"1983"^^ . . . "En geometria de Riemann, l'escalar de curvatura o escalar de Ricci \u00E9s la forma m\u00E9s simple per descriure la curvatura d'una varietat de Riemann. Aquest escalar assigna a cada punt de la varietat un \u00FAnic nombre real que caracteritza la curvatura intr\u00EDnseca de la varietat en aquest punt. En dues dimensions la curvatura escalar caracteritza completament la curvatura d'una varietat riemaniana. Tot i aix\u00ED, en dimensions iguals o superiors a 3, cal m\u00E9s informaci\u00F3 (vegeu \u00AB\u00BB per a una discussi\u00F3 m\u00E9s extensa). on"@ca . . . "Shen"@en . "In geometria differenziale la curvatura scalare (o scalare di Ricci) \u00E8 il pi\u00F9 semplice invariante di curvatura di una variet\u00E0 riemanniana. Ad ogni punto della variet\u00E0 essa associa un numero reale determinato dalla geometria intrinseca della variet\u00E0 intorno a quel punto. La curvatura scalare \u00E8 definita a partire dal tensore di curvatura di Ricci, che \u00E8 a sua volta definito a partire dal tensore di Riemann."@it . . . . . "Michelsohn"@en . "2004"^^ . "2006"^^ . . . "2016"^^ . . "2017"^^ . "Petersen"@en . . . "1989"^^ . "285622"^^ . . "345"^^ . "Section 2D"@en . "1987"^^ . "Perelman"@en . . . "\u6570\u91CF\u66F2\u7387"@zh . "1995"^^ . "35656"^^ . . . . . "Michelsohn"@en . "\uC2A4\uCE7C\uB77C \uACE1\uB960"@ko . . . . . . . "2008"^^ . "En matem\u00E1ticas, la curvatura escalar de una superficie es el doble de la familiar curvatura gaussiana. Para las variedades riemannianas de dimensi\u00F3n m\u00E1s alta (n > 2), es el doble de la suma de todas las curvaturas seccionales a lo largo de todos los 2-planos atravesados por un cierto marco ortonormal. Matem\u00E1ticamente, la curvatura escalar o escalar de curvatura, que suele designarse con las letras R o S, coincide tambi\u00E9n la traza total de la curvatura de Ricci as\u00ED como del tensor de curvatura."@es . . . . . "2016"^^ . "2017"^^ . . . "\uC2A4\uCE7C\uB77C \uACE1\uB960(scalar\u66F2\u7387, \uC601\uC5B4: scalar curvature \uB610\uB294 Ricci scalar)\uC740 \uB9AC\uCE58 \uACE1\uB960 \uD150\uC11C\uC758 \uB300\uAC01\uD569\uC774\uB2E4. \uB9AC\uB9CC \uB2E4\uC591\uCCB4\uC758 \uACE1\uB960\uC744 \uB098\uD0C0\uB0B4\uB294 \uC2A4\uCE7C\uB77C\uC7A5\uC774\uB2E4. \uAE30\uD638\uB294 \uB300\uAC1C \uC9C0\uD45C(index) \uD45C\uAE30\uBC95\uC5D0\uC11C\uB294 \uC774\uB098, \uC9C0\uD45C\uB97C \uC4F0\uC9C0 \uC54A\uB294 \uD45C\uAE30\uBC95\uC5D0\uC11C\uB294 \uB9AC\uB9CC \uACE1\uB960 \uD150\uC11C \uBC0F \uB9AC\uCE58 \uACE1\uB960 \uD150\uC11C\uC640 \uD63C\uB3D9\uB418\uBBC0\uB85C \uB610\uB294 \uB97C \uC4F0\uAE30\uB3C4 \uD55C\uB2E4."