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Foliation

In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation). The number p (the dimension of the leaves) is called the dimension of the foliation and q = n − p is called its codimension.

In some papers on general relativity by mathematical physicists, the term foliation (or slicing) is used to describe a situation where  the relevant Lorentz manifold (a (p+1)-dimensional spacetime) has been decomposed into hypersurfaces of dimension p, specified as the level sets of a real-valued smooth function (scalar field) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time-like, so that  its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these  hypersurface are often  called the leaves (or sometimes slices) of the foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets  of a function, they  generally cannot be expressed this way  globally, as a leaf may pass through  a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the 3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.

Foliated charts and atlases
In order to give a more precise definition of foliation, it is necessary to define some auxiliary elements. A rectangular neighborhood in Rn is an open subset of the form B = J1 × ⋅⋅⋅ × Jn, where Ji is a (possibly unbounded) relatively open interval in the ith coordinate axis. If J1 is of the form (a,0], it is said that B has boundary
 * $$\partial B = \left \{ \left(0,x^2, \ldots, x^n \right ) \in B \right \}.$$

In the following definition, coordinate charts are considered that have values in Rp × Rq, allowing the possibility of manifolds with boundary and (convex) corners.

A foliated chart on the n-manifold M of codimension q is a pair (U,φ), where U ⊆ M is open and $$\varphi: U \to B_{\tau} \times B_{\pitchfork}$$ is a diffeomorphism, $$B_{\pitchfork}$$ being a rectangular neighborhood in Rq and $$B_{\tau}$$ a rectangular neighborhood in Rp. The set Py = φ−1(Bτ × {y}), where $$y \in B_{\pitchfork}$$, is called a plaque of this foliated chart. For each x ∈ Bτ, the set Sx = φ−1({x} × $$B_{\pitchfork}$$) is called a transversal of the foliated chart. The set ∂τU = φ−1(Bτ × (∂$$B_{\pitchfork}$$)) is called the tangential boundary of U and $$\partial_{\pitchfork}U$$ = φ−1((∂Bτ) × $$B_{\pitchfork}$$) is called the transverse boundary of U.

The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation Bτ is read as "B-tangential" and $$B_{\pitchfork}$$ as "B-transverse". There are also various possibilities. If both $$B_{\pitchfork}$$ and Bτ have empty boundary, the foliated chart models codimension-q foliations of n-manifolds without boundary. If one, but not both of these rectangular neighborhoods has boundary, the foliated chart models the various possibilities for foliations of n-manifolds with boundary and without corners. Specifically, if ∂$$B_{\pitchfork}$$ ≠ ∅ = ∂Bτ, then ∂U = ∂τU is a union of plaques and the foliation by plaques is tangent to the boundary. If ∂Bτ ≠ ∅ = ∂$$B_{\pitchfork}$$, then ∂U = $$\partial_{\pitchfork}U$$ is a union of transversals and the foliation is transverse to the boundary. Finally, if ∂$$B_{\pitchfork}$$ ≠ ∅ ≠ ∂Bτ, this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.

A foliated atlas of codimension q and class Cr (0 ≤ r ≤ ∞) on the n-manifold M is a Cr-atlas $$\mathcal{U} = \{(U_{\alpha}, \varphi_{\alpha})\mid \alpha \in A\}$$ of foliated charts of codimension q which are coherently foliated in the sense that, whenever P and Q are plaques in distinct charts of $$\mathcal{U}$$, then P ∩ Q is open both in P and Q.

A useful way to reformulate the notion of coherently foliated charts is to write for w ∈ Uα ∩ Uβ
 * $$\varphi_{\alpha} (w) = \left ( x_{\alpha} (w), y_{\alpha} (w) \right ) \in B_{\tau}^{\alpha} \times B_{\pitchfork}^{\alpha},$$
 * $$\varphi_{\beta} (w) = \left ( x_{\beta} (w), y_{\beta} (w) \right ) \in B_{\tau}^{\beta} \times B_{\pitchfork}^{\beta}.$$

The notation (Uα,φα) is often written (Uα,xα,yα), with
 * $$x_{\alpha} = \left (x_{\alpha}^1, \dots,x_{\alpha}^p \right ), $$
 * $$y_{\alpha} = \left (y_{\alpha}^1, \dots,y_{\alpha}^q \right ). $$

On φβ(Uα ∩ Uβ) the coordinates formula can be changed as
 * $$g_{\alpha \beta} \left ( x_{\beta},y_{\beta} \right ) = \varphi_{\alpha} \circ \varphi_{\beta}^{-1} \left ( x_{\beta}, y_{\beta} \right ) = \left ( x_{\alpha} \left ( x_{\beta}, y_{\beta} \right ), y_{\alpha} \left ( x_{\beta}, y_{\beta} \right ) \right ).$$

The condition that (Uα,xα,yα) and (Uβ,xβ,yβ) be coherently foliated means that, if P ⊂ Uα is a plaque, the connected components of P ∩ Uβ lie in (possibly distinct) plaques of Uβ. Equivalently, since the plaques of Uα and Uβ are level sets of the transverse coordinates yα and yβ, respectively, each point z ∈ Uα ∩ Uβ has a neighborhood in which the formula
 * $$y_\alpha = y_\alpha(x_\beta, y_\beta) = y_\alpha(y_\beta)$$

is independent of xβ.

The main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation. For this and other purposes, the general definition of foliated atlas above is a bit clumsy. One problem is that a plaque of (Uα,φα) can meet multiple plaques of (Uβ,φβ). It can even happen that a plaque of one chart meets infinitely many plaques of another chart. However, no generality is lost in assuming the situation to be much more regular as shown below.

Two foliated atlases $$\mathcal{U}$$ and $$\mathcal{V}$$ on M of the same codimension and smoothness class Cr are coherent $$\left ( \mathcal{U} \thickapprox \mathcal{V} \right )$$ if $$\mathcal{U} \cup \mathcal{V}$$ is a foliated Cr-atlas. Coherence of foliated atlases is an equivalence relation.
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!Proof Reflexivity and symmetry are immediate. To prove transitivity let $$\mathcal{U} \thickapprox \mathcal{V}$$ and $$\mathcal{V} \thickapprox \mathcal{W}$$. Let (Uα,xα,yα) ∈ $$\mathcal{U}$$ and (Wλ,xλ,yλ) ∈ $$\mathcal{W}$$ and suppose that there is a point w ∈ Uα ∩ Wλ. Choose (Vδ,xδ,yδ) ∈ $$\mathcal{V}$$ such that w ∈ Vδ. By the above remarks, there is a neighborhood N of w in Uα ∩ Vδ ∩ Wλ such that
 * $$y_\delta = y_\delta( y_\lambda) \quad \text{on} \quad \varphi_\lambda ( N ),$$
 * $$y_\alpha = y_\alpha(y_\delta) \quad \text{on} \quad \varphi_\delta ( N ),$$

and hence
 * $$y_{\alpha} = y_\alpha \left ( y_\delta( y_\lambda) \right ) \quad \text{on} \quad \varphi_\delta ( N ).$$

Since w ∈ Uα ∩ Wλ is arbitrary, it can be concluded that yα(xλ,yλ) is locally independent of xλ. It is thus proven that $$\mathcal{U} \thickapprox \mathcal{W}$$, hence that coherence is transitive. Plaques and transversals defined above on open sets are also open. But one can speak also of closed plaques and transversals. Namely, if (U,φ) and (W,ψ) are foliated charts such that $$\overline{U}$$ (the closure of U) is a subset of W and φ = ψ|U then, if $$\varphi(U) = B_{\tau} \times B_{\pitchfork},$$ it can be seen that $$\psi|\overline{U}$$, written $$\overline{\varphi}$$, carries $$\overline{U}$$ diffeomorphically onto $$\overline{B}_{\tau} \times \overline{B}_{\pitchfork}.$$
 * }

A foliated atlas is said to be regular if
 * 1) for each α ∈ A, $$\overline{U}_{\alpha}$$ is a compact subset of a foliated chart (Wα,ψα) and φα = ψα|Uα;
 * 2) the cover {Uα | α ∈ A} is locally finite;
 * 3) if (Uα,φα) and (Uβ,φβ) are elements of the foliated atlas, then the interior of each closed plaque P ⊂ $$\overline{U}_{\alpha}$$ meets at most one plaque in $$\overline{U}_{\beta}.$$

By property (1), the coordinates xα and yα extend to coordinates $$\overline{x}_{\alpha}$$ and $$\overline{y}_{\alpha}$$ on $$\overline{U}_{\alpha}$$ and one writes $$\overline{\varphi}_{\alpha} = \left (\overline{x}_{\alpha},\overline{y}_{\alpha} \right ).$$ Property (3) is equivalent to requiring that, if Uα ∩ Uβ ≠ ∅, the transverse coordinate changes $$\overline{y}_{\alpha} = \overline{y}_{\alpha} \left ( \overline{x}_{\beta}, \overline{y}_{\beta} \right )$$ be independent of $$\overline{x}_{\beta}.$$ That is
 * $$\overline{g}_{\alpha \beta} = \overline{\varphi}_{\alpha} \circ \overline{\varphi}_{\beta}^{-1} : \overline{\varphi}_{\beta} \left ( \overline{U}_{\alpha} \cap \overline{U}_{\beta} \right ) \rightarrow \overline{\varphi}_{\alpha} \left ( \overline{U}_{\alpha} \cap \overline{U}_{\beta} \right )$$

has the formula
 * $$\overline{g}_{\alpha \beta} \left ( \overline{x}_{\beta}, \overline{y}_{\beta} \right ) = \left ( \overline{x}_{\alpha} \left ( \overline{x}_{\beta}, \overline{y}_{\beta} \right ), \overline{y}_{\alpha} \left ( \overline{y}_{\beta} \right ) \right ).$$

Similar assertions hold also for open charts (without the overlines). The transverse coordinate map yα can be viewed as a submersion
 * $$y_{\alpha} : U_{\alpha} \rightarrow \mathbb{R}^q$$

and the formulas yα = yα(yβ) can be viewed as diffeomorphisms
 * $$\gamma_{\alpha \beta} : y_{\beta} \left ( U_{\alpha} \cap U_{\beta} \right ) \rightarrow y_{\alpha} \left ( U_{\alpha} \cap U_{\beta} \right ).$$

These satisfy the cocycle conditions. That is, on yδ(Uα ∩ Uβ ∩ Uδ),
 * $$\gamma_{\alpha \delta} = \gamma_{\alpha \beta} \circ \gamma_{\beta \delta}$$

and, in particular,
 * $$\gamma_{\alpha \alpha} \equiv y_{\alpha} \left ( U_{\alpha} \right ),$$
 * $$\gamma_{\alpha \beta} = \gamma_{\beta \alpha}^{-1}.$$

Using the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent refinement that is regular.
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!Proof Fix a metric on M and a foliated atlas $$\mathcal{W}.$$ Passing to a subcover, if necessary, one can assume that $$\mathcal{W} = \left \{ W_j,\psi_j \right \}_{j=1}^l$$ is finite. Let ε > 0 be a Lebesgue number for $$\mathcal{W}.$$ That is, any subset X ⊆ M of diameter < ε lies entirely in some Wj. For each x ∈ M, choose j such that x ∈ Wj and choose a foliated chart (Ux, φx) such that
 * x ∈ Ux ⊆ $$\overline{U}_x$$ ⊂ Wj,
 * φx = ψj|Ux,
 * diam(Ux) < ε/2.

