diffeomorfisme

A diffeomorphism, sometimes written diffeomorfism in some languages, is a central concept in differential geometry. It is a smooth bijection f: M → N between smooth manifolds whose inverse f^{-1}: N → M is also smooth. When M = N, f is a diffeomorphism from the manifold to itself and is called a smooth automorphism.

Definition and properties: A map f is a diffeomorphism if it is bijective, smooth (infinitely differentiable), and its inverse is smooth. Equivalently, f is a smooth map with a smooth inverse. The derivative df_p at each point p ∈ M is a linear isomorphism between the tangent spaces, which means diffeomorphisms preserve the local differentiable structure. Diffeomorphisms are homeomorphisms that preserve orientation in the appropriate sense and form the group Diff(M) under composition.

Examples: On the real line, any smooth bijection with a nowhere-vanishing derivative is a diffeomorphism; for instance, f(x) = x^3 + x has f'(x) = 3x^2 + 1 > 0 for all x, so it is a diffeomorphism R → R. Translations and linear invertible maps with smooth inverses are diffeomorphisms of R^n. Coordinate changes between charts are local diffeomorphisms, and when pieced together into a global invertible smooth map, they yield global diffeomorphisms.

Group and applications: The set Diff(M) of all diffeomorphisms of a manifold M forms an (infinite-dimensional) Lie group under composition in suitable topologies. Diffeomorphisms are the preferred notion of smooth equivalence in geometry, and they play a key role in dynamics, topology, geometry, and physics, where they act as smooth coordinate changes and gauge symmetries.