﻿ "Orientable" related terms, short phrases and links Web keywen.com
Orientable       Article     History   Tree Map
 Encyclopedia of Keywords > Science > Mathematics > Topology > Manifold > Orientable Michael Charnine

 Keywords and Sections
 Review of Short Phrases and Links

This Review contains major "Orientable"- related terms, short phrases and links grouped together in the form of Encyclopedia article.

### DefinitionsUpDw('Definitions','-Abz-');

1. If E is orientable, the structure group can be reduced further to SO(n) by a choice of orientation.

### Tangent BundleUpDw('TANGENT_BUNDLE','-Abz-');

1. Note that as a manifold in its own right, the tangent bundle is always orientable, even over nonorientable manifolds.

### Non-Orientable SurfacesUpDw('NON-ORIENTABLE_SURFACES','-Abz-');

1. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.

### Top Homology GroupUpDw('TOP_HOMOLOGY_GROUP','-Abz-');

1. A compact n -manifold M is orientable if and only if the top homology group,, is isomorphic to.

### Oriented Orthonormal Frame BundleUpDw('ORIENTED_ORTHONORMAL_FRAME_BUNDLE','-Abz-');

1. If M is orientable, then one also has the oriented orthonormal frame bundle F SO M.

### Seifert Fiber SpaceUpDw('SEIFERT_FIBER_SPACE','-Abz-');

1. Any manifold modeled on is orientable, and is a circle bundle over some 2-dimensional hyperbolic orbifold (a Seifert fiber space).

### Non-OrientableUpDw('NON-ORIENTABLE','-Abz-');

1. A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established.

### Orientable IffUpDw('ORIENTABLE_IFF','-Abz-');

1. On Surfaces A manifold obtained by identification is orientable iff all face identifications relative to a consistent orientation are orientation-reversing.

### FamiliesUpDw('FAMILIES','-Abz-');

1. The surfaces in the first two families are orientable.

### ClaimUpDw('CLAIM','-Abz-');

1. But there's one other problem with your claim, which is that if your manifold isn't orientable then the integer top-dimensional homology is 0.
2. On the top of page 6 of the above given PDF file, he makes the claim that every orientable real 2-manifold can be turned into a Riemann surface.

### PossibleUpDw('POSSIBLE','-Abz-');

1. In mathematics, a manifold or space is orientable if and only if it is possible to define left- and right-directions globally throughout that space.

### SenseUpDw('SENSE','-Abz-');

1. For example, a s p i n c -structure on a manifold turns out to be precisely what is needed to be orientable in the sense of complex topological k-theory.

### TypeUpDw('TYPE','-Abz-');

1. Depending on the type of field, field labels may or may not be orientable.

### ClassificationUpDw('CLASSIFICATION','-Abz-');

1. In this talk, I give a classification of (orientable and nonorientable) regular embeddings of graphs of order \$pq\$, where \$p\$ and \$q\$ are primes.

### ColoredUpDw('COLORED','-Abz-');

1. An external vertex is colored with the field label of its incident edge (and if the field label is orientable, with a head or a tail).

### GeneralizationUpDw('GENERALIZATION','-Abz-');

1. The theories for which such generalization is possible are called complex orientable.

### EmbeddedUpDw('EMBEDDED','-Abz-');

1. A surface that is embedded in R 3 will be orientable in the sense if and only if it is orientable as an abstract surface.

### EquivalentUpDw('EQUIVALENT','-Abz-');

1. In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as above) is orientable.
2. Topologically speaking, both versions of this Riemann surface are equivalent - they are orientable two-dimensional surfaces of genus one.

### ExactlyUpDw('EXACTLY','-Abz-');

1. Such a choice is only possible if the surface is orientable, and in this case there are exactly two different orientations.

### HomeomorphismUpDw('HOMEOMORPHISM','-Abz-');

1. Then NI(h) = 0 for every homeomorphism h: M! X. Notice that X is orientable since H1(X) is torsion free, and so the normal i* *n- variant NI(h) is defined.

### GloballyUpDw('GLOBALLY','-Abz-');

1. The Gauss map can be defined (globally) if and only if the surface is orientable, but it is always defined locally (i.e.

### GeometryUpDw('GEOMETRY','-Abz-');

1. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable.
2. Finite volume manifolds with this geometry have the structure of a Seifert fiber space if they are orientable.

### SphereUpDw('SPHERE','-Abz-');

1. Sphere s, planes, and tori are orientable, for example.
2. Take for instance a sphere: we can distinguish between the inside and the outside, and therefore the sphere is orientable.
3. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are orientable manifolds.

### CoveringsUpDw('COVERINGS','-Abz-');

1. Orientable double coverings of manifolds.

### MeansUpDw('MEANS','-Abz-');

1. Nota Bene: This means that all complex projective spaces are orientable.

### IntuitivelyUpDw('INTUITIVELY','-Abz-');

1. In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides".

### FactUpDw('FACT','-Abz-');

1. Note that whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise.

### InfinitelyUpDw('INFINITELY','-Abz-');

1. Abstract: We show that contact homology distinguishes infinitely many tight contact structures on any orientable, toroidal, irreducible 3-manifold.

### Fundamental GroupUpDw('FUNDAMENTAL_GROUP','-Abz-');

1. It follows from the Synge theorem that the fundamental group of such manifold in even dimensions is 0, if orientable and otherwise.

### TrivialUpDw('TRIVIAL','-Abz-');

1. Since S n is orientable, ν is trivial.

### Vector BundleUpDw('VECTOR_BUNDLE','-Abz-');

1. If the vector bundle E is orientable then one can reduce the structure group of Cℓ(E) from O(n) to SO(n) in the natural manner.

