same dimension), there is no canonical way of constructing a regular value. But, Sard's theorem looks at this entire set and proves it is of full measure (and hence there exists atleast one regular value). This result enables us to prove many such results like (the transversality theorem for instance).
|
1 | [made Community Wiki] | ||
|
|
||||
