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Hirsch-Smale theory is the name now given to the study of regular homotopy classes of immersions and more generally the space of immersions via their derivative maps. It is one of the spectacular success stories of geometric topology and in particular the h-principle.
Definition 2.1. For a submanifold and a manifold , a pair is called an -immersion if
- is an immersion,
- is a linear map, and
- there exists an open neighborhood of in and an immersion such that and .
Theorem 2.3. Let be a smooth -immersion.If and , then can be extended to an -immersion .
[Hirsch1959], Theorem 3.9.