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67-645: The Foumban shear zone is a series of faults associated with major mylonite zones, a segment of the CASZ. The CASZ can be traced from the Sudan to the Adamawa plateau , after which its path is obscured by volcanoes. Based on reconstruction of the configuration of South America before it separated from Africa, the zone can be identified with the Pernambuco fault. The shear zone underlies a chain of active volcanoes, called

134-476: A U {\displaystyle {\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 F {\displaystyle {\mathcal {F}}} . Since Q

201-499: A , φ α } α ∈ A {\displaystyle {\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 U {\displaystyle {\mathcal {U}}} -plaque P 0 such that x , y ∈ P 0 or there

268-399: A , meets U in either the empty set or a countable collection of subspaces whose images under φ {\displaystyle \varphi } in φ ( M a ∩ U ) {\displaystyle \varphi (M_{a}\cap U)} are p -dimensional affine subspaces whose first n − p coordinates are constant. Locally, every foliation

335-406: A dimension - p foliation F {\displaystyle {\mathcal {F}}} of an n -dimensional manifold M that is a covered by charts U i together with maps such that for overlapping pairs U i , U j the transition functions φ ij  : R → R defined by take the form where x denotes the first q = n − p coordinates, and y denotes

402-420: A coherent refinement that is regular. Fix a metric on M and a foliated atlas W . {\displaystyle {\mathcal {W}}.} Passing to a subcover , if necessary, one can assume that W = { W j , ψ j } j = 1 l {\displaystyle {\mathcal {W}}=\left\{W_{j},\psi _{j}\right\}_{j=1}^{l}}

469-1226: A finite subatlas U = { U i , φ i } i = 1 N {\displaystyle {\mathcal {U}}=\left\{U_{i},\varphi _{i}\right\}_{i=1}^{N}} of {( U x , φ x ) | x ∈ M }. If U i ∩ U j ≠ 0, then diam( U i ∪ U j ) < ε, and so there is an index k such that U ¯ i ∪ U ¯ j ⊆ W k . {\displaystyle {\overline {U}}_{i}\cup {\overline {U}}_{j}\subseteq W_{k}.} Distinct plaques of U ¯ i {\displaystyle {\overline {U}}_{i}} (respectively, of U ¯ j {\displaystyle {\overline {U}}_{j}} ) lie in distinct plaques of W k . Hence each plaque of U ¯ i {\displaystyle {\overline {U}}_{i}} has interior meeting at most one plaque of U ¯ j {\displaystyle {\overline {U}}_{j}} and vice versa. By construction, U {\displaystyle {\mathcal {U}}}

536-549: A foliation. For a slightly more geometrical definition, p -dimensional foliation F {\displaystyle {\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 , φ ) {\displaystyle (U,\varphi )} with U homeomorphic to R containing x such that every leaf, M

603-600: A foliation. More specifically, if U {\displaystyle {\mathcal {U}}} and V {\displaystyle {\mathcal {V}}} are foliated atlases on M and if U {\displaystyle {\mathcal {U}}} is associated to a foliation F {\displaystyle {\mathcal {F}}} then U {\displaystyle {\mathcal {U}}} and V {\displaystyle {\mathcal {V}}} are coherent if and only if V {\displaystyle {\mathcal {V}}}

670-489: 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. In order to give

737-443: A monoclinic symmetry which is directly related to the orientations of the finite strain axes. Although structures like asymmetric folds and boudinages are also related to the orientations of the finite strain axes, these structures can form from distinct strain paths and are not reliable kinematic indicators. Foliation In mathematics ( differential geometry ), a foliation is an equivalence relation on an n -manifold ,

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804-472: A more precise definition of foliation, it is necessary to define some auxiliary elements. A rectangular neighborhood in R is an open subset of the form B = J 1 × ⋅⋅⋅ × J n , where J i is a (possibly unbounded) relatively open interval in the i th coordinate axis. If J 1 is of the form ( a ,0], it is said that B has boundary In the following definition, coordinate charts are considered that have values in R × R , allowing

