manu/mldr-zoomed-100char-total-queries-documents

chunk_id stringlengths 3 9	chunk stringlengths 1 100
64_136	which pushes forward to the fixed generator. Conversely, an oriented atlas determines such a
64_137	generator as compatible local orientations can be glued together to give a generator for the
64_138	homology group .
64_139	Orientation and cohomology
64_140	A manifold M is orientable if and only if the first Stiefel–Whitney class vanishes. In particular,
64_141	if the first cohomology group with Z/2 coefficients is zero, then the manifold is orientable.
64_142	Moreover if M is orientable and w1 vanishes, then parametrizes the choices of orientations. This
64_143	characterization of orientability extends to orientability of general vector bundles over M, not
64_144	just the tangent bundle.
64_145	The orientation double cover
64_146	Around each point of M there are two local orientations. Intuitively, there is a way to move from
64_147	a local orientation at a point to a local orientation at a nearby point : when the two points lie
64_148	in the same coordinate chart , that coordinate chart defines compatible local orientations at and
64_149	. The set of local orientations can therefore be given a topology, and this topology makes it into
64_150	a manifold.
64_151	More precisely, let O be the set of all local orientations of M. To topologize O we will specify a
64_152	subbase for its topology. Let U be an open subset of M chosen such that is isomorphic to Z.
64_153	Assume that α is a generator of this group. For each p in U, there is a pushforward function .
64_154	The codomain of this group has two generators, and α maps to one of them. The topology on O is
64_155	defined so that
64_156	is open.
64_157	There is a canonical map that sends a local orientation at p to p. It is clear that every point
64_158	of M has precisely two preimages under . In fact, is even a local homeomorphism, because the
64_159	preimages of the open sets U mentioned above are homeomorphic to the disjoint union of two copies
64_160	of U. If M is orientable, then M itself is one of these open sets, so O is the disjoint union of
64_161	two copies of M. If M is non-orientable, however, then O is connected and orientable. The
64_162	manifold O is called the orientation double cover.
64_163	Manifolds with boundary
64_164	If M is a manifold with boundary, then an orientation of M is defined to be an orientation of its
64_165	interior. Such an orientation induces an orientation of ∂M. Indeed, suppose that an orientation
64_166	of M is fixed. Let be a chart at a boundary point of M which, when restricted to the interior of
64_167	M, is in the chosen oriented atlas. The restriction of this chart to ∂M is a chart of ∂M. Such
64_168	charts form an oriented atlas for ∂M.
64_169	When M is smooth, at each point p of ∂M, the restriction of the tangent bundle of M to ∂M is
64_170	isomorphic to , where the factor of R is described by the inward pointing normal vector. The
64_171	orientation of Tp∂M is defined by the condition that a basis of Tp∂M is positively oriented if and
64_172	only if it, when combined with the inward pointing normal vector, defines a positively oriented
64_173	basis of TpM.
64_174	Orientable double cover
64_175	A closely related notion uses the idea of covering space. For a connected manifold M take M, the
64_176	set of pairs (x, o) where x is a point of M and o is an orientation at x; here we assume M is
64_177	either smooth so we can choose an orientation on the tangent space at a point or we use singular
64_178	homology to define orientation. Then for every open, oriented subset of M we consider the
64_179	corresponding set of pairs and define that to be an open set of M. This gives M a topology and the
64_180	projection sending (x, o) to x is then a 2-to-1 covering map. This covering space is called the
64_181	orientable double cover, as it is orientable. M is connected if and only if M is not orientable.
64_182	Another way to construct this cover is to divide the loops based at a basepoint into either
64_183	orientation-preserving or orientation-reversing loops. The orientation preserving loops generate a
64_184	subgroup of the fundamental group which is either the whole group or of index two. In the latter
64_185	case (which means there is an orientation-reversing path), the subgroup corresponds to a connected
64_186	double covering; this cover is orientable by construction. In the former case, one can simply take
64_187	two copies of M, each of which corresponds to a different orientation.
64_188	Orientation of vector bundles
64_189	A real vector bundle, which a priori has a GL(n) structure group, is called orientable when the
64_190	structure group may be reduced to , the group of matrices with positive determinant. For the
64_191	tangent bundle, this reduction is always possible if the underlying base manifold is orientable and
64_192	in fact this provides a convenient way to define the orientability of a smooth real manifold: a
64_193	smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector
64_194	bundle). Note that as a manifold in its own right, the tangent bundle is always orientable, even
64_195	over nonorientable manifolds.
64_196	Related concepts Lorentzian geometry
64_197	In Lorentzian geometry, there are two kinds of orientability: space orientability and time
64_198	orientability. These play a role in the causal structure of spacetime. In the context of general
64_199	relativity, a spacetime manifold is space orientable if, whenever two right-handed observers head
64_200	off in rocket ships starting at the same spacetime point, and then meet again at another point,
64_201	they remain right-handed with respect to one another. If a spacetime is time-orientable then the
64_202	two observers will always agree on the direction of time at both points of their meeting. In fact,
64_203	a spacetime is time-orientable if and only if any two observers can agree which of the two meetings
64_204	preceded the other.
64_205	Formally, the pseudo-orthogonal group O(p,q) has a pair of characters: the space orientation
64_206	character σ+ and the time orientation character σ−,
64_207	Their product σ = σ+σ− is the determinant, which gives the orientation character. A
64_208	space-orientation of a pseudo-Riemannian manifold is identified with a section of the associated
64_209	bundle
64_210	where O(M) is the bundle of pseudo-orthogonal frames. Similarly, a time orientation is a section
64_211	of the associated bundle
64_212	See also Curve orientation Orientation sheaf References
64_213	External links Orientation of manifolds at the Manifold Atlas.
64_214	Orientation covering at the Manifold Atlas.
64_215	Orientation of manifolds in generalized cohomology theories at the Manifold Atlas.
64_216	The Encyclopedia of Mathematics article on Orientation.
64_217	Differential topology Surfaces Articles containing video clips
64_218	de:Orientierung (Mathematik)#Orientierung einer Mannigfaltigkeit
65_0	The , usually translated as Elder, was one of the highest-ranking government posts under the
65_1	Tokugawa shogunate of Edo period Japan. The term refers either to individual Elders, or to the
65_2	Council of Elders as a whole; under the first two shōguns, there were only two Rōjū. The number was
65_3	then increased to five, and later reduced to four. The Rōjū were appointed from the ranks of the
65_4	fudai daimyōs with domains of between 25,000 and 50,000 koku.
65_5	Duties
65_6	The Rōjū had a number of responsibilities, most clearly delineated in the 1634 ordinance that
65_7	reorganized the government and created a number of new posts:
65_8	Relations with the Emperor, the Court, and the Prince-Abbots.
65_9	Supervision of those daimyō who controlled lands worth at least 10,000 koku.
65_10	Managing the forms taken by official documents in official communications.
65_11	Supervision of the internal affairs of the Shogun's domains. Coinage, public works, and enfiefment.
65_12	Governmental relations and supervision of monasteries and shrines.
65_13	Compilation of maps, charts, and other government records.
65_14	The Rōjū served not simultaneously, but in rotation, each serving the Shogun for a month at a time,
65_15	communicating with the Shogun through a chamberlain, called Soba-yōnin. However, the Rōjū also
65_16	served as members of the Hyōjōsho council, along with the Ō-Metsuke and representatives of various