@ko . . "Lawson"@en . "Sections 1G and 1H"@en . . . . . . "Aubin"@en . "Berline"@en . . . . . . . . "Zhu"@en . . "En g\u00E9om\u00E9trie riemannienne, la courbure scalaire (ou scalaire de Ricci) est un des outils de mesure de la courbure d'une vari\u00E9t\u00E9 riemannienne. Cet invariant riemannien est une fonction qui affecte \u00E0 chaque point m de la vari\u00E9t\u00E9 un simple nombre r\u00E9el not\u00E9 R(m) ou s(m), portant une information sur la courbure intrins\u00E8que de la vari\u00E9t\u00E9 en ce point. Ainsi, on peut d\u00E9crire le comportement infinit\u00E9simal des boules et des sph\u00E8res centr\u00E9es en m \u00E0 l'aide de la courbure scalaire. Dans un espace \u00E0 deux dimensions, la courbure scalaire caract\u00E9rise compl\u00E8tement la courbure de la vari\u00E9t\u00E9. En dimension sup\u00E9rieure \u00E0 3, cependant, il n'y suffit pas et d'autres invariants sont n\u00E9cessaires. La courbure scalaire est d\u00E9finie comme la trace du tenseur de Ricci relativement \u00E0 la m\u00E9trique (le point d'application m est souvent omis) On peut aussi \u00E9crire en coordonn\u00E9es locales et avec les conventions d'Einstein, , avec"@fr . . . "\u5728\u9ECE\u66FC\u51E0\u4F55\u4E2D\uFF0C\u6570\u91CF\u66F2\u7387\uFF08Scalar curvature\uFF09\u6216\u91CC\u5947\u6570\u91CF\uFF08Ricci scalar\uFF09\u662F\u4E00\u4E2A\u9ECE\u66FC\u6D41\u5F62\u6700\u7B80\u5355\u7684\u66F2\u7387\u4E0D\u53D8\u91CF\u3002\u5BF9\u9ECE\u66FC\u6D41\u5F62\u7684\u6BCF\u4E00\u70B9\uFF0C\u6570\u91CF\u66F2\u7387\u662F\u7531\u8BE5\u70B9\u9644\u8FD1\u7684\u5185\u8574\u51E0\u4F55\u786E\u5B9A\u7684\u4E00\u4E2A\u5B9E\u6570\u3002 \u5728 2 \u7EF4\u6570\u91CF\u66F2\u7387\u5B8C\u5168\u786E\u5B9A\u4E86\u9ECE\u66FC\u6D41\u5F62\u7684\u66F2\u7387\uFF1B\u5F53\u7EF4\u6570 \u2265 3\uFF0C\u66F2\u7387\u6BD4\u6570\u91CF\u66F2\u7387\u542B\u6709\u66F4\u591A\u7684\u4FE1\u606F\u3002\u53C2\u89C1\u4E2D\u5B8C\u6574\u7684\u8BA8\u8BBA\u3002 \u6570\u91CF\u66F2\u7387\u4E00\u822C\u8BB0\u4E3A S\uFF08\u5176\u5B83\u8BB0\u6CD5\u6709 Sc, R\uFF09\uFF0C\u5B9A\u4E49\u4E3A\u5173\u4E8E\u5EA6\u91CF\u7684\u91CC\u5947\u66F2\u7387\u5F20\u91CF\u7684\u8FF9\uFF1A \u8FD9\u4E2A\u8FF9\u548C\u5EA6\u91CF\u76F8\u5173\uFF0C\u56E0\u4E3A\u91CC\u5947\u5F20\u91CF\u662F\u4E00\u4E2A (0,2) \u578B\u5F20\u91CF\uFF1B\u5FC5\u987B\u5C06\u6307\u6807\u4E0A\u5347\u5F97\u5230\u4E00\u4E2A (1,1) \u578B\u5F20\u91CF\u624D\u80FD\u53D6\u8FF9\u3002\u5728\u5C40\u90E8\u5750\u6807\u4E2D\u6211\u4EEC\u53EF\u4EE5\u5199\u6210 \u8FD9\u91CC \u7ED9\u4E86\u4E00\u4E2A\u5750\u6807\u7CFB\u4E0E\u4E00\u4E2A\u5EA6\u91CF\u5F20\u91CF\uFF0C\u6570\u91CF\u66F2\u7387\u53EF\u4EE5\u8868\u793A\u4E3A\uFF1A \u8FD9\u91CC \u662F\u5EA6\u91CF\u7684\u514B\u91CC\u65AF\u6258\u8D39\u5C14\u7B26\u53F7\u3002 \u4E0D\u50CF\u9ECE\u66FC\u66F2\u7387\u5F20\u91CF\u6216\u91CC\u5947\u5F20\u91CF\u53EF\u4EE5\u5BF9\u4EFB\u4F55\u4EFF\u5C04\u8054\u7EDC\u81EA\u7136\u5730\u5B9A\u4E49\uFF0C\u6570\u91CF\u66F2\u7387\u53EA\u5728\u9ECE\u66FC\u51E0\u4F55\u5B58\u5728\uFF1B\u5176\u5B9A\u4E49\u4E0E\u5EA6\u91CF\u5BC6\u4E0D\u53EF\u5206\u3002"@zh . . "Section 4.4"@en . "Bao"@en . "Chern"@en . "Section 12.3.3"@en . . . . . . "Scalaire kromming"@nl . "Blackadar"@en . . "Remark 3.1.7"@en . . "Lafontaine"@en . "Lawson"@en . . . "\uC2A4\uCE7C\uB77C \uACE1\uB960(scalar\u66F2\u7387, \uC601\uC5B4: scalar curvature \uB610\uB294 Ricci scalar)\uC740 \uB9AC\uCE58 \uACE1\uB960 \uD150\uC11C\uC758 \uB300\uAC01\uD569\uC774\uB2E4. \uB9AC\uB9CC \uB2E4\uC591\uCCB4\uC758 \uACE1\uB960\uC744 \uB098\uD0C0\uB0B4\uB294 \uC2A4\uCE7C\uB77C\uC7A5\uC774\uB2E4. \uAE30\uD638\uB294 \uB300\uAC1C \uC9C0\uD45C(index) \uD45C\uAE30\uBC95\uC5D0\uC11C\uB294 \uC774\uB098, \uC9C0\uD45C\uB97C \uC4F0\uC9C0 \uC54A\uB294 \uD45C\uAE30\uBC95\uC5D0\uC11C\uB294 \uB9AC\uB9CC \uACE1\uB960 \uD150\uC11C \uBC0F \uB9AC\uCE58 \uACE1\uB960 \uD150\uC11C\uC640 \uD63C\uB3D9\uB418\uBBC0\uB85C \uB610\uB294 \uB97C \uC4F0\uAE30\uB3C4 \uD55C\uB2E4."@ko . "Getzler"@en . "Section 1.2.3"@en . . . . . . "\u30B9\u30AB\u30E9\u30FC\u66F2\u7387"@ja . . "Section 3.K.3"@en . . . "Definition 3.19"@en . . . "Hulin"@en . "Michelsohn"@en . . "En matem\u00E1ticas, la curvatura escalar de una superficie es el doble de la familiar curvatura gaussiana. Para las variedades riemannianas de dimensi\u00F3n m\u00E1s alta (n > 2), es el doble de la suma de todas las curvaturas seccionales a lo largo de todos los 2-planos atravesados por un cierto marco ortonormal. Matem\u00E1ticamente, la curvatura escalar o escalar de curvatura, que suele designarse con las letras R o S, coincide tambi\u00E9n la traza total de la curvatura de Ricci as\u00ED como del tensor de curvatura."@es . "In de differentiaalmeetkunde, en relativiteitstheorie, verwijst de term scalaire kromming naar de kromming van een Riemannse vari\u00EBteit. Het is een scalaire functie, die aangeeft in welke mate een oppervlak verschilt van de vlakke ruimte. De scalaire kromming zegt wel minder over een vari\u00EBteit dan de Ricci-kromming: het kan immers dat een niet-triviaal oppervlak scalaire kromming gelijk aan nul heeft, omdat het oppervlak in bepaalde richtingen positief gekromd is, en in andere richtingen negatief, zodat de totale kromming nul is. In zo een geval is de Ricci-kromming niet nul. Enkel in twee dimensies geeft de scalaire kromming evenveel informatie als de Ricci-kromming. In de algemene relativiteitstheorie is de kromming van een ruimte (op plaatsen waar er geen materie is) gerelateerd aan de k"@nl . . . . . "Cao"@en . "Lawson"@en . . . "Petersen"@en . . "\u0421\u043A\u0430\u043B\u044F\u0440\u043D\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430 (\u0430\u0431\u043E \u0441\u043A\u0430\u043B\u044F\u0440 \u0420\u0456\u0447\u0456) \u2014 \u043D\u0430\u0439\u043F\u0440\u043E\u0441\u0442\u0456\u0448\u0438\u0439 \u0437 \u043C\u043E\u0436\u043B\u0438\u0432\u0438\u0445 \u0456\u043D\u0432\u0430\u0440\u0456\u0430\u043D\u0442\u0456\u0432 \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u0438 \u0420\u0456\u043C\u0430\u043D\u043E\u0432\u0438\u0445 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0456\u0432. \u041A\u043E\u0436\u043D\u0456\u0439 \u0442\u043E\u0447\u0446\u0456 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443 \u0432\u043E\u043D\u0430 \u0441\u0442\u0430\u0432\u0438\u0442\u044C \u0443 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u0456\u0441\u0442\u044C \u043E\u0434\u043D\u0435 \u0434\u0456\u0439\u0441\u043D\u0435 \u0447\u0438\u0441\u043B\u043E, \u044F\u043A\u0435 \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u0432\u043D\u0443\u0442\u0440\u0456\u0448\u043D\u044C\u043E\u044E \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0454\u044E \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0430 \u0432 \u043E\u043A\u043E\u043B\u0438\u0446\u0456 \u0446\u0456\u0454\u0457 \u0442\u043E\u0447\u043A\u0438. \u0417\u043E\u043A\u0440\u0435\u043C\u0430, \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430 \u0432\u0438\u0440\u0430\u0436\u0430\u0454 \u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043E\u0431'\u0454\u043C\u0443 \u043D\u0430 \u044F\u043A\u0438\u0439 \u0432\u0456\u0434\u0440\u0456\u0437\u043D\u044F\u044E\u0442\u044C\u0441\u044F \u0433\u0435\u043E\u0434\u0435\u0437\u0438\u0447\u043D\u0456 \u043A\u0443\u043B\u0456 \u0443 \u0432\u0438\u043A\u0440\u0438\u0432\u043B\u0435\u043D\u043E\u043C\u0443 \u0440\u0456\u043C\u0430\u043D\u043E\u0432\u043E\u043C\u0443 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0456 \u0456 \u0432 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u043E\u043C\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0456. \u041E\u0442\u0440\u0438\u043C\u0443\u0454\u0442\u044C\u0441\u044F \u0437\u0433\u043E\u0440\u0442\u043A\u043E\u044E \u0442\u0435\u043D\u0437\u043E\u0440\u0430 \u0420\u0456\u0447\u0447\u0456 \u0437 \u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u043C \u0442\u0435\u043D\u0437\u043E\u0440\u043E\u043C"@uk . "In de differentiaalmeetkunde, en relativiteitstheorie, verwijst de term scalaire kromming naar de kromming van een Riemannse vari\u00EBteit. Het is een scalaire functie, die aangeeft in welke mate een oppervlak verschilt van de vlakke ruimte. De scalaire kromming zegt wel minder over een vari\u00EBteit dan de Ricci-kromming: het kan immers dat een niet-triviaal oppervlak scalaire kromming gelijk aan nul heeft, omdat het oppervlak in bepaalde richtingen positief gekromd is, en in andere richtingen negatief, zodat de totale kromming nul is. In zo een geval is de Ricci-kromming niet nul. Enkel in twee dimensies geeft de scalaire kromming evenveel informatie als de Ricci-kromming. In de algemene relativiteitstheorie is de kromming van een ruimte (op plaatsen waar er geen materie is) gerelateerd aan de kosmologische constante. Aangezien deze verschilt van nul, heeft ons universum een (positieve) kromming. In eerste benadering (als men de materie in ons heelal zou uitsmeren) is ons universum dus een homogene, isotrope, positief gekromde ruimte, welke beschreven kan worden met een de Sitter-metriek."