871-419: A neighborhood U and a system of local, class C coordinates x =( x , ⋅⋅⋅, x ) : U → R such that for each leaf L α , the components of U ∩ L α are described by the equations x =constant, ⋅⋅⋅, x =constant. A foliation is denoted by F {\displaystyle {\mathcal {F}}} ={ L α } α∈ A . The notion of leaves allows for an intuitive way of thinking about

938-488: A neighborhood in which the formula is independent of x β . 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 ( U α , φ α ) can meet multiple plaques of ( U β , φ β ). It can even happen that a plaque of one chart meets infinitely many plaques of another chart. However, no generality

1005-430: 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 F {\displaystyle {\mathcal {F}}} of codimension q and class C on M is a coherence class of foliated atlases of codimension q and class C on M . By Zorn's lemma , it is obvious that every coherence class of foliated atlases contains

1072-408: 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 F {\displaystyle {\mathcal {F}}} of codimension q . Let U = { U

1139-558: A rectangular neighborhood in R and B τ {\displaystyle B_{\tau }} a rectangular neighborhood in R . The set P y = φ ( B τ × { y }), where y ∈ B ⋔ {\displaystyle y\in B_{\pitchfork }} , is called a plaque of this foliated chart. For each x ∈ B τ , the set S x = φ ({ x } × B ⋔ {\displaystyle B_{\pitchfork }} )

1206-404: A regular, foliated atlas, P 0 meets only finitely many other plaques. That is, there are only finitely many plaque chains { P 0 , P i } of length 1. By induction on the length p of plaque chains that begin at P 0 , it is similarly proven that there are only finitely many of length ≤ p. Since every U {\displaystyle {\mathcal {U}}} -plaque in L is, by

1273-541: 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

1340-779: A subatlas, it is assumed that W = { W j , ψ j } j = 0 ∞ {\displaystyle {\mathcal {W}}=\left\{W_{j},\psi _{j}\right\}_{j=0}^{\infty }} is countable and a strictly increasing sequence { n l } l = 0 ∞ {\displaystyle \left\{n_{l}\right\}_{l=0}^{\infty }} of positive integers can be found such that W l = { W j , ψ j } j = 0 n l {\displaystyle {\mathcal {W}}_{l}=\left\{W_{j},\psi _{j}\right\}_{j=0}^{n_{l}}} covers K l . Let δ l denote

1407-399: A transition to diffusion creep can occur once the grain size is reduced sufficiently. Mylonites generally develop in ductile shear zones where high rates of strain are focused. They are the deep crustal counterparts to cataclastic brittle faults that create fault breccias . Determining the displacements that occur in mylonite zones depends on correctly determining the orientations of

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1474-553: A unique maximal foliated atlas. Thus, Definition. A foliation of codimension q and class C on M is a maximal foliated C -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

1541-474: Is has the formula Similar assertions hold also for open charts (without the overlines). The transverse coordinate map y α can be viewed as a submersion and the formulas y α = y α ( y β ) can be viewed as diffeomorphisms These satisfy the cocycle conditions . That is, on y δ ( U α ∩ U β ∩ U δ ), and, in particular, Using the above definitions for coherence and regularity it can be proven that every foliated atlas has

1608-597: 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 ( x ) for x ∈ Q . This definition describes

1675-623: Is a coherent refinement of W {\displaystyle {\mathcal {W}}} and is a regular foliated atlas. If M is not compact, local compactness and second countability allows one to choose a sequence { K i } i = 0 ∞ {\displaystyle \left\{K_{i}\right\}_{i=0}^{\infty }} of compact subsets such that K i ⊂ int K i +1 for each i ≥ 0 and M = ⋃ i = 1 ∞ K i . {\displaystyle M=\bigcup _{i=1}^{\infty }K_{i}.} Passing to