Suppose that Ux ⊂ Wk, k ≠ j, and write ψk = (xk,yk) as usual, where yk : Wk → Rq is the transverse coordinate map. This is a submersion having the plaques in Wk as level sets. Thus, yk restricts to a submersion yk : Ux → Rq.

This is locally constant in xj; so choosing Ux smaller, if necessary, one can assume that yk|$$\overline{U}_x$$ has the plaques of $$\overline{U}_x$$ as its level sets. That is, each plaque of Wk meets (hence contains) at most one (compact) plaque of $$\overline{U}_x$$. Since 1 < k < l < ∞, one can choose Ux so that, whenever Ux ⊂ Wk, distinct plaques of $$\overline{U}_x$$ lie in distinct plaques of Wk. Pass to a finite subatlas $$\mathcal{U} = \left \{U_i,\varphi_i \right \}_{i=1}^N$$ of {(Ux,φx) | x ∈ M}. If Ui ∩ Uj ≠ 0, then diam(Ui ∪ Uj) < ε, and so there is an index k such that $$\overline{U}_i \cup \overline{U}_j \subseteq W_k.$$ Distinct plaques of $$\overline{U}_i$$ (respectively, of $$\overline{U}_j$$) lie in distinct plaques of Wk. Hence each plaque of $$\overline{U}_i$$ has interior meeting at most one plaque of $$\overline{U}_j$$ and vice versa. By construction, $$\mathcal{U}$$ is a coherent refinement of $$\mathcal{W}$$ and is a regular foliated atlas.

If M is not compact, local compactness and second countability allows one to choose a sequence $$\left \{ K_i \right \}_{i=0}^{\infty}$$ of compact subsets such that Ki ⊂ int Ki+1 for each i ≥ 0 and $$M = \bigcup_{i=1}^{\infty} K_i.$$ Passing to a subatlas, it is assumed that $$\mathcal{W} = \left \{ W_j, \psi_j \right \}_{j=0}^{\infty}$$ is countable and a strictly increasing sequence $$ \left \{ n_l \right \}_{l=0}^{\infty}$$ of positive integers can be found such that $$\mathcal{W}_l = \left \{ W_j, \psi_j \right \}_{j=0}^{n_l}$$ covers Kl. Let δl denote the distance from Kl to ∂Kl+1 and choose εl > 0 so small that εl < min{δl/2,εl-1} for l ≥ 1, ε0 < δ0/2, and εl is a Lebesgue number for $$\mathcal{W}_l$$ (as an open cover of Kl) and for $$\mathcal{W}_{l+1}$$ (as an open cover of Kl+1). More precisely, if X ⊂ M meets Kl (respectively, Kl+1) and diam X < εl, then X lies in some element of $$\mathcal{W}_l$$ (respectively, $$\mathcal{W}_{l+1}$$). For each x ∈ Kl $$\diagdown$$ int Kl-1, construct (Ux,φx) as for the compact case, requiring that $$\overline{U}_x$$ be a compact subset of Wj and that φx = ψj|Ux, some j ≤ nl. Also, require that diam$$\overline{U}_x$$ < εl/2. As before, pass to a finite subcover $$\left \{U_i,\varphi_i \right \}_{i=n_{l-1} + 1}^{n_l}$$ of Kl $$\diagdown$$ int Kl-1. (Here, it is taken n−1 = 0.) This creates a regular foliated atlas $$\mathcal{U} = \left \{U_i,\varphi_i \right \}_{i=1}^{\infty}$$ that refines $$\mathcal{W}$$ and is coherent with $$\mathcal{W}.$$.
 * }

Foliation definitions
Several alternative definitions of foliation exist depending on the way through which the foliation is achieved. The most common way to achieve a foliation is through decomposition reaching to the following Definition. A p-dimensional, class Cr foliation of an n-dimensional manifold M is a decomposition of M into a union of disjoint connected submanifolds {Lα}α∈A, called the leaves of the foliation, with the following property: Every point in M has a neighborhood U and a system of local, class Cr coordinates x=(x1, ⋅⋅⋅, xn) : U→Rn such that for each leaf Lα, the components of U ∩ Lα are described by the equations xp+1=constant, ⋅⋅⋅, xn=constant. A foliation is denoted by $$\mathcal{F}$$={Lα}α∈A.

The notion of leaves allows for an intuitive way of thinking about a foliation. For a slightly more geometrical definition, $p$-dimensional foliation $$\mathcal{F}$$ of an $n$-manifold $M$ may be thought of as simply a collection ${M_{a}}$ of pairwise-disjoint, connected, immersed $p$-dimensional submanifolds (the leaves of the foliation) of $M$, such that for every point $x$ in $M$, there is a chart $$(U,\varphi)$$ with $U$ homeomorphic to $R^{n}$ containing $x$ such that every leaf, $M_{a}$, meets $U$ in either the empty set or a countable collection of subspaces whose images under $$\varphi$$ in $$\varphi (M_a \cap U)$$ are $p$-dimensional affine subspaces whose first $n − p$ coordinates are constant.

Locally, every foliation is a submersion allowing the following

Definition. Let M and Q be manifolds of dimension n and q≤n respectively, and let f : M→Q be a submersion, that is, suppose that the rank of the function differential (the Jacobian) is q. It follows from the Implicit Function Theorem that ƒ induces a codimension-q foliation on M where the leaves are defined to be the components of f−1(x) for x ∈ Q.

This definition describes a dimension-$p$ foliation $$\mathcal{F}$$ of an $n$-dimensional manifold $M$ that is a covered by charts $U_{i}$ together with maps

 * $$\varphi_i:U_i \to \mathbb{R}^n$$

such that for overlapping pairs $U_{i}, U_{j}$ the transition functions $φ_{ij} : R^{n} → R^{n}$ defined by

 * $$\varphi_{ij} =\varphi_j \varphi_i^{-1}$$

take the form

 * $$\varphi_{ij}(x,y) = (\varphi_{ij}^1(x),\varphi_{ij}^2(x,y))$$

where $x$ denotes the first $q$ = $n − p$ coordinates, and $y$ denotes the last $p$ co-ordinates. That is,

 * $$\begin{align}

\varphi_{ij}^1:{} &\mathbb{R}^q\to\mathbb{R}^q \\ \varphi_{ij}^2:{} &\mathbb{R}^n\to\mathbb{R}^p \end{align}$$ The splitting of the transition functions φij into $$\varphi_{ij}^1(x)$$ and $$\varphi_{ij}^2(x,y)$$ as a part of the submersion is completely analogous to the splitting of $$\overline{g}_{\alpha \beta}$$ into $$\overline{y}_{\alpha} \left ( \overline{y}_{\beta} \right )$$ and $$ \overline{x}_{\alpha} \left ( \overline{x}_{\beta}, \overline{y}_{\beta} \right )$$ as a part of the definition of a regular foliated atlas. This makes possible another definition of foliations in terms of regular foliated atlases. To this end, one has to prove first that every regular foliated atlas of codimension q is associated to a unique foliation $$\mathcal{F}$$ of codimension q.
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!Proof Let $$\mathcal{U} = \left \{U_a,\varphi_{\alpha} \right \}_{\alpha \in A}$$ be a regular foliated atlas of codimension q. Define an equivalence relation on M by setting x ~ y if and only if either there is a $$\mathcal{U}$$-plaque P0 such that x,y ∈ P0 or there is a sequence L = {P0,P1,⋅⋅⋅,Pp} of $$\mathcal{U}$$-plaques such that x ∈ P0, y ∈ Pp, and Pi ∩ Pi-1 ≠ ∅ with 1 ≤ i ≤ p. The sequence L will be called a plaque chain of length p connecting x and y. In the case that x,y ∈ P0, it is said that {P0} is a plaque chain of length 0 connecting x and y. The fact that ~ is an equivalence relation is clear. It is also clear that each equivalence class L is a union of plaques. Since $$\mathcal{U}$$-plaques can only overlap in open subsets of each other, L is locally a topologically immersed submanifold of dimension n − q. The open subsets of the plaques P ⊂ L form the base of a locally Euclidean topology on L of dimension n − q and L is clearly connected in this topology. It is also trivial to check that L is Hausdorff. The main problem is to show that L is second countable. Since each plaque is 2nd countable, the same will hold for L if it is shown that the set of $$\mathcal{U}$$-plaques in L is at most countably infinite. Fix one such plaque P0. By the definition of a regular, foliated atlas, P0 meets only finitely many other plaques. That is, there are only finitely many plaque chains {P0,Pi} of length 1. By induction on the length p of plaque chains that begin at P0, it is similarly proven that there are only finitely many of length ≤ p. Since every $$\mathcal{U}$$-plaque in L is, by the definition of ~, reached by a finite plaque chain starting at P0, the assertion follows. As shown in the proof, the leaves of the foliation are equivalence classes of plaque chains of length ≤ p which are also topologically immersed Hausdorff $p$-dimensional submanifolds. Next, it is shown that the equivalence relation of plaques on a leaf is expressed in equivalence of coherent foliated atlases in respect to their association with a foliation. More specifically, if $$\mathcal{U}$$ and $$\mathcal{V}$$ are foliated atlases on M and if $$\mathcal{U}$$ is associated to a foliation $$\mathcal{F}$$ then $$\mathcal{U}$$ and $$\mathcal{V}$$ are coherent if and only if $$\mathcal{V}$$ is also associated to $$\mathcal{F}$$.
 * }
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!Proof If $$\mathcal{V}$$ is also associated to $$\mathcal{F}$$, every leaf L is a union of $$\mathcal{V}$$-plaques and of $$\mathcal{U}$$-plaques. These plaques are open subsets in the manifold topology of L, hence intersect in open subsets of each other. Since plaques are connected, a $$\mathcal{U}$$-plaque cannot intersect a $$\mathcal{V}$$-plaque unless they lie in a common leaf; so the foliated atlases are coherent. Conversely, if we only know that $$\mathcal{U}$$ is associated to $$\mathcal{F}$$ and that $$\mathcal{V} \approx \mathcal{U}$$, let Q be a $$\mathcal{V}$$-plaque. If L is a leaf of $$\mathcal{F}$$ and w ∈ L ∩ Q, let P ∈ L be a $$\mathcal{U}$$-plaque with w ∈ P. Then P ∩ Q is an open neighborhood of w in Q and P ∩ Q ⊂ L ∩ Q. Since w ∈ L ∩ Q is arbitrary, it follows that L ∩ Q is open in Q. Since L is an arbitrary leaf, it follows that Q decomposes into disjoint open subsets, each of which is the intersection of Q with some leaf of $$\mathcal{F}$$. Since Q is connected, L ∩ Q = Q. Finally, Q is an arbitrary $$\mathcal{V}$$-plaque, and so $$\mathcal{V}$$ is associated to $$\mathcal{F}$$. It is now obvious that the correspondence between foliations on M and their associated foliated atlases induces a one-to-one correspondence between the set of foliations on M and the set of coherence classes of foliated atlases or, in other words, a foliation $$\mathcal{F}$$ of codimension q and class Cr on M is a coherence class of foliated atlases of codimension q and class Cr on M. By Zorn's lemma, it is obvious that every coherence class of foliated atlases contains a unique maximal foliated atlas. Thus,
 * }

