diffeomorphisms

A diffeomorphism is a central concept in differential geometry. Given smooth manifolds M and N, a diffeomorphism f: M → N is a bijection that is smooth and whose inverse f⁻¹: N → M is also smooth. Diffeomorphisms are the isomorphisms in the category of smooth manifolds, preserving the differentiable structure rather than just the topological one.

Equivalently, a diffeomorphism is a smooth bijection with a smooth inverse; in particular, in local coordinates the Jacobian matrix has nonzero determinant at every point, ensuring local invertibility by the inverse function theorem. When M = N, the diffeomorphisms form the diffeomorphism group Diff(M) under composition, with the identity map and inverses included.

Properties and variations: Diffeomorphisms are automatically homeomorphisms, and they are local diffeomorphisms with global invertibility. If M is oriented, a diffeomorphism may be orientation-preserving or orientation-reversing, depending on the sign of the determinant of its derivative. For manifolds with boundary, a diffeomorphism maps the boundary to itself and remains smooth with a smooth inverse.

Examples: Translations and rotations of Euclidean space R^n are diffeomorphisms. On the circle S^1, smooth bijections with smooth inverses are circle diffeomorphisms. A smooth bijection like f(x) = x^3 on R is not a diffeomorphism, since its inverse x^(1/3) fails to be smooth at 0.

Remarks: Diff(M) has rich structure in geometry and topology and can be studied as an infinite-dimensional group in suitable settings. Diffeomorphisms preserve the differentiable structure and are distinct from merely continuous or topological symmetries.