1742-642: Is a foliated C -atlas. Coherence of foliated atlases is an equivalence relation. Reflexivity and symmetry are immediate. To prove transitivity let U ≈ V {\displaystyle {\mathcal {U}}\thickapprox {\mathcal {V}}} and V ≈ W {\displaystyle {\mathcal {V}}\thickapprox {\mathcal {W}}} . Let ( U α , x α , y α ) ∈ U {\displaystyle {\mathcal {U}}} and ( W λ , x λ , y λ ) ∈ W {\displaystyle {\mathcal {W}}} and suppose that there

1809-599: 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 C (0 ≤ r ≤ ∞) on the n -manifold M is a C -atlas U = { ( U α , φ α ) ∣ α ∈ A } {\displaystyle {\mathcal {U}}=\{(U_{\alpha },\varphi _{\alpha })\mid \alpha \in A\}} of foliated charts of codimension q which are coherently foliated in

1876-400: Is a point w ∈ U α ∩ W λ . Choose ( V δ , x δ , y δ ) ∈ V {\displaystyle {\mathcal {V}}} such that w ∈ V δ . By the above remarks, there is a neighborhood N of w in U α ∩ V δ ∩ W λ such that and hence Since w ∈ U α ∩ W λ is arbitrary, it can be concluded that y α ( x λ , y λ )

1943-441: Is a sequence L = { P 0 , P 1 ,⋅⋅⋅, P p } of U {\displaystyle {\mathcal {U}}} -plaques such that x ∈ P 0 , y ∈ P p , and P i ∩ P i -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 ∈ P 0 , it is said that { P 0 } is a plaque chain of length 0 connecting x and y . The fact that ~

2010-810: Is a subset of W and φ = ψ | U then, if φ ( U ) = B τ × B ⋔ , {\displaystyle \varphi (U)=B_{\tau }\times B_{\pitchfork },} it can be seen that ψ | U ¯ {\displaystyle \psi |{\overline {U}}} , written φ ¯ {\displaystyle {\overline {\varphi }}} , carries U ¯ {\displaystyle {\overline {U}}} diffeomorphically onto B ¯ τ × B ¯ ⋔ . {\displaystyle {\overline {B}}_{\tau }\times {\overline {B}}_{\pitchfork }.} A foliated atlas

2077-491: Is a union of plaques and the foliation by plaques is tangent to the boundary. If ∂B τ ≠ ∅ = ∂ B ⋔ {\displaystyle B_{\pitchfork }} , then ∂U = ∂ ⋔ U {\displaystyle \partial _{\pitchfork }U} is a union of transversals and the foliation is transverse to the boundary. Finally, if ∂ B ⋔ {\displaystyle B_{\pitchfork }} ≠ ∅ ≠ ∂B τ , this

Foumban Shear Zone - Misplaced Pages Continue

2144-473: Is also associated to F {\displaystyle {\mathcal {F}}} . If V {\displaystyle {\mathcal {V}}} is also associated to F {\displaystyle {\mathcal {F}}} , every leaf L is a union of V {\displaystyle {\mathcal {V}}} -plaques and of U {\displaystyle {\mathcal {U}}} -plaques. These plaques are open subsets in

2211-438: Is an equivalence relation is clear. It is also clear that each equivalence class L is a union of plaques. Since U {\displaystyle {\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

2278-421: Is associated to F {\displaystyle {\mathcal {F}}} and that V ≈ U {\displaystyle {\mathcal {V}}\approx {\mathcal {U}}} , let Q be a V {\displaystyle {\mathcal {V}}} -plaque. If L is a leaf of F {\displaystyle {\mathcal {F}}} and w ∈ L ∩ Q , let P ∈ L be