64_136

which pushes forward to the fixed generator. Conversely, an oriented atlas determines such a

64_137

generator as compatible local orientations can be glued together to give a generator for the

64_138

homology group .

64_139

Orientation and cohomology

64_140

A manifold M is orientable if and only if the first Stiefel–Whitney class vanishes. In particular,

64_141

if the first cohomology group with Z/2 coefficients is zero, then the manifold is orientable.

64_142

Moreover if M is orientable and w1 vanishes, then parametrizes the choices of orientations. This

64_143

characterization of orientability extends to orientability of general vector bundles over M, not

64_144

just the tangent bundle.

64_145

The orientation double cover

64_146

Around each point of M there are two local orientations. Intuitively, there is a way to move from

64_147

a local orientation at a point to a local orientation at a nearby point : when the two points lie

64_148

in the same coordinate chart , that coordinate chart defines compatible local orientations at and

64_149

. The set of local orientations can therefore be given a topology, and this topology makes it into

64_150

a manifold.

64_151

More precisely, let O be the set of all local orientations of M. To topologize O we will specify a

64_152

subbase for its topology. Let U be an open subset of M chosen such that is isomorphic to Z.

64_153

Assume that α is a generator of this group. For each p in U, there is a pushforward function .

64_154

The codomain of this group has two generators, and α maps to one of them. The topology on O is

64_155

defined so that

64_156

is open.

64_157

There is a canonical map that sends a local orientation at p to p. It is clear that every point

64_158

of M has precisely two preimages under . In fact, is even a local homeomorphism, because the

64_159

preimages of the open sets U mentioned above are homeomorphic to the disjoint union of two copies

64_160

of U. If M is orientable, then M itself is one of these open sets, so O is the disjoint union of

64_161

two copies of M. If M is non-orientable, however, then O is connected and orientable. The

64_162

manifold O is called the orientation double cover.

64_163

Manifolds with boundary

64_164

If M is a manifold with boundary, then an orientation of M is defined to be an orientation of its

64_165

interior. Such an orientation induces an orientation of ∂M. Indeed, suppose that an orientation

64_166

of M is fixed. Let be a chart at a boundary point of M which, when restricted to the interior of

64_167

M, is in the chosen oriented atlas. The restriction of this chart to ∂M is a chart of ∂M. Such

64_168

charts form an oriented atlas for ∂M.

