It is closely related to the concept of Reebless foliation.

This metamorphism causes changes in the mineral composition of the rocks ; creates a foliation, or planar surface, that is related to mineral growth under stress ; and can remove signs of the original textures of the rocks, such as bedding in sedimentary rocks, flow features of lavas, and crystal patterns in crystalline rocks.

The crystalline structure of mica forms layers that can be split or delaminated into thin sheets usually causing foliation in rocks.

* The proof that every Haefliger structure on a manifold can be integrated to a foliation ( this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one ).

The relation between nonlocality and preferred foliation can be better understood as follows.

In this way, the need for a preferred foliation can be avoided and relativistic covariance can be saved.

More generally, if the map is merely a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back.

For example, in the codimension 1 case, we can define the tangent bundle of the foliation as, for some ( non-canonical ) ( i. e. a non-zero co-vector field ).

One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup-but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies.

With use of this function d we can construct two mathematical models, where the second is generated by applying d to proper elements of the first, such that the two models are identical prior to the time t = 0, where t is a time function created by a foliation of spacetime, but differ after t = 0.

With use of this function we can construct two mathematical models, where the second is generated by applying to proper elements of the first, such that the two models are identical prior to the time, where is a time function created by a foliation of spacetime, but differ after.

With the aid of offset markers such as displaced layering and dykes, or the deflection ( bending ) of layering / foliation into a shear zone, one can additionally determine the sense of shear.

Some of the types of information that can only be obtained from bedrock outcrops, or through precise drilling and coring operations, are ; structural geology features orientations ( e. g. bedding planes, fold axes, foliation ), depositional features orientations ( e. g. paleo-current directions, grading, facies changes ), paleomagnetic orientations.

can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is

Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.

# the vorticity tensor represents any tendency of the initial sphere to rotate ; the vorticity vanishes if and only if the world lines in the congruence are everywhere orthogonal to the spatial hypersurfaces in some foliation of the spacetime, in which case, for a suitable coordinate chart, each hyperslice can be considered as a surface of ' constant time '.

In cases where there is a bedding-plane foliation caused by burial metamorphism or diagenesis this may be enumerated as S0a.

For example an F < sub > 2 </ sub > fold, with an S < sub > 2 </ sub > axial plane foliation would be the result of a D < sub > 2 </ sub > deformation.

Stretching lineations may be difficult to quantify, especially in highly stretched ductile rocks where minimal foliation information is preserved.

A-dimensional foliation of an-manifold may be thought of as simply a collection of pairwise-disjoint, connected, immersed-dimensional submanifolds ( the leaves of the foliation ) of, such that for every point in, there is a chart with homeomorphic to containing such that for every leaf, meets in either the empty set or a countable collection of subspaces whose images under in are-dimensional affine subspaces whose first coordinates are constant.

In particular, if and is a homeomorphism of F, then the suspension foliation of is defined to be the suspension foliation of the representation given by.

This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution ( i. e. an dimensional subbundle of the tangent bundle of a manifold ) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under Lie bracket.

In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields ( satisfying an integrability condition ) in much the same way as an integral curve may be assigned to a single vector field.

In this context, the Frobenius theorem relates integrability to foliation ; to state the theorem, both concepts must be clearly defined.

A subbundle may also be defined to arise from a foliation of a manifold.

Let be a submanifold that is a leaf of a foliation.

A Pfaffian system is said to be completely integrable if N admits a foliation by maximal integral manifolds.

( Note that the foliation need not be regular ; i. e. the leaves of the foliation might not be embedded submanifolds.

In mathematics, a taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf.

Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface.

A taut foliation cannot have a Reeb component, since the component would act like a " dead-end " from which a transverse curve could never escape ; consequently, the boundary torus of the Reeb component has no transverse circle puncturing it.

The existence of a taut foliation implies various useful properties about a closed 3-manifold.

For example, a closed, orientable 3-manifold, which admits a taut foliation with no sphere leaf, must be irreducible, covered by, and have negatively curved fundamental group.

A flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are

In the vector field formulation, the theorem states that a subbundle of the tangent bundle of a manifold is integrable ( or involutive ) if and only if it arises from a regular foliation.

If TN is exactly E restricted to N, then one says that E arises from a regular foliation of M. Again, this definition is purely local: the foliation is defined only on charts.

Given the above definitions, Frobenius ' theorem states that a subbundle E is integrable if and only if it arises from a regular foliation of M.

The layering within metamorphic rocks is called foliation ( derived from the Latin word folia, meaning " leaves "), and it occurs when a rock is being shortened along one axis during recrystallization.

These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.

The notion of leaves allows for a more intuitive way of thinking about a foliation.

* A foliation of a manifold is a decomposition of the manifold into a union of submanifolds called leaves

( Note that the leaves of the associated foliation are necessarily space-like hypersurfaces.

* When g is pseudo-Anosov, there is an element of its mapping class that preserves a pair of transverse singular foliations of S, stretching the leaves of one ( the stable foliation ) while contracting the leaves of the other ( the unstable foliation ).

In mathematics, a Lagrangian foliation or polarization is a foliation of a symplectic manifold, whose leaves are Lagrangian submanifolds.

There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation.

The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is

leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called " angle " variables.

Essentially, these distinctions correspond to the dimensions of the leaves of the foliation.