达布定理 (微分几何)

达布定理数学领域微分几何中关于微分形式的一个定理,部分地推广了弗罗贝尼乌斯定理。它是包括辛几何在内多个领域的基石。这个定理以让·加斯东·达布[1] 命名,他在解 Pfaff 问题[2] 时建立了这个定理。

这个定理的推论之一是任何两个同维数的辛流形是局部辛同胚的。这就是说,任何 2n-维辛流形能局部的看作带标准辛形式的线性辛空间 Cn。应用于切触几何也有类似的结论。

定理的陈述和第一个推论

定理准确的陈述如下。[3] 设 θ 是一个 n 维流形上的 1-形式,使得 dθ 有常秩 p 。如果任一点都有

θ ∧ (dθ)p = 0 ,

那么有一个局部的坐标系 x1,...,xn-p, y1, ..., yp ,在这个坐标系下

θ = x1 dy1 + ... + xp : θ = x1 dy1 + ... + xp dyp + dxp+1

dyp。 另一个方面,如果任一点有

θ ∧ (dθ)p ≠ 0 任何处,

那么有一个局部坐标系 x1,...,xn-p, y1, ..., yp 使得

θ = x1 dy1 + ... + xp dyp + dxp+1.

特别的,设 ω 是 n=2m 维流形 M 上的一个辛 2-形式。M 上任一点 p 的局部,由 庞加莱引理,总有一个 1-形式 θ 满足 dθ=ω 。进一步 θ 满足达布定理的第一个假设,从而局部存在一个 p 附近的坐标卡 U 使得

θ = x1 dy1 + ... + xm dym

外导数便有

ω = dθ = dx1 ∧ dy1 + ... + dxm ∧ dym

坐标卡 U 称为 p 附近的达布坐标卡。[4] 流形 M 能被这样的卡覆盖

换一种方式叙述,将 R2mCm 等同起来,令 zj = xj + i yj。如果 φ : UCn是一个达布坐标卡,那么 ω 是标准辛形式 ω0Cn 上的拉回:

和黎曼几何的比较

这个结论意味着辛几何没有局部不变性:在任何一点附近,总能取一个达布基。这和黎曼几何具有显著的不同,高斯绝妙定理指出曲率是黎曼几何的一个局部不变量。曲率阻碍了将度量局部写成一个平方和。

必须要强调的是,达布定理是说 ω 能在 p 附近的“整个邻域”写成一个标准形式。黎曼几何中,度量总能在给定一“点”写成一个标准形式,但一般不能在那个点的邻域,除非局部为欧氏空间。

又见

注释

  1. ^ Darboux (1882).
  2. ^ Pfaff (1814-1815).
  3. ^ Sternberg (1964) p. 140-141.
  4. ^ Cf. with McDuff and Salamon (1998) p. 96.

参考文献

外部链接

Read other articles:

7th season of top-tier football league in Vyshcha Liha Football league seasonVyshcha LihaSeason1997–98ChampionsDynamo Kyiv6th titleRelegatedTorpedo, ChornomoretsChampions LeagueDynamo KyivCup Winners' CupCSKA KyivUEFA CupShakhtar DonetskTop goalscorer(22) Serhii Rebrov (Dynamo)← 1996–97 1998–99 → The 1997–98 Vyshcha Liha season was the 7th since its establishment. FC Dynamo Kyiv were the defending champions. Teams Promotions Metalurh Donetsk, the champion of the 1996–97 Ukrainia...

 

 

Jeju Air제주항공 Jeju Hanggong IATA ICAO Kode panggil 7C JJA JEJUAIR Didirikan2005PenghubungBandar Udara Internasional GimpoPenghubung sekunderBandar Udara Internasional GimpoProgram penumpang setiaJoin & Joy ClubArmada6 (+6 orders)Tujuan5SloganJoin & JoyPerusahaan indukAekyung GroupKantor pusat Jeju, Korea SelatanTokoh utamaKo, Young Sup (Direktur)Situs webhttp://www.jejuair.net/ Jeju AirHangul제주항공 Hanja濟州航空 Alih AksaraJeju HanggongMcCune–ReischauerCheju Hanggon...

