# \(h\)-Principles and Flexibility in Geometry

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*Hansjörg Geiges*

The notion of homotopy principle or \(h\)-principle is
one of the key concepts in an elegant language developed by Gromov to deal with
a host of questions in geometry and topology. Roughly speaking, for a certain
differential geometric problem to satisfy the \(h\)-principle is
equivalent to saying that a solution to the problem exists whenever certain
obvious topological obstructions vanish.

The foundational examples for applications of Gromov's ideas
include

- (i) Hirsch-Smale immersion theory,
- (ii) Nash-Kuiper \(C^1\)–isometric immersion theory,
- (iii) existence of symplectic and contact structures on open manifolds.

Gromov has developed several powerful methods that allow one to prove \(h\)-principles. These notes, based on lectures given in the Graduiertenkolleg of Leipzig University, present two such methods which are strong enough to deal with applications (i) and (iii).