Definition. A foliation of codimension q and class Cr on M is a maximal foliated Cr-atlas of codimension q on M.

In practice, a relatively small foliated atlas is generally used to represent a foliation. Usually, it is also required this atlas to be regular.

In the chart $U_{i}$, the stripes $x = constant$ match up with the stripes on other charts $U_{j}$. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.

If one shrinks the chart $U_{i}$ it can be written as $U_{ix} × U_{iy}$, where $U_{ix} ⊂ R^{n−p}, U_{iy} ⊂ R^{p}$, $U_{iy}$ is homeomorphic to the plaques, and the points of $U_{ix}$ parametrize the plaques in $U_{i}$. If one picks $y_{0}$ in $U_{iy}$, then $U_{ix} × {y_{0}} |undefined$ is a submanifold of $U_{i}$ that intersects every plaque exactly once. This is called a local transversal section of the foliation. Note that due to monodromy global transversal sections of the foliation might not exist.

The case r = 0 is rather special. Those C0 foliations that arise in practice are usually "smooth-leaved". More precisely, they are of class Cr,0, in the following sense.

Definition. A foliation $$\mathcal{F}$$ is of class Cr,k, r > k ≥ 0, if the corresponding coherence class of foliated atlases contains a regular foliated atlas {Uα,xα,yα}α∈A such that the change of coordinate formula
 * $$g_{\alpha \beta}(x_\beta, y_\beta) = ( x_\alpha( x_\beta, y_\beta), y_\alpha ( y_\beta)).$$

is of class Ck, but xα is of class Cr in the coordinates xβ and its mixed xβ partials of orders ≤ r are Ck in the coordinates (xβ,yβ).

The above definition suggests the more general concept of a foliated space or abstract lamination. One relaxes the condition that the transversals be open, relatively compact subsets of Rq, allowing the transverse coordinates yα to take their values in some more general topological space Z. The plaques are still open, relatively compact subsets of Rp, the change of transverse coordinate formula yα(yβ) is continuous and xα(xβ,yβ) is of class Cr in the coordinates xβ and its mixed xβ partials of orders ≤ r are continuous in the coordinates (xβ,yβ). One usually requires M and Z to be locally compact, second countable and metrizable. This may seem like a rather wild generalization, but there are contexts in which it is useful.

Holonomy
Let (M, $$\mathcal{F}$$) be a foliated manifold. If L is a leaf of $$\mathcal{F}$$ and s is a path in L, one is interested in the behavior of the foliation in a neighborhood of s in M. Intuitively, an inhabitant of the leaf walks along the path s, keeping an eye on all of the nearby leaves. As they (hereafter denoted by s(t)) proceed, some of these leaves may "peel away", getting out of visual range, others may suddenly come into range and approach L asymptotically, others may follow along in a more or less parallel fashion or wind around L laterally, etc. If s is a loop, then s(t) repeatedly returns to the same point s(t0) as t goes to infinity and each time more and more leaves may have spiraled into view or out of view, etc. This behavior, when appropriately formalized, is called the holonomy of the foliation.

Holonomy is implemented on foliated manifolds in various specific ways: the total holonomy group of foliated bundles, the holonomy pseudogroup of general foliated manifolds, the germinal holonomy groupoid of general foliated manifolds, the germinal holonomy group of a leaf, and the infinitesimal holonomy group of a leaf.

Foliated bundles
The easiest case of holonomy to understand is the total holonomy of a foliated bundle. This is a generalization of the notion of a Poincaré map. The term "first return (recurrence) map" comes from the theory of dynamical systems. Let Φt be a nonsingular Cr flow (r ≥ 1) on the compact n-manifold M. In applications, one can imagine that M is a cyclotron or some closed loop with fluid flow. If M has a boundary, the flow is assumed to be tangent to the boundary. The flow generates a 1-dimensional foliation $$\mathcal{F}$$. If one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, etc.), the underlying foliation $$\mathcal{F}$$ is said to be oriented. Suppose that the flow admits a global cross section N. That is, N is a compact, properly embedded, Cr submanifold of M of dimension n – 1, the foliation $$\mathcal{F}$$ is transverse to N, and every flow line meets N. Because the dimensions of N and of the leaves are complementary, the transversality condition is that
 * $$T_y (M) = T_y(\mathcal{F}) \oplus T_y(N) \text{ for each } y\in N.$$

Let y ∈ N and consider the ω-limit set ω(y) of all accumulation points in M of all sequences $$\left \{\Phi_{t_k}(y)\right\}_{k=1}^\infty$$, where tk goes to infinity. It can be shown that ω(y) is compact, nonempty, and a union of flow lines. If $$z = \lim_{k \rightarrow \infty} \Phi_{t_k} \in \omega(y),$$ there is a value t* ∈ R such that Φt*(z) ∈ N and it follows that
 * $$\lim_{k \to \infty} \Phi_{t_k + t^\ast} (y) = \Phi_{t^\ast}(z) \in N.$$

Since N is compact and $$\mathcal{F}$$ is transverse to N, it follows that the set {t > 0 | Φt(y) ∈ N} is a monotonically increasing sequence $$\{\tau_k(y)\}_{k=1}^\infty$$ that diverges to infinity.

As y ∈ N varies, let τ(y) = τ1(y), defining in this way a positive function τ ∈ Cr(N) (the first return time) such that, for arbitrary y ∈ N, Φt(y) ∉ N, 0 < t < τ(y), and Φτ(y)(y) ∈ N.

Define f : N → N by the formula f(y) = Φτ(y)(y). This is a Cr map. If the flow is reversed, exactly the same construction provides the inverse f−1; so f ∈ Diffr(N). This diffeomorphism is the first return map and τ is called the first return time. While the first return time depends on the parametrization of the flow, it should be evident that f depends only on the oriented foliation $$\mathcal{F}$$. It is possible to reparametrize the flow Φt, keeping it nonsingular, of class Cr, and not reversing its direction, so that τ ≡ 1.

The assumption that there is a cross section N to the flow is very restrictive, implying that M is the total space of a fiber bundle over S1. Indeed, on R × N, define ~f to be the equivalence relation generated by
 * $$(t,y) \sim_f (t-1,f(y)).$$

Equivalently, this is the orbit equivalence for the action of the additive group Z on R × N defined by
 * $$k \cdot (t,y) = (t - k,f^k(y) ),$$

for each k ∈ Z and for each (t,y) ∈ R × N. The mapping cylinder of f is defined to be the Cr manifold
 * $$M_f = (\mathbb{R} \times N)/{\sim_f}.$$

By the definition of the first return map f and the assumption that the first return time is τ ≡ 1, it is immediate that the map
 * $$\Phi : \mathbb{R} \times N \rightarrow M.$$

defined by the flow, induces a canonical Cr diffeomorphism
 * $$\varphi : M_f \rightarrow M.$$

If we make the identification Mf = M, then the projection of R × N onto R induces a Cr map
 * $$\pi : M \rightarrow \mathbb{R} / \mathbb{Z} = S^1$$

that makes M into the total space of a fiber bundle over the circle. This is just the projection of S1 × D2 onto S1. The foliation $$\mathcal{F}$$ is transverse to the fibers of this bundle and the bundle projection $\pi$, restricted to each leaf L, is a covering map π : L → S1. This is called a foliated bundle.

Take as basepoint x0 ∈ S1 the equivalence class 0 + Z; so π−1(x0) is the original cross section N. For each loop s on S1, based at x0, the homotopy class [s] ∈ π1(S1,x0) is uniquely characterized by deg s ∈ Z. The loop s lifts to a path in each flow line and it should be clear that the lift sy that starts at y ∈ N ends at fk(y) ∈ N, where k = deg s. The diffeomorphism fk ∈ Diffr(N) is also denoted by hs and is called the total holonomy of the loop s. Since this depends only on [s], this is a definition of a homomorphism
 * $$h : \pi_1(S^1,x_0) \rightarrow \operatorname{Diff}^{\,r}(N),$$

called the total holonomy homomorphism for the foliated bundle.

Using fiber bundles in a more direct manner, let (M,$$\mathcal{F}$$) be a foliated n-manifold of codimension q. Let π : M → B be a fiber bundle with q-dimensional fiber F and connected base space B. Assume that all of these structures are of class Cr, 0 ≤ r ≤ ∞, with the condition that, if r = 0, B supports a C1 structure. Since every maximal C1 atlas on B contains a C∞ subatlas, no generality is lost in assuming that B is as smooth as desired. Finally, for each x ∈ B, assume that there is a connected, open neighborhood U ⊆ B of x and a local trivialization
 * $$\begin{matrix}

\pi^{-1}(U) & \xrightarrow{\varphi} & U\times{F} \\ \scriptstyle{\pi} \Bigg\downarrow & {\qquad} & \Bigg\downarrow{\scriptstyle{p}} \\ U & \xrightarrow{\text{id}} & U \end{matrix} $$ where φ is a Cr diffeomorphism (a homeomorphism, if r = 0) that carries $ \mathcal{F} \mid \pi^{-1}(U) $ to the product foliation {U × {y}}y ∈ F. Here, $ \mathcal{F} \mid \pi^{-1}(U) $  is the foliation with leaves the connected components of L ∩ π−1(U), where L ranges over the leaves of $$\mathcal{F}$$. This is the general definition of the term "foliated bundle" (M,$$\mathcal{F}$$,π) of class Cr.

$$\mathcal{F}$$ is transverse to the fibers of π (it is said that $$\mathcal{F}$$ is transverse to the fibration) and that the restriction of π to each leaf L of $$\mathcal{F}$$ is a covering map π : L → B. In particular, each fiber Fx = π−1(x) meets every leaf of $$\mathcal{F}$$. The fiber is a cross section of $$\mathcal{F}$$ in complete analogy with the notion of a cross section of a flow.