 

 

Politeknik Negeri KetapangNama sebelumnyaPoliteknik Pangeran IranataJenisPerguruan Tinggi Negeri, PoliteknikDirekturIrianto Sastro Prawiro, S.ST., M.M.A.AlamatJalan Rangge Sentap, Desa Dalong, Kel. Sukaharja, Kec. Delta Pawan, Kabupaten Ketapang, Kalimantan Barat, IndonesiaSitus webhttps://politap.ac.id/ Politeknik Negeri Ketapang atau POLITAP adalah satu-satunya perguruan tinggi negeri di Kabupaten Ketapang, Provinsi Kalimantan Barat. Sejarah Politeknik Negeri Ketapang dirintis pendiriannya ...

Piala Liga Inggris 2005–20062005–06 Football League CupNegara Inggris WalesTanggal penyelenggaraan22 Agustus 2005 s.d. 26 Februari 2006Jumlah peserta92Juara bertahanChelseaJuaraManchester United(gelar ke-2)Tempat keduaWigan AthleticPencetak gol terbanyakLouis Saha(6 gol)← 2004–2005 2006–2007 → Piala Liga Inggris 2005–2006 adalah edisi ke-46 penyelenggaraan Piala Liga Inggris, sebuah kompetisi dengan sistem gugur untuk 92 tim terbaik di Inggris. Edisi ini dimenangkan ol...

 

 

Pokémon gym leader and character in the Pokémon anime series Fictional character BrockPokémon characterBrock in Pokémon the Series: Diamond and PearlFirst appearancePokémon Red and Blue (1996)Created byKen SugimoriDesigned byKen SugimoriAtsuko Nishida (anime)Portrayed byDennis KenneyVoiced by English Eric Stuart (anime, Season 1–8)Bill Rogers (anime, Season 9–present)Johnny Yong Bosch (Pokémon Origins)Tommy Arciniega (Pokémon Masters) Japanese Yūji Ueda (anime)Tomokazu Sugita (Pok...

 

 

2015 video game 2015 video gamePokkén TournamentPromotional art used in Wii U version's cover art depicting Pikachu and Lucario in its mega-evolved formDeveloper(s)Bandai Namco StudiosPublisher(s)JP: Bandai Namco Entertainment (arcade release)JP: The Pokémon Company (consoles)WW: Nintendo (consoles)Director(s)Haruki SuzakiProducer(s)Masaaki HoshinoKatsuhiro HaradaDesigner(s)Yasuhito KobayashiProgrammer(s)Sei NakataniArtist(s)Hiroyasu HosoyaHiromi WatanabeComposer(s) Hiroki Hashimoto Hiroyuk...

هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (سبتمبر 2019) فيصل فوزي معلومات شخصية الاسم الكامل فيصل فوزي أنور أمين تاريخ الميلاد 25 أغسطس 1996 (العمر 27 سنة) الطول 1.75 مركز اللعب لاعب وسط الجنسية  السعودية معلومات الن�...

 

 

Masjid Sultan Omar Ali Saifuddinمسجد السلطان عمر علي سيف الدينMasjid Sultan Omar Ali SaifuddinPandangan luar masjid pada siang hariAgamaAfiliasiIslam SunniLokasiLokasiBandar Seri Begawan, Brunei DarussalamNegara BruneiShown within BruneiKoordinat4°53′22″N 114°56′21″E / 4.8894°N 114.9392°E / 4.8894; 114.9392Koordinat: 4°53′22″N 114°56′21″E / 4.8894°N 114.9392°E / 4.8894; 114.9392ArsitekturTipe...

 

 

Iraqi politician Salah Al-Mukhtarصلاح المختارSecretary General of the National Command of the Arab Socialist Ba'ath PartyIncumbentAssumed office 3 November 2020Preceded byIzzat Ibrahim al-Douri Personal detailsBorn1944 (age 79–80)Baghdad, Kingdom of IraqPolitical partyIraqi Ba'athOccupationPolitician, diplomatMilitary serviceAllegiance IraqBranch/service Iraqi ArmyYears of servicec. late 1970s - c. late 1980sRank Colonel Salah Al-Mukhtar (Arabic: صلاح �...

La nieta elegidaGenreTelenovelaPembuat Julio Jiménez Iván Martínez Sutradara Rodrigo Lalinde Consuelo González Pemeran Consuelo Luzardo Francisca Estévez Carlos Torres Juliette Pardau Penata musikAlejandro DíazNegara asalKolombiaBahasa asliSpanyolJmlh. musim1Jmlh. episode72ProduksiProduser eksekutifNana VelásquezRumah produksiRCN TelevisiónRilis asliJaringanCanal RCNRilis27 September 2021 (2021-09-27) –1 Februari 2022 (2022-2-1) La nieta elegida adalah telenove...