@nl . . "Lafontaine"@en . "Petersen"@en . . . . . . "30"^^ . . . "Section 24.4"@en . . "O'Neill"@en . . "Sections II.8 and IV.3"@en . . . "Gilkey"@en . . "\u5728\u9ECE\u66FC\u51E0\u4F55\u4E2D\uFF0C\u6570\u91CF\u66F2\u7387\uFF08Scalar curvature\uFF09\u6216\u91CC\u5947\u6570\u91CF\uFF08Ricci scalar\uFF09\u662F\u4E00\u4E2A\u9ECE\u66FC\u6D41\u5F62\u6700\u7B80\u5355\u7684\u66F2\u7387\u4E0D\u53D8\u91CF\u3002\u5BF9\u9ECE\u66FC\u6D41\u5F62\u7684\u6BCF\u4E00\u70B9\uFF0C\u6570\u91CF\u66F2\u7387\u662F\u7531\u8BE5\u70B9\u9644\u8FD1\u7684\u5185\u8574\u51E0\u4F55\u786E\u5B9A\u7684\u4E00\u4E2A\u5B9E\u6570\u3002 \u5728 2 \u7EF4\u6570\u91CF\u66F2\u7387\u5B8C\u5168\u786E\u5B9A\u4E86\u9ECE\u66FC\u6D41\u5F62\u7684\u66F2\u7387\uFF1B\u5F53\u7EF4\u6570 \u2265 3\uFF0C\u66F2\u7387\u6BD4\u6570\u91CF\u66F2\u7387\u542B\u6709\u66F4\u591A\u7684\u4FE1\u606F\u3002\u53C2\u89C1\u4E2D\u5B8C\u6574\u7684\u8BA8\u8BBA\u3002 \u6570\u91CF\u66F2\u7387\u4E00\u822C\u8BB0\u4E3A S\uFF08\u5176\u5B83\u8BB0\u6CD5\u6709 Sc, R\uFF09\uFF0C\u5B9A\u4E49\u4E3A\u5173\u4E8E\u5EA6\u91CF\u7684\u91CC\u5947\u66F2\u7387\u5F20\u91CF\u7684\u8FF9\uFF1A \u8FD9\u4E2A\u8FF9\u548C\u5EA6\u91CF\u76F8\u5173\uFF0C\u56E0\u4E3A\u91CC\u5947\u5F20\u91CF\u662F\u4E00\u4E2A (0,2) \u578B\u5F20\u91CF\uFF1B\u5FC5\u987B\u5C06\u6307\u6807\u4E0A\u5347\u5F97\u5230\u4E00\u4E2A (1,1) \u578B\u5F20\u91CF\u624D\u80FD\u53D6\u8FF9\u3002\u5728\u5C40\u90E8\u5750\u6807\u4E2D\u6211\u4EEC\u53EF\u4EE5\u5199\u6210 \u8FD9\u91CC \u7ED9\u4E86\u4E00\u4E2A\u5750\u6807\u7CFB\u4E0E\u4E00\u4E2A\u5EA6\u91CF\u5F20\u91CF\uFF0C\u6570\u91CF\u66F2\u7387\u53EF\u4EE5\u8868\u793A\u4E3A\uFF1A \u8FD9\u91CC \u662F\u5EA6\u91CF\u7684\u514B\u91CC\u65AF\u6258\u8D39\u5C14\u7B26\u53F7\u3002 \u4E0D\u50CF\u9ECE\u66FC\u66F2\u7387\u5F20\u91CF\u6216\u91CC\u5947\u5F20\u91CF\u53EF\u4EE5\u5BF9\u4EFB\u4F55\u4EFF\u5C04\u8054\u7EDC\u81EA\u7136\u5730\u5B9A\u4E49\uFF0C\u6570\u91CF\u66F2\u7387\u53EA\u5728\u9ECE\u66FC\u51E0\u4F55\u5B58\u5728\uFF1B\u5176\u5B9A\u4E49\u4E0E\u5EA6\u91CF\u5BC6\u4E0D\u53EF\u5206\u3002"@zh . . . "\u0421\u043A\u0430\u043B\u044F\u0440\u043D\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430"@uk . . . . "Chavel"@en . . "Section 1J"@en . . . . . . . . . "Section 1I"@en . "88"^^ . . "146"^^ . "Section XII.8"@en . . "Sections 4.4 and 4.5"@en . "144"^^ . "Section 6.1"@en . . . "O'Neill"@en . "Curvatura escalar de Ricci"@es . "Lemmas 81.1 and 81.2"@en . "135"^^ . "Courbure scalaire"@fr . "Kleiner"@en . "\u0421\u043A\u0430\u043B\u044F\u0440\u043D\u0430\u044F \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u0430"@ru . "O'Neill"@en . . . "Escalar de Ricci"@ca . "En g\u00E9om\u00E9trie riemannienne, la courbure scalaire (ou scalaire de Ricci) est un des outils de mesure de la courbure d'une vari\u00E9t\u00E9 riemannienne. Cet invariant riemannien est une fonction qui affecte \u00E0 chaque point m de la vari\u00E9t\u00E9 un simple nombre r\u00E9el