Diffeomorphism

A diffeomorphism is a smooth bijection between smooth manifolds whose inverse is also smooth.

For a diffeomorphism, pull a type $(r,s)$ tensor field back by the inverse differential on vector slots and the dual differential on covector slots. Invertibility distinguishes this construction from pullback of purely covariant tensors along an arbitrary smooth map. This operation preserves tensor products and tensor contractions, supplying the flow definition of the Lie derivative of a tensor field.

On a compact smooth manifold without boundary, every smooth self-map sufficiently $C^1$-close to a diffeomorphism is a diffeomorphism. Reducing to a map near the identity, invertibility of its differential follows uniformly by compactness. It is then a proper local diffeomorphism, hence a finite covering map. Sufficient $C^0$ closeness gives a homotopy to the identity on each component; its induced map on the fundamental group is surjective. The covering therefore has one sheet on each component, proving the assertion. This lets a nearby Lagrangian submanifold transverse to cotangent fibers be represented by a single global differential one-form graph.

A large diffeomorphism does not approach the identity at the relevant boundary or cannot be continuously deformed to the identity within the allowed gauge transformations. It may therefore act nontrivially on physical states or boundary data.

A large spatial diffeomorphism changes spatial coordinates by a transformation that remains nonzero at large distance. In cosmology, suitable large spatial diffeomorphisms generate zero-momentum adiabatic modes.