 

 

この項目には、一部のコンピュータや閲覧ソフトで表示できない文字が含まれています(詳細)。 数字の大字(だいじ)は、漢数字の一種。通常用いる単純な字形の漢数字(小字)の代わりに同じ音の別の漢字を用いるものである。 概要 壱万円日本銀行券(「壱」が大字) 弐千円日本銀行券(「弐」が大字) 漢数字には「一」「二」「三」と続く小字と、「壱」「�...

 

 

此条目序言章节没有充分总结全文内容要点。 (2019年3月21日)请考虑扩充序言,清晰概述条目所有重點。请在条目的讨论页讨论此问题。 哈萨克斯坦總統哈薩克總統旗現任Қасым-Жомарт Кемелұлы Тоқаев卡瑟姆若马尔特·托卡耶夫自2019年3月20日在任任期7年首任努尔苏丹·纳扎尔巴耶夫设立1990年4月24日(哈薩克蘇維埃社會主義共和國總統) 哈萨克斯坦 哈萨克斯坦政府...

Національний військовий медичний центр імені Волтера Ріда Тип військовий шпитальЗасновано 1940Кількість ліжок 1200[1][2], 2464[1][2] і 244[3]Країна  СШААдреса Бетесда, МерілендКоординати 39°00′07″ пн. ш. 77°05′37″ зх. д. / 39.001944444472222528° �...

 

 

For other uses, see Clos (disambiguation). Standard method combination in ANSI common lisp The Common Lisp Object System (CLOS) is the facility for object-oriented programming in ANSI Common Lisp. CLOS is a powerful dynamic object system which differs radically from the OOP facilities found in more static languages such as C++ or Java. CLOS was inspired by earlier Lisp object systems such as MIT Flavors and CommonLoops, although it is more general than either. Originally proposed as an add-on...

 

 

Dislike or fear of Russia, its people or its culture For opposition towards the current government of Russia, see Opposition to Vladimir Putin in Russia. Anti-Russian sentiment or Russophobia, is dislike or fear or hatred of Russia, Russian people, Russian culture,[1] or Russian policy.[2] The Collins English Dictionary defines it as intense and often irrational hatred of Russia.[3] The opposite of Russophobia is Russophilia. Historically, Russophobia has included stat...

Hard object produced within a living shelled mollusc For the programming language, see Perl. For other uses, see Pearl (disambiguation). PearlVarious pearlsGeneralCategoryCarbonate mineral, proteinFormula(repeating unit) Calcium carbonate, CaCO3 Conchiolin Strunz classification05.8Crystal systemOrthorhombic[1]IdentificationColorWhite, pink, silver, cream, brown, green, blue, black, yellow, orange, red, gold, purple, iridescentCleavageNone[1]FractureUneven, variousMohs scale ha...

 

 

1526–1791 series of wars in Europe This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Ottoman–Habsburg wars – news · newspapers · books · scholar · JSTOR (November 2016) (Learn how and when to remove this message) Ottoman–Habsburg warsPart of the Ottoman wars in EuropeLeft to right from the top: the Siege...

 

 

Questa voce o sezione sull'argomento competizioni calcistiche non è ancora formattata secondo gli standard. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto di riferimento. FA Cup 1966-1967 Competizione FA Cup Sport Calcio Edizione 86ª Organizzatore FA Luogo  Inghilterra Risultati Vincitore  Tottenham(5º titolo) Secondo  Chelsea Semi-finalisti  Nottingham Forest Leeds Utd Cronologia della competizione 1965-66 1...

1945 United Kingdom general election 1945 United Kingdom general election ← 1935 5 July 1945 1950 → ← outgoing memberselected members →All 640 seats in the House of Commons321 seats needed for a majorityOpinion pollsTurnout72.8%, 1.7%   First party Second party   Leader Clement Attlee Winston Churchill Party Labour Conservative Leader since 25 October 1935 9 October 1940 Leader's seat Limehouse Woodford Last election 15...

 

 

Island of Labuan, Malaysia Daat IslandNative name: Pulau DaatGeographyLocationLabuan, MalaysiaCoordinates5°16′25.0″N 115°19′05.1″E / 5.273611°N 115.318083°E / 5.273611; 115.318083Total islands1Area2.4 km2 (0.93 sq mi) The Daat Island (Malay: Pulau Daat) is an island in Labuan, Malaysia. History In 1856, Labuan Governor George Warren Edwardes issued a grant to John Gavaron Treacher and Clarence Cooper to be the owner of the island. Since then,...