2345-436: Is called a transversal of the foliated chart. The set ∂ τ U = φ ( B τ × ( ∂ B ⋔ {\displaystyle B_{\pitchfork }} )) is called the tangential boundary of U and ∂ ⋔ U {\displaystyle \partial _{\pitchfork }U} = φ (( ∂B τ ) × B ⋔ {\displaystyle B_{\pitchfork }} )

2412-452: Is called the transverse boundary of U . The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation B τ is read as " B -tangential" and B ⋔ {\displaystyle B_{\pitchfork }} as " B -transverse". There are also various possibilities. If both B ⋔ {\displaystyle B_{\pitchfork }} and B τ have empty boundary,

2479-409: 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 U {\displaystyle {\mathcal {U}}} -plaques in L is at most countably infinite. Fix one such plaque P 0 . By the definition of

2546-410: Is connected, L ∩ Q = Q . Finally, Q is an arbitrary V {\displaystyle {\mathcal {V}}} -plaque, and so V {\displaystyle {\mathcal {V}}} is associated to F {\displaystyle {\mathcal {F}}} . It is now obvious that the correspondence between foliations on M and their associated foliated atlases induces

2613-603: Is equivalent to requiring that, if U α ∩ U β ≠ ∅, the transverse coordinate changes y ¯ α = y ¯ α ( x ¯ β , y ¯ β ) {\displaystyle {\overline {y}}_{\alpha }={\overline {y}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)} be independent of x ¯ β . {\displaystyle {\overline {x}}_{\beta }.} That

2680-453: Is finite. Let ε > 0 be a Lebesgue number for W . {\displaystyle {\mathcal {W}}.} That is, any subset X ⊆ M of diameter < ε lies entirely in some W j . For each x ∈ M , choose j such that x ∈ W j and choose a foliated chart ( U x , φ x ) such that Suppose that U x ⊂ W k , k ≠ j , and write ψ k = ( x k , y k ) as usual, where y k  : W k → R

2747-571: Is generally agreed that crystal-plastic deformation must have occurred, and that fracturing and cataclastic flow are secondary processes in the formation of mylonites. Mechanical abrasion of grains by milling does not occur, although this was originally thought to be the process that formed mylonites, which were named from the Greek μύλος mylos , meaning mill. Mylonites form at depths of no less than 4 km. There are many different mechanisms that accommodate crystal-plastic deformation. In crustal rocks

Foumban Shear Zone - Misplaced Pages Continue

2814-513: Is locally independent of x λ . It is thus proven that U ≈ W {\displaystyle {\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 U ¯ {\displaystyle {\overline {U}}} (the closure of U )

2881-566: Is lost in assuming the situation to be much more regular as shown below. Two foliated atlases U {\displaystyle {\mathcal {U}}} and V {\displaystyle {\mathcal {V}}} on M of the same codimension and smoothness class C are coherent ( U ≈ V ) {\displaystyle \left({\mathcal {U}}\thickapprox {\mathcal {V}}\right)} if U ∪ V {\displaystyle {\mathcal {U}}\cup {\mathcal {V}}}

2948-796: Is said to be regular if By property (1), the coordinates x α and y α extend to coordinates x ¯ α {\displaystyle {\overline {x}}_{\alpha }} and y ¯ α {\displaystyle {\overline {y}}_{\alpha }} on U ¯ α {\displaystyle {\overline {U}}_{\alpha }} and one writes φ ¯ α = ( x ¯ α , y ¯ α ) . {\displaystyle {\overline {\varphi }}_{\alpha }=\left({\overline {x}}_{\alpha },{\overline {y}}_{\alpha }\right).} Property (3)

3015-532: Is taken n −1 = 0.) This creates a regular foliated atlas U = { U i , φ i } i = 1 ∞ {\displaystyle {\mathcal {U}}=\left\{U_{i},\varphi _{i}\right\}_{i=1}^{\infty }} that refines W {\displaystyle {\mathcal {W}}} and is coherent with W . {\displaystyle {\mathcal {W}}.} . Several alternative definitions of foliation exist depending on

