A foliation decomposes a topological manifold into connected leaves which locally look like parallel slices in product coordinate charts.
For a cooriented integrable codimension-one distribution, choose a nonvanishing defining form and an auxiliary form with . Changing the auxiliary form to changes by . Replacing by changes the auxiliary form by , changing the three-form by the exact differential . Stokes' theorem proves invariance on a closed oriented three-manifold. For the weak stable foliation of a curvature-minus-one surface geodesic flow, take , to obtain in the positive base/fibre orientation.
A leaf is a maximal connected integral manifold of the smooth constant-rank integrable distribution defining a regular foliation. It carries its intrinsic manifold topology and is immersed in the ambient manifold. It need not be embedded or closed. Local Frobenius theorem coordinates make each leaf segment a plaque with constant transverse coordinates; global confinement to an immersed leaf follows by joining such plaques along tangent curves.
A codimension-one foliation of a closed three-manifold is taut if every leaf meets a closed transverse curve. In particular, the fibers of a bundle over form a taut foliation.
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Foliation is a term that can refer to different concepts depending on the context, primarily in geology and botany. Here are the key meanings: 1. **Geology**: In geology, foliation refers to the parallel layering that can occur in metamorphic rocks due to the alignment of mineral grains under directional pressure. This structure is typically produced by processes such as metamorphism, where heat and pressure cause the minerals in the rock to recrystallize and realign.