The foliation $$\mathcal{F}$$ being transverse to the fibers does not, of itself, guarantee that the leaves are covering spaces of B. A simple version of the problem is a foliation of R2, transverse to the fibration
 * $$\pi : \mathbb{R}^2 \rightarrow \mathbb{R},$$
 * $$\pi(x,y) = x,$$

but with infinitely many leaves missing the y-axis. In the respective figure, it is intended that the "arrowed" leaves, and all above them, are asymptotic to the axis x = 0. One calls such a foliation incomplete relative to the fibration, meaning that some of the leaves "run off to infinity" as the parameter x ∈ B approaches some x0 ∈ B. More precisely, there may be a leaf L and a continuous path s : [0,a) → L such that limt→a−π(s(t)) = x0 ∈ B, but limt→a−s(t) does not exist in the manifold topology of L. This is analogous to the case of incomplete flows, where some flow lines "go to infinity" in finite time. Although such a leaf L may elsewhere meet π−1(x0), it cannot evenly cover a neighborhood of x0, hence cannot be a covering space of B under π. When F is compact, however, it is true that transversality of $$\mathcal{F}$$ to the fibration does guarantee completeness, hence that $(M,\mathcal{F},\pi)$ is a foliated bundle.

There is an atlas $$\mathcal{U}$$ = {Uα,xα}α∈A on B, consisting of open, connected coordinate charts, together with trivializations φα : π−1(Uα) → Uα × F that carry $$\mathcal{F}$$|π−1(Uα) to the product foliation. Set Wα = π−1(Uα) and write φα = (xα,yα) where (by abuse of notation) xα represents xα ∘ π and yα : π−1(Uα) → F is the submersion obtained by composing φα with the canonical projection Uα × F → F.

The atlas $$\mathcal{W}$$ = {Wα,xα,yα}α∈A plays a role analogous to that of a foliated atlas. The plaques of Wα are the level sets of yα and this family of plaques is identical to F via yα. Since B is assumed to support a C∞ structure, according to the Whitehead theorem one can fix a Riemannian metric on B and choose the atlas $$\mathcal{U}$$ to be geodesically convex. Thus, Uα ∩ Uβ is always connected. If this intersection is nonempty, each plaque of Wα meets exactly one plaque of Wβ. Then define a holonomy cocycle $$\gamma = \left \{ \gamma_{\alpha \beta} \right \}_{\alpha,\beta \in A}$$ by setting
 * $$\gamma_{\alpha \beta} = y_\alpha \circ y_\beta^{-1} : F \rightarrow F.$$

Flat space
Consider an $n$-dimensional space, foliated as a product by subspaces consisting of points whose first $n − p$ coordinates are constant. This can be covered with a single chart. The statement is essentially that $R^{n} = R^{n−p} × R^{p}$ with the leaves or plaques $R^{p}$ being enumerated by $R^{n−p}$. The analogy is seen directly in three dimensions, by taking $n = 3$ and $p = 2$: the 2-dimensional leaves of a book are enumerated by a (1-dimensional) page number.

Bundles
A rather trivial example of foliations are products $M = B × F$, foliated by the leaves $F_{b} = {b} × F, b ∈ B$. (Another foliation of $M$ is given by $B_{f} = B × { f },  f ∈ F$.)

A more general class are flat $G$-bundles with $G = Homeo(F)$ for a manifold $F$. Given a representation $ρ : \pi_{1}(B) → Homeo(F)$, the flat $Homeo(F)$-bundle with monodromy $ρ$ is given by $$M=\left(\widetilde{B}\times F\right)/\pi_1B$$, where $\pi_{1}(B)$ acts on the universal cover $$\widetilde{B}$$ by deck transformations and on $F$ by means of the representation $ρ$.

Flat bundles fit into the framework of fiber bundles. A map $\pi : M → B$ between manifolds is a fiber bundle if there is a manifold F such that each $b ∈ B$ has an open neighborhood $U$ such that there is a homeomorphism $$\varphi:\pi^{-1}(U)\to U\times F$$ with $$\pi = p_1 \varphi $$, with $p_{1} : U × F → U$ projection to the first factor. The fiber bundle yields a foliation by fibers $$F_b:=\pi^{-1}(\{b\}), b\in B$$. Its space of leaves L is homeomorphic to $B$, in particular L is a Hausdorff manifold.

Coverings
If $M → N$ is a covering map between manifolds, and $F$ is a foliation on $N$, then it pulls back to a foliation on $M$. 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.

Submersions
If $M^{n} → N^{q}, (q ≤ n)$ is a submersion of manifolds, it follows from the inverse function theorem that the connected components of the fibers of the submersion define a codimension $q$ foliation of $M$. Fiber bundles are an example of this type.

An example of a submersion, which is not a fiber bundle, is given by

 * $$\begin{cases} f:[-1,1]\times \mathbb{R}\to \mathbb{R} \\f(x,y)=(x^2-1) e^y\end{cases}$$

This submersion yields a foliation of $[−1, 1] × R$ which is invariant under the $Z$-actions given by
 * $$z(x, y)= (x,y+n ), \quad \text{or} \quad z(x, y)=\left((-1)^nx, y\right)$$

for $(x, y) ∈ [−1, 1] × R$ and $n ∈ Z$. The induced foliations of $Z \ ([−1, 1] × R)$ are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliation (of the Möbius band). Their leaf spaces are not Hausdorff.

Reeb foliations
Define a submersion

 * $$\begin{cases} f:D^{n}\times \mathbb{R}\to \mathbb{R} \\ f(r,\theta,t):=(r^2-1)e^t\end{cases}$$

where $(r, θ) ∈ [0, 1] × S^{n−1}$ are cylindrical coordinates on the $n$-dimensional disk $D^{n}$. This submersion yields a foliation of $D^{n} × R$ which is invariant under the $Z$-actions given by

 * $$z(x,y)=(x,y+z)$$

for $(x, y) ∈ D^{n} × R, z ∈ Z$. The induced foliation of $Z \ (D^{n} × R)$ is called the $n$-dimensional Reeb foliation. Its leaf space is not Hausdorff.

For $n = 2$, this gives a foliation of the solid torus which can be used to define the Reeb foliation of the 3-sphere by gluing two solid tori along their boundary. Foliations of odd-dimensional spheres $S^{2n+1}$ are also explicitly known.

Lie groups
If $G$ is a Lie group, and $H$ is a Lie subgroup, then $G$ is foliated by cosets of $H$. When $H$ is closed in $G$, the quotient space $G$/$H$ is a smooth (Hausdorff) manifold turning $G$ into a fiber bundle with fiber $H$ and base $G$/$H$. This fiber bundle is actually principal, with structure group $H$.

Lie group actions
Let $G$ be a Lie group acting smoothly on a manifold $M$. If the action is a locally free action or free action, then the orbits of $G$ define a foliation of $M$.

Linear and Kronecker foliations
If $$\tilde{X}$$ is a nonsingular (i.e., nowhere zero) vector field, then the local flow defined by $$\tilde{X}$$ patches together to define a foliation of dimension 1. Indeed, given an arbitrary point x ∈ M, the fact that $$\tilde{X}$$ is nonsingular allows one to find a coordinate neighborhood (U,x1,...,xn) about x such that
 * $$- \varepsilon < x^i < \varepsilon, \quad 1 \le i \le n,$$

and
 * $$\frac{\partial}{\partial x^1} = \tilde{X}\mid U.$$

Geometrically, the flow lines of $$\tilde{X} \mid U$$ are just the level sets
 * $$x^i = c^i, \quad 2 \le i \le n,$$

where all $$|c^i| < \varepsilon.$$ Since by convention manifolds are second countable, leaf anomalies like the "long line" are precluded by the second countability of M itself. The difficulty can be sidestepped by requiring that $$\tilde{X}$$ be a complete field (e.g., that M be compact), hence that each leaf be a flow line. An important class of 1-dimensional foliations on the torus T2 are derived from projecting constant vector fields on T2. A constant vector field
 * $$\tilde{X} \equiv \begin{bmatrix}a \\ b \end{bmatrix}$$

on R2 is invariant by all translations in R2, hence passes to a well-defined vector field X when projected on the torus $T^{2}= R^{2}/Z^{2}$. It is assumed that a ≠ 0. The foliation $$\mathcal{\tilde{F}}$$ on R2 produced by $$\tilde{X}$$ has as leaves the parallel straight lines of slope θ = b/a. This foliation is also invariant under translations and passes to the foliation $$\mathcal{F}$$ on T2 produced by X.

Each leaf of $$\mathcal{\tilde{F}}$$ is of the form
 * $$\tilde{L} = \{(x_0 + ta,y_0 + tb)\}_{t \in \mathbb{R}}.$$

If the slope is rational then all leaves are closed curves homeomorphic to the circle. In this case, one can take a,b ∈ Z. For fixed t ∈ R, the points of $$\tilde{L}$$ corresponding to values of t ∈ t0 + Z all project to the same point of T2; so the corresponding leaf L of $$\mathcal{F}$$ is an embedded circle in T2. Since L is arbitrary, $$\mathcal{F}$$ is a foliation of T2 by circles. It follows rather easily that this foliation is actually a fiber bundle π : T2 → S1. This is known as a linear foliation.