3082-970: Is the transverse coordinate map. This is a submersion having the plaques in W k as level sets. Thus, y k restricts to a submersion y k  : U x → R . This is locally constant in x j ; so choosing U x smaller, if necessary, one can assume that y k | U ¯ x {\displaystyle {\overline {U}}_{x}} has the plaques of U ¯ x {\displaystyle {\overline {U}}_{x}} as its level sets. That is, each plaque of W k meets (hence contains) at most one (compact) plaque of U ¯ x {\displaystyle {\overline {U}}_{x}} . Since 1 < k < l < ∞, one can choose U x so that, whenever U x ⊂ W k , distinct plaques of U ¯ x {\displaystyle {\overline {U}}_{x}} lie in distinct plaques of W k . Pass to

3149-555: The Cameroon Volcanic Line . In August 1986 a magnitude 5 earthquake with epicenter near Lake Nyos indicated that the shear zone may be again reactivating. 6°37′0″N 12°52′0″E  /  6.61667°N 12.86667°E  / 6.61667; 12.86667 This palaeogeography article is a stub . You can help Misplaced Pages by expanding it . Mylonite Mylonite is a fine-grained, compact metamorphic rock produced by dynamic recrystallization of

3216-554: The equivalence classes being connected, injectively immersed submanifolds , all of the same dimension p , modeled on the decomposition of the real coordinate space R into the cosets x + R of the standardly embedded subspace R . 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 C ), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In

3283-421: 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 , U iy ⊂ R , 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 } is a submanifold of U i that intersects every plaque exactly once. This

3350-748: The compact case, requiring that U ¯ x {\displaystyle {\overline {U}}_{x}} be a compact subset of W j and that φ x = ψ j | U x , some j ≤ n l . Also, require that diam U ¯ x {\displaystyle {\overline {U}}_{x}} < ε l /2. As before, pass to a finite subcover { U i , φ i } i = n l − 1 + 1 n l {\displaystyle \left\{U_{i},\varphi _{i}\right\}_{i=n_{l-1}+1}^{n_{l}}} of K l ╲ {\displaystyle \diagdown } int K l -1 . (Here, it

3417-407: The constituent minerals resulting in a reduction of the grain size of the rock. Mylonites can have many different mineralogical compositions; it is a classification based on the textural appearance of the rock. Mylonites are ductilely deformed rocks formed by the accumulation of large shear strain , in ductile fault zones. There are many different views on the formation of mylonites, but it

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3484-493: The coordinates formula can be changed as The condition that ( U α , x α , y α ) and ( U β , x β , y β ) be coherently foliated means that, if P ⊂ U α is a plaque, the connected components of P ∩ U β lie in (possibly distinct) plaques of U β . Equivalently, since the plaques of U α and U β are level sets of the transverse coordinates y α and y β , respectively, each point z ∈ U α ∩ U β has

3551-453: The definition of ~, reached by a finite plaque chain starting at P 0 , 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

3618-926: The distance from K l to ∂ K l +1 and choose ε l > 0 so small that ε l < min{δ l /2,ε l -1 } for l ≥ 1, ε 0 < δ 0 /2, and ε l is a Lebesgue number for W l {\displaystyle {\mathcal {W}}_{l}} (as an open cover of K l ) and for W l + 1 {\displaystyle {\mathcal {W}}_{l+1}} (as an open cover of K l +1 ). More precisely, if X ⊂ M meets K l (respectively, K l +1 ) and diam X < ε l , then X lies in some element of W l {\displaystyle {\mathcal {W}}_{l}} (respectively, W l + 1 {\displaystyle {\mathcal {W}}_{l+1}} ). For each x ∈ K l ╲ {\displaystyle \diagdown } int K l -1 , construct ( U x , φ x ) as for

