leafwise

Leafwise is an English term formed from leaf and -wise. In general usage, it means “along or with respect to a leaf” or “in a way pertaining to leaves.” In mathematics, and especially in differential geometry and foliation theory, leafwise refers to structures or operations that are defined on, or restricted to, the individual leaves of a foliation of a manifold. A foliation partitions the space into submanifolds called leaves; leafwise notions ignore variations transverse to the leaves and focus on tangent directions within each leaf.

Leafwise concepts are used to describe geometric and analytic structures that live on each leaf independently of the transverse directions. Examples include the leafwise tangent bundle, which consists of vectors tangent to the leaves; leafwise differential forms, which are forms restricted to the leaves; and the leafwise exterior derivative d_F, which differentiates along leaves and yields leafwise de Rham cohomology. A leafwise Riemannian metric assigns a metric on each leaf, varying smoothly in the transverse direction. Similar leafwise constructions exist for Laplacians, connections, and flows, where operators act only along leafwise directions. In dynamical systems and ergodic theory, leafwise notions describe properties such as leafwise ergodicity or leafwise measures with respect to the foliation.

Outside mathematics, the term is rarely used, and when used it generally means “acting on a leaf” but is not a standard technical term in botany or plant sciences. The concept emphasizes localization to a leaf rather than the global structure.