When the slope θ = b/a is irrational, the leaves are noncompact, homeomorphic to the non-compactified real line, and dense in the torus (cf Irrational rotation). The trajectory of each point (x0,y0) never returns to the same point, but generates an "everywhere dense" winding about the torus, i.e. approaches arbitrarily close to any given point. Thus the closure to the trajectory is the entire two-dimensional torus. This case is named Kronecker foliation, after Leopold Kronecker and his

Kronecker's Density Theorem. If the real number θ is distinct from each rational multiple of π, then the set {einθ | n ∈ Z} is dense in the unit circle.
 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Proof To see this, note first that, if a leaf $$\tilde{L}$$ of $$\mathcal{\tilde{F}}$$ does not project one-to-one into T2, there must be a real number t ≠ 0 such that ta and tb are both integers. But this would imply that b/a ∈ Q. In order to show that each leaf L of $$\mathcal{F}$$ is dense in T2, it is enough to show that, for every v ∈ R2, each leaf $$\tilde{L}$$ of $$\mathcal{\tilde{F}}$$ achieves arbitrarily small positive distances from suitable points of the coset v + Z2. A suitable translation in R2 allows one to assume that v = 0; so the task is reduced to showing that $$\tilde{L}$$ passes arbitrarily close to suitable points (n,m) ∈ Z2. The line $$\tilde{L}$$ has slope-intercept equation
 * $$y = \theta x + c.$$

So it will suffice to find, for arbitrary η > 0, integers n and m such that
 * $$|\theta n + c - m| < \eta.$$

Equivalently, c ∈ R being arbitrary, one is reduced to showing that the set {θn − m}m,n∈Z is dense in R. This is essentially the criterion of Eudoxus that θ and 1 be incommensurable (i.e., that θ be irrational). A similar construction using a foliation of $R^{n}$ by parallel lines yields a 1-dimensional foliation of the $n$-torus $R^{n}/Z^{n}$ associated with the linear flow on the torus.
 * }

Suspension foliations
A flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are
 * $$L_f:= \left\{p\left(\tilde{b},f\right): \tilde{b}\in\widetilde{B}\right\}, \quad \mbox{ for }f\in F,$$

where $$p:\widetilde{B}\times F\to M$$ is the canonical projection. This foliation is called the suspension of the representation $ρ : \pi_{1}(B) → Homeo(F)$.

In particular, if $B = S^{1}$ and $$\varphi:F\to F$$ is a homeomorphism of $F$, then the suspension foliation of $$\varphi$$ is defined to be the suspension foliation of the representation $ρ : Z → Homeo(F)$ given by $ρ(z) = Φ^{z}$. Its space of leaves is $L = $\mathcal{F}$/~$, where $x ~ y$ whenever $y = Φ^{n}(x)$ for some $n ∈ Z$.

The simplest example of foliation by suspension is a manifold X of dimension q. Let f : X → X be a bijection. One defines the suspension M = S1 ×f X as the quotient of [0,1] × X by the equivalence relation (1,x) ~ (0,f(x)).
 * M = S1 ×f X = [0,1] × X

Then automatically M carries two foliations: $$\mathcal{F}$$2 consisting of sets of the form F2,t = {(t,x)~ : x ∈ X} and $$\mathcal{F}$$1 consisting of sets of the form F2,x 0 = {(t,x) : t ∈ [0,1] ,x ∈ Ox 0 }, where the orbit Ox 0 is defined as
 * Ox 0 = {..., f−2(x0), f−1(x0), x0, f(x0), f2(x0), ...},

where the exponent refers to the number of times the function f is composed with itself. Note that Ox 0 = Of(x 0) = Of−2(x 0), etc., so the same is true for F1,x 0. Understanding the foliation $$\mathcal{F}$$1 is equivalent to understanding the dynamics of the map f. If the manifold X is already foliated, one can use the construction to increase the codimension of the foliation, as long as f maps leaves to leaves.

The Kronecker foliations of the 2-torus are the suspension foliations of the rotations $R_{α} : S^{1} → S^{1}$ by angle $α ∈ [0, 2\pi).$ More specifically, if Σ = Σ2 is the two-holed torus with C1,C2 ∈ Σ the two embedded circles let $$\mathcal{F}$$ be the product foliation of the 3-manifold M = Σ × S1 with leaves Σ × {y}, y ∈ S1. Note that Ni = Ci × S1 is an embedded torus and that $$\mathcal{F}$$ is transverse to Ni, i = 1,2. Let Diff+(S1) denote the group of orientation-preserving diffeomorphisms of S1 and choose f1,f2 ∈ Diff+(S1). Cut M apart along N1 and N2, letting $$N_i^{+}$$ and $$N_i^{-}$$ denote the resulting copies of Ni, i = 1,2. At this point one has a manifold M' = Σ' × S1 with four boundary components $$\left \{N_i^{\pm} \right \}_{i=1,2}.$$ The foliation $$\mathcal{F}$$ has passed to a foliation $$\mathcal{F^{\prime}}$$ transverse to the boundary ∂M' , each leaf of which is of the form Σ' × {y}, y ∈ S1.

This leaf meets ∂M'  in four circles $$C_i^{\pm} \times \{y\} \subset N_i^{\pm}.$$ If z ∈ Ci, the corresponding points in $$C_i^{\pm}$$ are denoted by z± and $$N_i^{-}$$ is "reglued" to $$N_i^{+}$$ by the identification
 * $$(z^{-},y) \equiv (z^{+},f_i(y)), \quad i = 1,2.$$

Since f1 and f2 are orientation-preserving diffeomorphisms of S1, they are isotopic to the identity and the manifold obtained by this regluing operation is homeomorphic to M. The leaves of $$\mathcal{F^{\prime}}$$, however, reassemble to produce a new foliation $$\mathcal{F}$$(f1,f2) of M. If a leaf L of $$\mathcal{F}$$(f1,f2) contains a piece Σ' × {y0}, then
 * $$L = \bigcup_{g \in G} \Sigma^{\prime} \times \{ g (y_0) \},$$

where G ⊂ Diff+(S1) is the subgroup generated by {f1,f2}. These copies of Σ' are attached to one another by identifications
 * (z−,g(y0)) ≡ (z+,f1(g(y0))) for each z ∈ C1,
 * (z−,g(y0)) ≡ (z+,f2(g(y0))) for each z ∈ C2,

where g ranges over G. The leaf is completely determined by the G-orbit of y0 ∈ S1 and can he simple or immensely complicated. For instance, a leaf will be compact precisely if the corresponding G-orbit is finite. As an extreme example, if G is trivial (f1 = f2 = idS1), then $$\mathcal{F}$$(f1,f2) = $$\mathcal{F}$$. If an orbit is dense in S1, the corresponding leaf is dense in M. As an example, if f1 and f2 are rotations through rationally independent multiples of 2π, every leaf will be dense. In other examples, some leaf L has closure $$\bar{L}$$ that meets each factor {w} × S1 in a Cantor set. Similar constructions can be made on Σ × I, where I is a compact, nondegenerate interval. Here one takes f1,f2 ∈ Diff+(I) and, since ∂I is fixed pointwise by all orientation-preserving diffeomorphisms, one gets a foliation having the two components of ∂M as leaves. When one forms M'  in this case, one gets a foliated manifold with corners. In either case, this construction is called the suspension of a pair of diffeomorphisms and is a fertile source of interesting examples of codimension-one foliations.

Foliations and integrability
There is a close relationship, assuming everything is smooth, with vector fields: given a vector field $X$ on $M$ that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension $n − 1$ foliation).

This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution (i.e. an $n − p$ 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. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from $GL(n)$ to a reducible subgroup.

The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist. For example, in the codimension 1 case, we can define the tangent bundle of the foliation as $ker(α)$, for some (non-canonical) $α ∈ Ω^{1}$ (i.e. a non-zero co-vector field). A given $α$ is integrable iff $α ∧ dα = 0$ everywhere.

There is a global foliation theory, because topological constraints exist. For example, in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0. There are many deep connections with contact topology, which is the "opposite" concept, requiring that the integrability condition is never satisfied.

Existence of foliations
gave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution. showed that any compact manifold with a distribution has a foliation of the same dimension.

title: Foliation


text:
Bashkir language

Bashkir or Bashkort (, ; Башҡорт теле, ) is a Turkic language belonging to the Kipchak branch. It is co-official with Russian in Bashkortostan. It is spoken by 1.09 million native speakers in Russia, as well as in Ukraine, Belarus, Kazakhstan, Uzbekistan, Estonia and other neighboring post-Soviet states, and among the Bashkir diaspora. It has three dialect groups: Southern, Eastern and Northwestern.

Speakers
Speakers of Bashkir mostly live in the republic of Bashkortostan (a republic within the Russian Federation). Many speakers also live in Tatarstan, Chelyabinsk, Orenburg, Tyumen, Sverdlovsk and Kurgan Oblasts and other regions of Russia. Minor Bashkir groups also live in Kazakhstan and other countries.

Classification
Bashkir together with Tatar belongs to the Kipchak-Bulgar (кыпчакско-булгарская) subgroup of the Kipchak languages. These languages have a similar vocabulary by 94.9%, and they not only have common origin, but also a common ancestor in the written language — Volga Turki. However, Bashkir differs from Tatar in several important ways:
 * Bashkir has dental fricatives and  in the place of Tatar (and other Turkic)  and  . Bashkir  and, however, cannot begin a word (there are exceptions: ҙур – "zur"  'big', and the particle/conjunction ҙа – "za"  or ҙә – "zə" . The only other Turkic language with a similar feature is Turkmen. However, in Bashkir,  and  are two independent phonemes, distinct from  and , whereas in Turkmen [θ] and [ð] are the two main realizations of the common Turkic  and . In other words, there are no  and  phonemes in Turkmen, unlike Bashkir which has both  and  and  and.
 * The word-initial and morpheme-initial is turned into . An example of both features can be Tatar сүз (süz) and Bashkir һүҙ (hüz), both meaning "word".
 * Common Turkic (Tatar ) is turned into Bashkir, e.g., Turkish ağaç , Tatar агач ağaç  and Bashkir ағас – ağas , all meaning "tree".
 * The word-initial in Tatar always corresponds to  in Standard Bashkir, e.g., Tatar җылы cılı  and Bashkir йылы – yılı, both meaning "warm". However, the eastern and northern dialects of Bashkir have the  > /~/ shift.

The Bashkir orthography is more explicit. and are written with their own letters Ҡ ҡ and Ғ ғ, whereas in Tatar they are treated as positional allophones of  and, written К к and Г г.

Labial vowel harmony in Bashkir is written explicitly, e.g. Tatar тормышым tormışım and Bashkir тормошом – tormoşom, both pronounced, meaning "my life".

Latin script
Bible in Bashkir

1: İsrailda xakimdar idara itkәn osorźa ildә yot başlandı. Şul arqala Yәhüźә bilәmәhendәge Beyt-Lәxәm tigәn qalanan ber keşe qatını hәm ike ulı menәn Moav yerenә küsende.

2: Bıl keşeneñ iseme – Әlimәlәx, qatınınıqı – Noğomi, ә uldarınıqı Maxlon menәn Kilyon ine. Yәhüźә bilәmәhendәge Beyt-Lәxәm qalahınan, Әfraś yerźәrenәn bulğan bıl ğailә Moav ilenә küsep kilep, şunda töyәklәnde.