not\u00E9 R(m) ou s(m), portant une information sur la courbure intrins\u00E8que de la vari\u00E9t\u00E9 en ce point. Ainsi, on peut d\u00E9crire le comportement infinit\u00E9simal des boules et des sph\u00E8res centr\u00E9es en m \u00E0 l'aide de la courbure scalaire. On peut aussi \u00E9crire en coordonn\u00E9es locales et avec les conventions d'Einstein, , avec"@fr . "Section 4.2.3"@en . "Section 11.2"@en . . "Vergne"@en . . . "160"^^ . . . "Section 3.1.5"@en . "88"^^ . . . . . . "In geometria differenziale la curvatura scalare (o scalare di Ricci) \u00E8 il pi\u00F9 semplice invariante di curvatura di una variet\u00E0 riemanniana. Ad ogni punto della variet\u00E0 essa associa un numero reale determinato dalla geometria intrinseca della variet\u00E0 intorno a quel punto. La curvatura scalare \u00E8 definita a partire dal tensore di curvatura di Ricci, che \u00E8 a sua volta definito a partire dal tensore di Riemann."@it . "\u0421\u043A\u0430\u043B\u044F\u0440\u043D\u0430\u044F \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u0430 \u2014 \u043E\u0434\u0438\u043D \u0438\u0437 \u0438\u043D\u0432\u0430\u0440\u0438\u0430\u043D\u0442\u043E\u0432 \u0440\u0438\u043C\u0430\u043D\u043E\u0432\u0430 \u043C\u043D\u043E\u0433\u043E\u043E\u0431\u0440\u0430\u0437\u0438\u044F, \u043F\u043E\u043B\u0443\u0447\u0430\u0435\u043C\u044B\u0439 \u0441\u0432\u0451\u0440\u0442\u043A\u043E\u0439 \u0442\u0435\u043D\u0437\u043E\u0440\u0430 \u0420\u0438\u0447\u0447\u0438 \u0441 \u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u043C \u0442\u0435\u043D\u0437\u043E\u0440\u043E\u043C.\u041E\u0431\u044B\u0447\u043D\u043E \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u0442\u0441\u044F \u0438\u043B\u0438 ."@ru . . . . . . "Jost"@en . "Section 3C"@en . . "Gallot"@en . "In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations. Furthermore, this scalar curvature is the Lagrangian density for the Einstein\u2013Hilbert action, the Euler\u2013Lagrange equations of which are the Einstein field equations in vacuum. The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques. Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature. In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case."@en . . . "107"^^ . "Hulin"@en . "Berger"@en . . "Escalar de curvatura de Ricci"@pt . "92"^^ . . . . . "Section 4.1"@en . . "Definition 1.22"@en . "Section IV.5"@en . . . . "135"^^ . "Section II.8"@en . "Jost"@en . . . . "\u30EA\u30FC\u30DE\u30F3\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\uFF08\u3059\u304B\u3089\u30FC\u304D\u3087\u304F\u308A\u3064\u3001\u82F1: Scalar curvature\uFF09\u307E\u305F\u306F\u30EA\u30C3\u30C1\u30B9\u30AB\u30E9\u30FC\uFF08\u82F1: Ricci