Foliationlike

Foliationlike is a term used informally in differential geometry and topology to describe structures that resemble a foliation but do not meet all of the formal requirements of a foliation. In the standard notion, a foliation on a smooth manifold M partitions M into disjoint connected immersed submanifolds called leaves, of fixed dimension, arranged so that locally M looks like a product of a leaf with a transverse space. A foliationlike structure may share some of these features but may also include singularities, varying leaf dimensions, or only partial local regularity.

In formal terms, a genuine foliation requires a constant leaf dimension and a locally coherent transverse structure. A foliationlike structure, by contrast, may arise from partitions induced by group actions, level sets of functions with critical points, or distributions that are not of uniform rank. Singular foliations provide a well-developed framework for such phenomena: leaves can have different dimensions, and the structure is described by a distribution that may be non-constant in rank but is integrable in a generalized sense. One common approach for singular foliations is the Stefan–Sussmann formulation, where a sheaf of vector fields closed under the Lie bracket generates the integral leaves.

Examples of foliationlike situations include orbit decompositions of smooth group actions with varying stabilizers and stratifications arising in singular spaces. These structures often retain local product-like behavior away from singularities but admit a more intricate global topology than regular foliations.

See also Foliation, singular foliation, partition of a manifold, Lie group action, and orbit decomposition.