3: Noğomiźıñ ire Әlimәlәx oşonda vafat buldı, qatın uldarı menәn genә qaldı.

4: İke ulı la moav qıźźarına öylәnde: ularźıñ bereheneñ iseme – Ğorpa, ikenseheneke Rut ine. Un yıl samahı unda yәşәgәs,

5: Maxlon menän Kilyon da ülep kitte. Noğomi ike ulınan da, irenәn dә mәxrüm qaldı.

6: Rabbınıñ Üź xalqına iltifat kürhәtep, ikmәk birgәnen işetkәs, Noğomi kilendәre menәn Moavtan tıwğan ilenә qaytırğa buldı.

7: Yәşәp yatqan urındarın taşlap, ul kilendәre menәn Yәhüźә tarafına yünәlde. Kitep barışlay,

8: Noğomi kilendәrenә: – Barığıź, әsәyźәregeź yortona qaytığıź. Heź mәrxümdәrgә hәm miñә qarata igelekle bulğan kewek, Rabbı heźgә qarata la mәrxәmәtle bulhın.

9: Rabbı hәr beregeźgә yañı ir yortonda rәxәtlek tabırğa nasip ithen, – tine. Ul xuşlaşıp kilendәren üpte, lәkin ular qısqırıp ilay-ilay:

10: – Beź hineñ menәn barabıź, hineñ xalqıñ arahında yәşәybeź, – tinelәr.

Source: https://ibt.org.ru/ru/text?m=BSK&l=Ruth.1.1.1&g=0

Orthography
After the adoption of Islam, which began in the 10th century and lasted for several centuries, the Bashkirs began to use Turki as a written language. Turki was written in a variant of the Arabic script.

In 1923, a writing system based on the Arabic script was specifically created for the Bashkir language. At the same time, the Bashkir literary language was created, moving away from the older written Turkic influences. At first, it used a modified Arabic alphabet. In 1930 it was replaced with the Unified Turkic Latin Alphabet, which was in turn replaced with an adapted Cyrillic alphabet in 1939.

The modern alphabet used by Bashkir is based on the Russian alphabet, with the addition of the following letters: Ә|Ә ә, Ө|Ө ө , Ү|Ү ү , Ғ|Ғ ғ , Ҡ|Ҡ ҡ , Ң|Ң ң , Ҙ|Ҙ ҙ , Ҫ|Ҫ ҫ , Һ|Һ һ.

Vowels
Bashkir has nine native vowels, and three or four loaned vowels (mainly in Russian loanwords).

Phonetically, the native vowels are approximately thus (with the Cyrillic letter followed by the usual Latin romanization in angle brackets): In Russian loans there are also, , and , written the same as the native vowels: ы, е/э, о, а respectively.

Historical shifts
Historically, the Old Turkic mid vowels have raised from mid to high, whereas the Old Turkic high vowels have become the Bashkir reduced mid series. (The same shifts have also happened in Tatar.)

Consonants

 * Notes
 * The phonemes, , are found only in loanwords, and, in the case of , in a few native onomatopoeic words.
 * is an intervocal allophone of, and it is distinct from . is an allophone of  in back vowel contexts.  and  occur as allophones of  and  before , and both occur only in front vowel contexts.

 * are dental, and is apical alveolar . The exact place of articulation of the other dental/alveolar consonants is unclear.

Grammar
A member of the Turkic language family, Bashkir is an agglutinative, SOV language. A large part of the Bashkir vocabulary has Turkic roots; and there are many loan words in Bashkir from Russian, Arabic and Persian sources.

Plurality
The form of the plural suffix is heavily dependent on the letter which comes immediately before it. When it's a consonant, there is a four-way distinction between "л" (l), "т" (t), "ҙ" (ź) and "д" (d); The vowel's distinction is two-way between "а" (after back vowels "а" (a), "ы" (ı), "о" (o), "у" (u)) and "ә" (after front vowels "ә" (ə), "е" (e), "и" (i), "ө" (ö), "ү" (ü)). Some nouns are also less likely to be used with their plural forms such as "һыу" (hıw, "water") or "ҡом" (qom, "sand").

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Bashkir language


text:
User talk:Evjohn

Hello there Evjohn welcome to Wikipedia! I hope you like the place and decide to stay. If you ever need editing help visit How does one edit a page or how to format them visit our manual of style. You can have a look on the changes of calcareous sponge for comparison. If you need pointers on how we title pages visit Naming conventions. If you have any other questions about the project then check out Help or add a question to the Village pump or my talk page. Have fun!, Muriel Gottrop

Image source needed
Howdy - can you please post the source of your image at Image:Sponge1.jpg - we need to know the copyright status so we can keep it. --Mperkins 02:16, 12 Dec 2003 (UTC)

title: User talk:Evjohn


text:
Don Carlos Buell

Don Carlos Buell (March 23, 1818 – November 19, 1898) was a United States Army officer who fought in the Seminole War, the Mexican–American War, and the American Civil War. Buell led Union armies in two great Civil War battles—Shiloh and Perryville. The nation was angry at his failure to defeat the outnumbered Confederates after Perryville, or to secure East Tennessee. Historians generally concur that he was a brave and industrious master of logistics, but was too cautious and too rigid to meet the great challenges he faced in 1862. Buell was relieved of field command in late 1862 and made no more significant military contributions until his resignation in 1864.

Early life and education
Don Carlos Buell was born in Lowell, Ohio, the eldest of nine children born to Salmon and Elizabeth Buell. He was a first cousin of George P. Buell, also a Union general. Buell's father died when he was 8 years old, and his uncle took him in and raised him. As a child, Buell had a difficult time making friends due to his distant, introverted personality and was often made fun of by other children. After winning a fight with a neighborhood bully, he became awakened to the idea that discipline and determination could overcome any obstacle. Buell's uncle sent him to a Presbyterian school which stressed duty, self-discipline, patriotism, and belief in a Supreme Being.

George Buell obtained for his nephew an appointment to West Point, but despite his high intelligence and good math skills, he accumulated numerous demerits and disciplinary problems and graduated in 1841 32nd in his class of 52. After graduation, Buell was commissioned a second lieutenant in the 3rd U.S. Infantry regiment and sent to fight in the Seminole Wars in Florida but did not see any combat. After the 3rd Infantry was transferred to Illinois, Buell found himself court-martialed for getting into an argument with an enlisted man and beating him over the head with the blunt end of his sword. However, an Army tribunal cleared him of any wrongdoing. There was considerable opposition to the verdict, and even General Winfield Scott felt that Buell needed to be punished for his actions, but the court would not retry the case.

In the Mexican–American War, he served under both Zachary Taylor and Winfield Scott. He was brevetted three times for bravery and was wounded at Churubusco. Between the wars he served in the U.S. Army Adjutant General's office and as an adjutant in California, reaching the rank of captain in 1851 and lieutenant colonel by the time the Civil War began.

Initial postings
At the start of the Civil War, Buell sought an important command, but instead his friend George McClellan emerged as the champion of the Union war effort. Buell himself was sent all the way out to California. After the Union defeat at Bull Run, McClellan summoned him back east where he was quickly promoted to brigadier general of volunteers, to rank from May 17, 1861. Buell received offers to take a command in Kentucky, but instead he stayed in Washington helping organize the nascent Army of the Potomac and being appointed as a division commander. In November, McClellan succeeded Winfield Scott as general-in-chief of the Army, and decided to post Buell out west, dividing the trans-Appalachia theater between him and Maj. Gen Henry Halleck. In November 1861, he succeeded Brig. Gen. William T. Sherman in command at Louisville, Kentucky, as commander of the newly formed Army of the Ohio, which at this time was a barely-disciplined rabble. Buell immediately set himself to work shaping the raw recruits into a fighting force. Although the Lincoln administration pressured Buell to occupy Eastern Tennessee, an area of strong Unionist sentiment, Buell was in no hurry, and even McClellan became impatient with his slow progress. Buell's excuse was that the railroad network in this area was poor, and he would have to rely on wagons for army supply that could be vulnerable to Confederate cavalry. Instead, he proposed a coordinated effort between him and Halleck to cut off Nashville. Halleck reluctantly agreed to the plan, which was helped along by Grant's capture of Fort Henry and Fort Donelson. Although the city fell to the Army of the Ohio on February 25, 1862, Halleck's relationship with Buell was strained. The same month, Andrew Johnson was made military governor of Tennessee and developed a lasting grudge against Buell for failing to liberate Eastern Tennessee. On March 21, Buell was promoted to major general of volunteers, but at the same time, Halleck rose to department commander which made Buell subordinate to him.

Shiloh
At the start of April, Buell was ordered to reinforce Grant's Army of the Tennessee, then encamped at Pittsburg Landing next to the Tennessee River. On the morning of April 6, the Confederates launched a surprise attack on Grant's army, beginning one of the largest and bloodiest battles of the war. After the Army of the Ohio arrived the next day, the combined Union forces repulsed the Confederates. Although Buell was the junior of the two generals in rank, he insisted that he was acting independently and would not accept orders from Grant. Buell considered himself the victor of Shiloh and denigrated Grant's contribution, writing after the war that he had no "marked influence that he exerted upon the fortune of the day." Contemporary historians, such as Larry Daniels and Kenneth W. Noe, consider that Grant actually saved himself by the conclusion of the first day of battle and that the rivalry between Grant and Buell hampered the conduct of battle on the second day. The commanders operated almost completely independently of each other and Buell "proved slow and hesitant to commit himself."

Following Shiloh, Governor Johnson objected to Buell's plans to withdraw the Nashville garrison on the grounds that Confederate sympathy in the city was still strong. However, Halleck sided with Buell and insisted that all available troops in the department were needed for the assault on Corinth. Henry Halleck arrived in person to take command of Grant and Buell's armies. The combined Union force, 100,000 men strong, began an extremely sluggish pursuit against P.G.T. Beauregard's Army of Mississippi, which had retreated into northern Mississippi. Despite a more than 2-1 numerical advantage, Halleck moved slowly. However, Buell was even slower and quickly caught the ire of Halleck. During the march to Corinth, Buell took extensive pauses to repair the railroad lines and would not entertain suggestions of allowing his army to live off the land. To that end, he court-martialed Col. John Turchin for allowing his soldiers to loot area homes. This action was not popular either with Buell's troops or the War Department, and President Lincoln ultimately overturned the verdict, while Turchin eventually got promoted to brigadier general.