3685-431: The finite strain axis and inferring how those orientations change with respect to the incremental strain axis. This is referred to as determining the shear sense. It is common practice to assume that the deformation is plane strain simple shear deformation. This type of strain field assumes that deformation occurs in a tabular zone where displacement is parallel to the shear zone boundary. Furthermore, during deformation

3752-537: The finite strain axes with respect to the incremental strain axes. Because of the constraints imposed by simple shear, displacement is assumed to occur in the foliation plane in a direction parallel to the mineral stretching lineation. Therefore, a plane parallel to the lineation and perpendicular to the foliation is viewed to determine the shear sense. The most common shear sense indicators are C/S fabrics, asymmetric porphyroclasts, vein and dike arrays, mantled porphyroclasts and mineral fibers. All of these indicators have

3819-415: 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 ⋔ {\displaystyle B_{\pitchfork }} ≠ ∅ = ∂B τ , then ∂U = ∂ τ U

3886-399: The incremental strain axis maintains a 45-degree angle to the shear zone boundary. The finite strain axes are initially parallel to the incremental axis, but rotate away during progressive deformation. Kinematic indicators are structures in mylonites that allow the sense of shear to be determined. Most kinematic indicators are based on deformation in simple shear and infer sense of rotation of

3953-1056: The last p co-ordinates. That is, The splitting of the transition functions φ ij into φ i j 1 ( x ) {\displaystyle \varphi _{ij}^{1}(x)} and φ i j 2 ( x , y ) {\displaystyle \varphi _{ij}^{2}(x,y)} as a part of the submersion is completely analogous to the splitting of g ¯ α β {\displaystyle {\overline {g}}_{\alpha \beta }} into y ¯ α ( y ¯ β ) {\displaystyle {\overline {y}}_{\alpha }\left({\overline {y}}_{\beta }\right)} and x ¯ α ( x ¯ β , y ¯ β ) {\displaystyle {\overline {x}}_{\alpha }\left({\overline {x}}_{\beta },{\overline {y}}_{\beta }\right)} as

4020-414: 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

4087-438: The manifold topology of L , hence intersect in open subsets of each other. Since plaques are connected, a U {\displaystyle {\mathcal {U}}} -plaque cannot intersect a V {\displaystyle {\mathcal {V}}} -plaque unless they lie in a common leaf; so the foliated atlases are coherent. Conversely, if we only know that U {\displaystyle {\mathcal {U}}}

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4154-535: The misorientation across that subgrain boundary will increase until the boundary becomes a high-angle boundary and the subgrain effectively becomes a new grain. This process, sometimes referred to as subgrain rotation recrystallization , acts to reduce the mean grain size. Volume and grain-boundary diffusion, the critical mechanisms in diffusion creep, become important at high temperatures and small grain sizes. Thus some researchers have argued that as mylonites are formed by dislocation creep and dynamic recrystallization,

4221-411: The most important case of differentiable foliation of class C it is usually understood that r ≥ 1 (otherwise, C 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

4288-501: The most important processes are dislocation creep and diffusion creep . Dislocation generation acts to increase the internal energy of crystals. This effect is compensated through grain-boundary-migration recrystallization which reduces the internal energy by increasing the grain boundary area and reducing the grain volume, storing energy at the mineral grain surface. This process tends to organize dislocations into subgrain boundaries . As more dislocations are added to subgrain boundaries,

4355-480: 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 φ : U → B τ × B ⋔ {\displaystyle \varphi :U\to B_{\tau }\times B_{\pitchfork }} is a diffeomorphism , B ⋔ {\displaystyle B_{\pitchfork }} being

4422-422: The sense that, whenever P and Q are plaques in distinct charts of U {\displaystyle {\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 ∈ U α ∩ U β The notation ( U α , φ α ) is often written ( U α , x α , y α ), with On φ β ( U α ∩ U β )

4489-422: 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 C foliation of an n -dimensional manifold M is a decomposition of M into a union of disjoint connected submanifolds { L α } α∈ A , called the leaves of the foliation, with the following property: Every point in M has

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