scalar\uFF09\u306F\u3001\u30EA\u30FC\u30DE\u30F3\u591A\u69D8\u4F53\u306E\u6700\u3082\u5358\u7D14\u306A\u66F2\u7387\u4E0D\u5909\u91CF\u3067\u3042\u308B\u3002\u30EA\u30FC\u30DE\u30F3\u591A\u69D8\u4F53\u306E\u5404\u70B9\u306B\u3001\u305D\u306E\u8FD1\u508D\u306B\u304A\u3051\u308B\u591A\u69D8\u4F53\u306E\u5185\u5728\u7684\u306A\u5F62\u72B6\u304B\u3089\u5B9A\u307E\u308B\u5358\u4E00\u306E\u5B9F\u6570\u3092\u5BFE\u5FDC\u3055\u305B\u308B\u3002 2\u6B21\u5143\u306B\u304A\u3044\u3066\u306F\u3001\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F\u30EA\u30FC\u30DE\u30F3\u591A\u69D8\u4F53\u306E\u66F2\u7387\u3092\u5B8C\u5168\u306B\u7279\u5FB4\u4ED8\u3051\u308B\u3002\u3057\u304B\u3057\u3001\u6B21\u5143\u304C3\u4EE5\u4E0A\u306E\u5834\u5408\u306F\u3001\u66F2\u7387\u306E\u6C7A\u5B9A\u306B\u306F\u3055\u3089\u306B\u60C5\u5831\u304C\u5FC5\u8981\u3067\u3042\u308B\u3002\u8A73\u3057\u3044\u8B70\u8AD6\u306F(en) \u3092\u53C2\u7167\u3002 \u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F\u3057\u3070\u3057\u3070 S (\u305D\u306E\u4ED6\u306E\u8868\u8A18\u3068\u3057\u3066Sc, R)\u3068\u8868\u3055\u308C\u3001\u8A08\u91CF\u30C6\u30F3\u30BD\u30EB g \u306B\u95A2\u3059\u308B\u30EA\u30C3\u30C1\u66F2\u7387 Ric \u306E\u30C8\u30EC\u30FC\u30B9 \u3068\u3057\u3066\u5B9A\u7FA9\u3055\u308C\u308B\u3002\u30EA\u30C3\u30C1\u30C6\u30F3\u30BD\u30EB\u306F (0,2)-\u578B\u30C6\u30F3\u30BD\u30EB\u3067\u3042\u308A\u3001\u30C8\u30EC\u30FC\u30B9\u3092\u3068\u308B\u305F\u3081\u306B\u306F\u6700\u521D\u306E\u6DFB\u5B57\u3092\u4E0A\u3052\u3066 (1,1)-\u578B\u30C6\u30F3\u30BD\u30EB\u3068\u3057\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\u304B\u3089\u3001\u3053\u306E\u30C8\u30EC\u30FC\u30B9\u306F\u8A08\u91CF\u306E\u53D6\u308A\u65B9\u306B\u4F9D\u5B58\u3059\u308B\u3002\u5C40\u6240\u5EA7\u6A19\u7CFB\u3092\u7528\u3044\u3066 \u3068\u66F8\u304D\u8868\u3059\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u305F\u3060\u3057 \u3067\u3042\u308B\u3002\u5EA7\u6A19\u7CFB\u3068\u8A08\u91CF\u30C6\u30F3\u30BD\u30EB\u304C\u4E0E\u3048\u3089\u308C\u305F\u3068\u304D\u3001\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F \u306E\u3088\u3046\u306B\u8868\u793A\u3067\u304D\u308B\u3002\u3053\u3053\u3067 \u0393abc \u306F\u8A08\u91CF\u306E\u30AF\u30EA\u30B9\u30C8\u30C3\u30D5\u30A7\u30EB\u8A18\u53F7\u3067\u3042\u308B\u3002 \u4EFB\u610F\u306E\u30A2\u30D5\u30A3\u30F3\u63A5\u7D9A\u306B\u5BFE\u3057\u3066\u81EA\u7136\u306B\u5B9A\u7FA9\u3055\u308C\u308B\u30EA\u30FC\u30DE\u30F3\u66F2\u7387\u30C6\u30F3\u30BD\u30EB\u3084\u30EA\u30C3\u30C1\u30C6\u30F3\u30BD\u30EB\u3068\u306F\u7570\u306A\u308A\u3001\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F\uFF08\u305D\u306E\u5B9A\u7FA9\u304C\u307E\u3055\u306B\u8A08\u91CF\u3068\u4E0D\u53EF\u5206\u306A\u65B9\u6CD5\u3067\u4E0E\u3048\u3089\u308C\u305F\u3053\u3068\u3092\u601D\u3048\u3070\uFF09\u5B8C\u5168\u306B\u30EA\u30FC\u30DE\u30F3\u5E7E\u4F55\u5B66\u306E\u9818\u57DF\u306B\u7279\u6709\u306E\u6982\u5FF5\u3067\u3042\u308B\u3053\u3068\u304C\u5206\u304B\u308B\u3002"@ja . . "Section 1.5.2"@en . "Section 3.1.5"@en . . "90"^^ . "160"^^ . "Jost"@en . . . "2016"^^ . "2017"^^ . "Curvatura