The Siege of Corinth ended in the Confederates abandoning the city on May 25. Afterwards, Halleck split up the two armies and sent Buell eastward to capture Chattanooga while Grant remained in the Corinth area. In July, Halleck was summoned back to Washington to replace George McClellan as commander-in-chief of all Union armies, thus effectively returning the two Western armies to independent action. Buell's advance towards Chattanooga nearly rivaled the earlier march on Corinth for sluggishness, with extensive pauses to stop and repair railroad lines. When Nathan Bedford Forrest's cavalry then ransacked the Army of Ohio's supply lines, Buell all but terminated the effort to take Chattanooga. Thus after a busy winter and spring, activity in the Western theater during the summer of 1862 almost completely ground to a halt. On July 17, Buell was promoted to colonel in the regular army.

The summer months were increasingly frustrating for the Army of the Ohio, which was averaging barely a mile a day. Bored soldiers took to uncontrollable looting of the countryside and harassment of slaves. These infractions in their turn were punished harshly by Buell and by mid-August, morale in the army had almost collapsed. Despite protests from the War Department to move faster, Buell insisted that he could not hold Chattanooga for any length of time without proper caution and preparedness.

Grant, despite his professional rivalry following Shiloh, addressed these charges against Buell in his memoirs, writing:

"General Buell was a brave, intelligent officer, with as much professional pride and ambition of a commendable sort as I ever knew. ... [He] became an object of harsh criticism later, some going so far as to challenge his loyalty. No one who knew him ever believed him capable of a dishonorable act, and nothing could be more dishonorable than to accept high rank and command in war and then betray the trust. When I came into command of the army in 1864, I requested the Secretary of War to restore General Buell to duty. ... The opportunity frequently occurred for me to defend General Buell against what I believed to be most unjust charges. On one occasion a correspondent put in my mouth the very charge I had so often refuted—of disloyalty. This brought from General Buell a very severe retort, which I saw in the New York World some time before I received the letter itself. I could very well understand his grievance at seeing untrue and disgraceful charges apparently sustained by an officer who, at the time, was at the head of the army. I replied to him, but not through the press. I kept no copy of my letter, nor did I ever see it in print; neither did I receive an answer."

Kentucky
In September, Confederate armies under Edmund Kirby Smith and Braxton Bragg invaded Kentucky and Buell was forced to take action. Buell wired Halleck that he planned to march on Louisville, but Halleck, already frustrated with his glacial movements in Tennessee, replied back that he did not care where Buell marched just as long as he was doing something to take the fight to the enemy. The Kentucky campaign did have the effect of re-energizing Buell's demoralized soldiers who were excited to finally be going somewhere and march into a state that had been mostly untouched by war. Louisville was occupied by the Army of the Ohio on September 25, but despite learning that Bragg's army was in nearby Munfordville, Buell, convinced that he was outnumbered, declined to pursue Bragg. A single corps of Buell's army was attacked by Bragg at the Battle of Perryville on October 8, 1862, while Buell, a couple of miles behind the action, was not aware that a battle was taking place until late in the day and thus did not effectively engage the full strength of his army to defeat the smaller enemy force. Buell was urged by his officers to counterattack the next day, but he refused on the grounds that he did not know exactly how many Confederates he was facing. By morning, Bragg ordered a retreat from the field. Although Perryville was tactically indecisive, it halted the Confederate invasion of Kentucky and forced their withdrawal back into Tennessee.

Although the battle ended with the Union army in possession of the field, the Confederates had escaped to fight another day, and Buell had not engaged most of his army in spite of having nearly 60,000 men to face a mere 16,000 Confederates. Even some of Buell's own officers and enlisted men began suspecting him of disloyalty, in part because he was one of a handful of Union generals to have owned slaves prior to the war. An Indiana artillery officer wrote "After Perryville, I became convinced that the sooner Buell was relieved of command of the Army of the Ohio, the better." When President Lincoln urged an immediate pursuit of Bragg, he was told by Buell that the route directly south from Perryville into Eastern Tennessee was rough, wooded country with few roads and would be too difficult to maneuver through. He said that the only sensible route was to go back west to Nashville, then travel east across Tennessee to Chattanooga. Halleck wired Buell telling him "The president does not understand why we cannot march as the enemy marches, live as he lives, and fight as he fights, unless we admit that there is some inherent defect in our generals and soldiers."

On October 24, Buell was relieved from command of the Army of the Ohio and replaced by Maj. Gen William Rosecrans. A military committee investigated Buell's conduct during and after Perryville, but came to no conclusions, and Buell considered his reputation vindicated as he did not compromise his principles in waging war. After his dismissal, he was ordered to Indianapolis to await future assignments, but none came. When Grant was appointed general-in-chief of the army in March 1864, he offered Buell a possible assignment but he refused to serve under either Sherman or George Thomas on the grounds that he outranked both of them. In his memoirs, Grant called this "the worst excuse a soldier can make for declining service". On May 23, Buell's volunteer commission expired and he reverted to the regular army rank of colonel. Unable to tolerate this demotion, he resigned from the army on June 1.

Later life and death
Following the war Buell lived again in Indiana, and then in Kentucky, employed in the iron and coal industry as president of the Green River Iron Company. He continued to be a target of criticism for his conduct during the war. Although Buell did not write any memoirs, he did produce a series of newspaper articles defending himself and criticizing Grant, particularly for the events at Shiloh, and Buell to his dying day maintained that he was the hero of the battle. The death of his wife in 1881 was very hard on him, and his final years were marked by poverty and ill health. By 1898 he was an invalid, and he died on November 19. He was buried in Bellefontaine Cemetery in St. Louis.

Legacy
Buell earned the nickname "The McClellan of the West" for his cautious approach and desire for a limited war that would not disrupt civilian life in the South or interfere with slavery. Although he staunchly opposed secession, he was never able to reconcile himself with the Lincoln administration. Buell's wife had owned slaves prior to the war, and their marriage, although she freed them shortly after the Confederate attack on Fort Sumter. Buell had no personal animosity towards slavery or the Southern way of life. He continued to be highly regarded by fellow generals, many of whom felt that the White House had been unjust towards him. William T. Sherman wrote to his brother John, a Congressman, "You have driven off McClellan, and is Burnside any better? You have driven off Buell, and is Rosecrans any better?" His harsh discipline and inability to relate to his soldiers on a personal level may have also contributed to his downfall.

Buell Armory on the University of Kentucky campus in Lexington, Kentucky, is named after Buell.

Buell Island in his hometown of Lowell is named after him.

His personal papers, maps, correspondence, uniform, West Point diploma and other possessions are held at the University of Notre Dame.

title: Don Carlos Buell


text:
Prussia (disambiguation)

Prussia (Preußen) was a German state that formed the German Empire in 1871.

Prussia or Prussian may also refer to:

 * Prussia (region), a historical region on the south-eastern coast of the Baltic Sea that lent its name to the later German state

Countries

 * State of the Teutonic Order (1230–1525), founded in the Prussian region by the Teutonic Order
 * Prussian Confederation (1440–1466), an alliance of German Hanseatic cities in Prussia who rebelled against the Teutonic Knights
 * Royal Prussia (1466–1772), a province of the Polish Crown, created by the separation of Prussia into two parts
 * Duchy of Prussia (1525–1618), a duchy established in the eastern part of Prussia
 * Brandenburg-Prussia (1618–1701), a state created by the personal union of the Duchy of Prussia and the Margraviate of Brandenburg

Post-1800

 * Kingdom of Prussia (1701–1918), a kingdom established in Brandenburg-Prussia
 * Province of Prussia (1829–1878), a province of the Kingdom of Prussia, created by a union of the Provinces of East and West Prussia
 * Free State of Prussia (1918–1945), a state of Germany after the abolition of the Kingdom of Prussia

Subdivisions

 * East Prussia, the eastern part of the region of Prussia
 * West Prussia, the western part of the region of Prussia
 * Province of West Prussia, a province of the Kingdom of Prussia, annexed from Poland in 1772 (former Royal Prussia), re-established 1878
 * Province of East Prussia, a province of the Kingdom of Prussia, created from former Ducal Prussia and Warmia in 1773, re-established 1878; an exclave of Germany after World War I
 * Posen-West Prussia, a province of the Free State of Prussia (1922–1938) (the western parts of former West Prussia that were retained by Germany)
 * Regierungsbezirk West Prussia, administrative region of the Province of East Prussia (1920–1938) (the eastern parts of former West Prussia that were retained by Germany)

People and languages

 * Prussians, an ethnic subgroup of Germans
 * Citizens of the state of Prussia
 * Old Prussians, Baltic tribes formerly inhabiting the region of Prussia
 * Old Prussian language, language of the Old Prussians, now extinct
 * High Prussian, German dialect in East Prussia
 * Low Prussian, German dialect in East Prussia

Institutions

 * Prussian Army, army of the Kingdom of Prussia
 * Prussian Cultural Heritage Foundation, a foundation (est. 1957) holding museums and libraries left behind by the former Free State of Prussia
 * Evangelical State Church in Prussia, a former Protestant church body under changing names (1817–2004)
 * Prussian Navy, navy of the Kingdom of Prussia
 * Prussian Military Academy
 * Prussian Trust, German expellee company

Cities

 * Prussia in Saskatchewan, Canada, was renamed Leader in 1917.
 * Prusia, near Pozuzo, Oxapampa, in the central jungle of Peru
 * Prussia, Iowa, in the United States.
 * King of Prussia, Pennsylvania, in the United States.

title: Prussia (disambiguation)


text:
Talk:Zionism/Archive 2

I obliterated much of the content, but I don't have time to explain my deletions here. I will later (in a few days, actually). If you want to restore some of the prose, that's fine, I'll delete it again later and explain then. I hope that I have laid out a little bit of a more sane framework for the discussion of Zionism as an ideology, its history, etc., although I'm nowhere close to done on just the framework questions. The thing that amazes me is there's so much marvelous fact to go through, so many books, so many records, and yet we have next to nothing about the history of Zionism, its pioneers or its institutions here. DanKeshet

 * Your edits are fine. I just am restoring some material that emphasizes the diversity of views among Zionists.  One of my big concerns, not just on this issue but on science and philosophy issues, is correcting widely-held misperceptions.  In this case, many people believe that Zionists all have a common agenda, while in reality little could be further from the truth. (e.g. capitalists vs. communists, left-wing Peace Now types versus Greater Israel expansionists, etc.) RK 00:42 14 Jun 2003 (UTC)

 * RK, I agree with you that it's very important to present truth in situations where there a lot of misconceptions. But I would rather show the differences in Zionist ideologies than tell the reader that they exist.  For example, our text should describe Ber Borochov and his views and also describe Vladimir Jabotinsky and his views.  This would do the reader a far greater service than merely saying that there are a variety of views.