64_169

When M is smooth, at each point p of ∂M, the restriction of the tangent bundle of M to ∂M is

64_170

isomorphic to , where the factor of R is described by the inward pointing normal vector. The

64_171

orientation of Tp∂M is defined by the condition that a basis of Tp∂M is positively oriented if and

64_172

only if it, when combined with the inward pointing normal vector, defines a positively oriented

64_173

basis of TpM.

64_174

Orientable double cover

64_175

A closely related notion uses the idea of covering space. For a connected manifold M take M, the

64_176

set of pairs (x, o) where x is a point of M and o is an orientation at x; here we assume M is

64_177

either smooth so we can choose an orientation on the tangent space at a point or we use singular

64_178

homology to define orientation. Then for every open, oriented subset of M we consider the

64_179

corresponding set of pairs and define that to be an open set of M. This gives M a topology and the

64_180

projection sending (x, o) to x is then a 2-to-1 covering map. This covering space is called the

64_181

orientable double cover, as it is orientable. M is connected if and only if M is not orientable.

64_182

Another way to construct this cover is to divide the loops based at a basepoint into either

64_183

orientation-preserving or orientation-reversing loops. The orientation preserving loops generate a

64_184

subgroup of the fundamental group which is either the whole group or of index two. In the latter

64_185

case (which means there is an orientation-reversing path), the subgroup corresponds to a connected

64_186

double covering; this cover is orientable by construction. In the former case, one can simply take

64_187

two copies of M, each of which corresponds to a different orientation.

64_188

Orientation of vector bundles

64_189

A real vector bundle, which a priori has a GL(n) structure group, is called orientable when the

64_190

structure group may be reduced to , the group of matrices with positive determinant. For the

64_191

tangent bundle, this reduction is always possible if the underlying base manifold is orientable and

64_192

in fact this provides a convenient way to define the orientability of a smooth real manifold: a

64_193

smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector

64_194

bundle). Note that as a manifold in its own right, the tangent bundle is always orientable, even

64_195

over nonorientable manifolds.

64_196

Related concepts Lorentzian geometry

64_197

In Lorentzian geometry, there are two kinds of orientability: space orientability and time

64_198

orientability. These play a role in the causal structure of spacetime. In the context of general

64_199

relativity, a spacetime manifold is space orientable if, whenever two right-handed observers head

64_200

off in rocket ships starting at the same spacetime point, and then meet again at another point,

64_201

they remain right-handed with respect to one another. If a spacetime is time-orientable then the

64_202

two observers will always agree on the direction of time at both points of their meeting. In fact,

64_203

a spacetime is time-orientable if and only if any two observers can agree which of the two meetings

64_204

preceded the other.

64_205

Formally, the pseudo-orthogonal group O(p,q) has a pair of characters: the space orientation

64_206

character σ+ and the time orientation character σ−,

64_207

Their product σ = σ+σ− is the determinant, which gives the orientation character. A

64_208

space-orientation of a pseudo-Riemannian manifold is identified with a section of the associated

64_209

bundle

64_210

where O(M) is the bundle of pseudo-orthogonal frames. Similarly, a time orientation is a section

64_211

of the associated bundle

64_212

See also Curve orientation Orientation sheaf References

64_213

External links Orientation of manifolds at the Manifold Atlas.

64_214

Orientation covering at the Manifold Atlas.

64_215

Orientation of manifolds in generalized cohomology theories at the Manifold Atlas.

64_216

The Encyclopedia of Mathematics article on Orientation.

64_217

Differential topology Surfaces Articles containing video clips

64_218

de:Orientierung (Mathematik)#Orientierung einer Mannigfaltigkeit

65_0

The , usually translated as Elder, was one of the highest-ranking government posts under the

65_1

Tokugawa shogunate of Edo period Japan. The term refers either to individual Elders, or to the

65_2

Council of Elders as a whole; under the first two shōguns, there were only two Rōjū. The number was

65_3

then increased to five, and later reduced to four. The Rōjū were appointed from the ranks of the

65_4

fudai daimyōs with domains of between 25,000 and 50,000 koku.

65_5

Duties

65_6

The Rōjū had a number of responsibilities, most clearly delineated in the 1634 ordinance that

65_7

reorganized the government and created a number of new posts:

65_8

Relations with the Emperor, the Court, and the Prince-Abbots.

65_9

Supervision of those daimyō who controlled lands worth at least 10,000 koku.

65_10

Managing the forms taken by official documents in official communications.

65_11

Supervision of the internal affairs of the Shogun's domains. Coinage, public works, and enfiefment.

65_12

Governmental relations and supervision of monasteries and shrines.

65_13

Compilation of maps, charts, and other government records.

65_14

The Rōjū served not simultaneously, but in rotation, each serving the Shogun for a month at a time,

65_15

communicating with the Shogun through a chamberlain, called Soba-yōnin. However, the Rōjū also

65_16

served as members of the Hyōjōsho council, along with the Ō-Metsuke and representatives of various