### Differentiable ManifoldUpDw('DIFFERENTIABLE_MANIFOLD','-Abz-');

1. It also defines a measure, but exists on any differentiable manifold, orientable or not.

### Cotangent BundleUpDw('COTANGENT_BUNDLE','-Abz-');

1. As a result, the cotangent bundle, considered as a manifold itself, is always orientable.
2. We showed that the cotangent bundle was orientable, but the only fact we used was that it was symplectic.

### Klein BottleUpDw('KLEIN_BOTTLE','-Abz-');

1. Not all manifolds are orientable, as exemplified by the Möbius strip and the Klein bottle, illustrated above.
2. Like the Möbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable.
3. Like the M�bius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable.

### Volume FormUpDw('VOLUME_FORM','-Abz-');

1. Thus 1-densities are a generalization of the notion of a volume form that does not necessarily require the manifold to be oriented or even orientable.
2. In fact, a manifold has a volume form if and only if it is orientable.

### Symplectic ManifoldUpDw('SYMPLECTIC_MANIFOLD','-Abz-');

1. So, every symplectic surface must be orientable, but in fact this is true for any symplectic manifold.
2. Also, this means that any symplectic manifold must be orientable.

### ClosedUpDw('CLOSED','-Abz-');

1. A 3-manifold is Haken if it is closed, orientable, and has no essential sphere, but has an incompressible surface.

### CoverUpDw('COVER','-Abz-');

1. This covering space is called the orientable double cover, as it is orientable.

### SmoothUpDw('SMOOTH','-Abz-');

1. If the manifold is smooth and orientable, the ﬁeld Im (x) is single-valued.

### BundleUpDw('BUNDLE','-Abz-');

1. Thus, w 1(E) = 0 if and only if the bundle E → X is orientable.

### Euler CharacteristicUpDw('EULER_CHARACTERISTIC','-Abz-');

1. The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic.

### Complex ManifoldsUpDw('COMPLEX_MANIFOLDS','-Abz-');

1. For instance, complex manifolds, including varieties, and also symplectic manifolds are orientable.

### Smooth ManifoldUpDw('SMOOTH_MANIFOLD','-Abz-');

1. The tangent bundle of a smooth manifold is orientable as a manifold (regardless of whether or not the manifold itself is).
2. Then X is an orientable smooth manifold of real dimension 2 n, so its cohomology groups lie in degrees zero through 2 n.

### CorollaryUpDw('COROLLARY','-Abz-');

1. As a corollary, every Lie group is orientable.
2. As a corollary, any symplectic manifold is orientable (indeed, oriented).

### CompactUpDw('COMPACT','-Abz-');

1. Abstract: Every finitely presented group G can be realized as the fundamental group of a (smooth, compact, connected, orientable) 4-dimensional manifold.
2. This has complex dimension n, but topological dimension, as a real manifold, 2 n, and is compact, connected, and orientable.
3. Corollary: A compact, orientable, even-dimensional manifold with positive sectional curvatures is simply connected.

### OrientationUpDw('ORIENTATION','-Abz-');

1. A bundle is called orientable if there exists an orientation.
2. The Riemann surface is orientable, since every orientation of a sheet is carried over to the sheet next to it.
3. The orientation character is thus: the non-trivial loop in acts as (- 1) n + 1 on orientation, so is orientable iff n+1 is even, i.e., n is odd.

### OrientationsUpDw('ORIENTATIONS','-Abz-');

1. For example, not every manifold is orientable, and those that are orientable admit exactly two orientations.

### CategoriesUpDw('Categories','-Abz-');

1. Science > Mathematics > Topology > Manifold
2. Riemann Surface
3. Cotangent Bundle
4. Two-Dimensional
5. Tangent Bundle

### Related Keywords

* Manifold * Manifolds * Orientable Manifold * Orientable Surface * Riemann Surface * Riemann Surfaces * Surface * Surfaces * Three-Dimensional Space
1. Books about "Orientable" in Amazon.com Short phrases about "Orientable"   Originally created: April 04, 2011.   Please send us comments and questions by this Online Form   Please click on to move good phrases up.
0.0201 sec. a=1..
{"ft_lang_label":"__label__en","ft_lang_prob":0.87458706,"math_prob":0.70894545,"size":7660,"snap":"2022-27-2022-33","text_gpt3_token_len":1812,"char_repetition_ratio":0.2088558,"word_repetition_ratio":0.04552469,"special_character_ratio":0.21762402,"punctuation_ratio":0.11513859,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95878816,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-19T02:07:35Z\",\"WARC-Record-ID\":\"<urn:uuid:5b15ef0f-a193-4c35-830f-c0f9d462e902>\",\"Content-Length\":\"31858\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:001515c1-55ff-435d-a14b-4bde38d5ade1>\",\"WARC-Concurrent-To\":\"<urn:uuid:5c666ec4-965a-4bd1-93ac-5be2b87031da>\",\"WARC-IP-Address\":\"83.149.227.181\",\"WARC-Target-URI\":\"http://keywen.com/en/ORIENTABLE\",\"WARC-Payload-Digest\":\"sha1:AAWNXUKPHUS2HX2BXEPZN7W5ZDKY57IA\",\"WARC-Block-Digest\":\"sha1:BSDS7V7442VU2NCSKEOC7CAD7AYLBQKX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573540.20_warc_CC-MAIN-20220819005802-20220819035802-00187.warc.gz\"}"}