 * Also, I don't think it's appropriate to classify Zionist thought on the capitalist-socialist political axis. Much of Zionist (and Israeli) political thought revolves around different issues (e.g. the Arab question, or secular vs. religious), so "right-left" serves to obfuscate rather than illustrate the issues involved.  Where does Shinui fit on the American political axis?  Nowhere.  Even the "socialist" Zionists have a different understanding of socialism than most other socialists.

 * Finally, just as a style point: I note that in some of the text (e.g. the first paragraph), you have a tendency to repeat the same point in consecutive sentences. In each of these cases, I have tended to combine the sentences, or simply delete one of them, in the interests of concise writing.  Peace, DanKeshet

I am wondering how, or even if, any of the following should be used within this article: RK 00:57 14 Jun 2003 (UTC)

 * Zionism recognizes that Jewish peoplehood is characterized by certain common values relating to religion, culture, language, history and basic ideals and aspirations, although secular and religious Zionists emphasize these aspects differently. Zionists hold that Israel should be a historically, culturally and religiously Jewish state. This does not mean that all citizens must be Jews, nor does it mean that Jewish citizens must be religiously observant. As defined by the documents and practices of Zionists over the past century, what Zionism means in practice is that Israel is Jewish in much the same way that Saudi Arabia is Arab. About 80% of Israelis are Jews; the rest are Muslim Arabs, Christian Arabs, Druze and Bedouin, with other ethnic and religious minorities.

DanKeshet, why are you deleting material on the variety of beliefs and programs that exist within Zionism? This is very pertinent to the topic, and it corrects some widely held misperceptions. This is the second time you have deleted nearly all of this material without comment, and without offering an alternative. I am very willing to work with you; please comment. RK 23:23 15 Jun 2003 (UTC)

I have proposed an alternative to this article at User:Adam Carr/Zionism Adam 15:12, 18 Oct 2003 (UTC)

title: Talk:Zionism/Archive 2


text:
KIIS 101.1

KIIS 1011 (official callsign: 3TTT) is a commercial FM radio station broadcasting in Melbourne, Victoria, Australia, on a frequency of 101.1 MHz and is the Melbourne affiliate of ARN's KIIS Network. The station was formerly known as 3DB and 3TT, broadcasting on 1026 kHz AM, before converting to FM in 1990.

3DB
3DB was awarded its broadcast licence in 1925 but did not immediately come on air due to debates in Federal Parliament as to whether or not Australia should proceed with the then-unique system of A Class (later ABC) and B Class (later commercial) stations. The station was originally owned by Druleigh Business and Technical College Pty Ltd, hence the callsign DB. 3DB eventually made its on-air debut on 21 February 1927 in a fairly low-key manner.

From 1929 until 1987, 3DB was owned and operated by the Herald & Weekly Times (HWT), owners of The Herald and The Sun News-Pictorial, then separate newspapers.

For most of this era 3DB was one of Melbourne's most popular stations, topping ratings surveys for some decades. 3DB was extremely popular in all fields in which it broadcast. Music ranged from pop music and Top 40 to classical, some of it live. The station was also a big producer of radio drama (including soap operas), live variety programs, quizzes, etc. 3DB also transmitted many sporting events, particularly horse racing, VFL football, and test cricket from England. 3DB's women's programs and children's session were also popular.

One of the most important people in 3DB's history was David Thomas Worrall, who was appointed manager of the station by the Herald and Weekly Times' Sir Keith Murdoch in 1929, and who was to remain as manager until 1958. He was responsible for introducing many big budget programs.

From 1942 onwards, 3DB was the radio broadcaster of the Royal Children's Hospital's Good Friday Appeal and played a major part in promoting the appeal, with the sums raised annually increasing dramatically from 1942 onwards.

After purchasing the Geelong Advertiser and its subsidiary 3GL in 1987, the Herald and Weekly Times owned more than one station in the Melbourne/Geelong region and, thus, was required by the Australian Broadcasting Tribunal to divest itself of one station. 3DB was sold and actually had four owners for short periods in the 12 months of 1987.

J Albert & Son, a Sydney based music company and owners of 2UW, purchased 3DB in late 1987 and changed the call sign to 3TT. The last program to be broadcast under the 3DB call sign, in the very early hours of 2 April 1988, was a repeat of Bert Newton's 1987 documentary on the 60th anniversary of 3DB.

3TT
The 3TT station name and a classic hits format were launched at 5.00 am on 2 April 1988. The first song played was The Doobie Brothers "Listen to the Music". The major reason cited for the change of name was the fact that 3DB had recently broadcast under the slogan 3DB - The New Beginning and it was felt that that would have made it difficult to dramatically change the format whilst retaining the name, 3DB. However, the new owners had actually considered a few other new call-signs, particularly names which included the same initial being repeated twice, such as 3BB or 3MM.

Lawrence Costin who had been on the 3DB announcing staff for over 30 years was the only 3DB on-air personality to be retained by 3TT; as a newsreader and reporter.

From AM to FM
At midnight on 23-24 June, the station converted to FM broadcasting on 101.1 MHz. 3DB/3TT's old AM frequency of 1026 AM was returned to the government, and from 1991, it was used by the ABC for 3PB, which broadcast intermittently with live coverage of federal parliament. In 1994, 3PB became a full-time news and parliament station ABC NewsRadio.

The events leading up to the transference of 3TT's licence to the FM radio band are quite complicated and political. Broadcasting on the FM band commenced in the US as early as 1933. There had been Australian experiments on the FM band in earlier years, but in the 1970s the Whitlam Labor government decided to issue the first permanent FM licences in Australia to community radio stations. The first such licence in Melbourne was issued to 3MBS in 1975. The Fraser coalition government continued the Whitlam policy and expanded the range of FM broadcasters, firstly by introducing the Australia-wide ABC FM network. This was followed by a number of brand new commercial stations; the first to open in Melbourne being EON FM.

Not surprisingly, there were protests from commercial stations on the AM band because they were not able to bid for stations on the FM band. This was seen by many as a big problem as FM was then seen as the future of radio broadcasting, and many stations who had pioneered the commercial side of the Australian industry saw themselves as being left out of its future development, even though many of them - including 3TT - had a history dating back to the 1920s. The Fraser government was not prepared to allocate FM licences to all AM stations, which is what the industry wanted, and what had occurred in many overseas countries. However, under some pressure, in the late 1980s the government decided to transfer one or two stations in the major cities from the AM to FM band, using a much criticised auction-style system in which those stations wishing to transfer bands were to put in bids, with the highest bids being accepted. In Melbourne the two highest bids were to be accepted, with six of the seven existing AM stations placing bids. (The exception being 3AW which argued that its older listening-base preferred the AM band to the FM band.)

The three highest bids were received from: 3KZ - $32 million; 3AK - $22 million; and 3TT - $11 million. The vast disparity in the bids suggests that there was no collusion between stations, and virtually no industrial espionage. Further, 3XY which had put in an even lower bid was quite convinced that it would get one of the two FM licences being offered.

In 1987, Kerry Packer of Australian Consolidated Press had sold the Nine Television Network to Alan Bond, and 3AK was included in the deal, even though Bond had made it clear that he was interested in the TV stations but not in radio - he therefore defaulted on his payment for an FM licence. The second Melbourne FM licence on offer was, therefore, eventually given to the third-highest bidder, 3TT!

TT-FM
Initially from opening on 24 June 1990, 3TTT retained the format of 3TT, using the on-air name 101.1 TT-FM. The first song that aired on TT-FM was "In the Air Tonight" by Phil Collins. In 1993, TT-FM was relaunched as an adult contemporary music station.

Mix 101.1
In June 2001, the station rebranded to Mix 101.1 after joining the Mix Network. The station was similar to Mix 106.5 in Sydney in creating an emphasis on more music and less talk.

The last song that aired on Mix 101.1 was "Bye Bye Bye" by NSYNC.

KIIS 101.1
In November 2014, the Australian Radio Network announced that Mix 101.1 would rebrand to KIIS 101.1 on 19 January 2015 with a new breakfast and drive show. Matt & Jane replaced Chrissie & Jane in the station's breakfast timeslot, with existing presenter Jane Hall to remain alongside former Fox FM breakfast and Triple M drive host Matt Tilley. Rosso on Drive was replaced by Hughesy & Kate, hosted by Dave Hughes and Kate Langbroek and formerly of Nova 100 breakfast. Both shows began on 19 January 2015, with the latter being networked across the KIIS Network of stations.

In October 2015, Jane Hall announced that she would be leaving the station and Matt & Jane at the end of the year to return to acting. She was replaced by Nova 100 breakfast co-host Meshel Laurie.

In October 2017, ARN announced that Matt Tilley, Meshel Laurie, Dave Hughes and Kate Langbroek would leave the station at the end of the year. Matt & Meshel was replaced by Jason 'Jase' Hawkins and Polly 'PJ' Harding. Will McMahon and Woody Whitelaw from Hit 92.9 have been announced as the new drive hosts. Hughesy & Kate moved to Southern Cross Austereo's Hit Network to replace Hamish & Andy.

In July 2021, ARN announced that Jason 'Jase' Hawkins would be joined by Lauren Phillips with Jase & Lauren starting 9 August following Polly 'PJ' Harding's departure from the station on July 23, 2021.

In November 2023, after months of media speculation, it was announced that The Kyle and Jackie O Show would be networked from KIIS 106.5 Sydney in 2024 effectively ending Jase & Lauren. The final show aired on Friday 1 December 2023. In March 2024, Jase & Lauren replaced Ben, Liam & Belle to host breakfast on Nova 100.

In March 2024, ARN announced changes to its daytime programming with Gordie Waters being networked from Sydney into Melbourne and KIIS 97.3 in Brisbane from April. Byron Cooke had previously announced his resignation.

Studios
3DB's original studio was on the 10th floor of Capitol House, above the Capitol Theatre, Swanston Street, next to the Druleigh Business and Technical College, original owners of the station. In 1929, a short while after the HWT purchased 3DB, the studios and offices were moved to 74 Flinders Street, to the east of the HWT building on the corner of Flinders and Exhibition Streets. In 1966 the HWT truck dock was expanded, and in the process 74 Flinders Street was pulled down, with the station then moving to 61 Flinders Lane, immediately behind the HWT building.

When 3DB became 3TT the station continued to broadcast from 61 Flinders Lane, but within a short time, in December 1988, 3TT moved to new studios in Queensbridge Street, South Melbourne.

The station is now located in the inner suburb of Richmond in the Pelaco building, sharing facilities with sister station Gold 104.3.

title: KIIS 101.1