scalare"@it . . . . "Parker"@en . . . "Section 1.5.2"@en . "1117702557"^^ . . "Example 2.4.3"@en . "200"^^ . "Lee"@en . . "Besse"@en . "1992"^^ . . "Lott"@en . . "10"^^ . "1995"^^ . . . "1998"^^ . "1984"^^ . "Section 24.3"@en . . "1987"^^ . "En geometria de Riemann, l'escalar de curvatura o escalar de Ricci \u00E9s la forma m\u00E9s simple per descriure la curvatura d'una varietat de Riemann. Aquest escalar assigna a cada punt de la varietat un \u00FAnic nombre real que caracteritza la curvatura intr\u00EDnseca de la varietat en aquest punt. En dues dimensions la curvatura escalar caracteritza completament la curvatura d'una varietat riemaniana. Tot i aix\u00ED, en dimensions iguals o superiors a 3, cal m\u00E9s informaci\u00F3 (vegeu \u00AB\u00BB per a una discussi\u00F3 m\u00E9s extensa). La curvatura escalar s'acostuma a denotar per S (altres notacions s\u00F3n Sc, R). Es defineix com la tra\u00E7a del tensor de respecte a la m\u00E8trica: La tra\u00E7a dep\u00E8n de la m\u00E8trica, ja que el tensor de Ricci \u00E9s un tensor (0,2); primer s'ha de contreure amb la m\u00E8trica per obtenir un tensor (1,1) de cara a obtenir la tra\u00E7a. En termes de coordenades locals podem escriure: on"@ca . "1989"^^ . "Em matem\u00E1tica, a curvatura escalar de uma superf\u00EDcie \u00E9 a familiar curvatura gaussiana. Para as variedades riemannianas de dimens\u00E3o mais alta (n > 2), \u00E9 o dobro da soma de todas as curvaturas seccionais ao longo de todos os 2-planos atravessados por um certo marco ortonormal. Matematicamente a curvatura escalar coincide tamb\u00E9m o tra\u00E7o total da curvatura de Ricci assim como do tensor de curvatura."@pt . "Section 1K"@en . . "2000"^^ . . "34"^^ . "2003"^^ . . "2004"^^ . "Gilkey"@en . . "do Carmo"@en . "In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor."@en . . "Besse"@en . . . . . . . . "Scalar curvature"@en . . "1983"^^ . "Section 1F"@en . "1989"^^ . "Corollary 7.4.4"@en . . . . . . . "Em matem\u00E1tica, a curvatura escalar de uma superf\u00EDcie \u00E9 a familiar curvatura gaussiana. Para as variedades riemannianas de dimens\u00E3o mais alta (n > 2), \u00E9 o dobro da soma de todas as curvaturas seccionais ao longo de todos os 2-planos atravessados por um certo marco ortonormal. Matematicamente a curvatura escalar coincide tamb\u00E9m o tra\u00E7o total da curvatura de Ricci assim como do tensor de curvatura."@pt . . . . . "\u0421\u043A\u0430\u043B\u044F\u0440\u043D\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430 (\u0430\u0431\u043E \u0441\u043A\u0430\u043B\u044F\u0440 \u0420\u0456\u0447\u0456) \u2014 \u043D\u0430\u0439\u043F\u0440\u043E\u0441\u0442\u0456\u0448\u0438\u0439 \u0437 \u043C\u043E\u0436\u043B\u0438\u0432\u0438\u0445 \u0456\u043D\u0432\u0430\u0440\u0456\u0430\u043D\u0442\u0456\u0432 \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u0438 \u0420\u0456\u043C\u0430\u043D\u043E\u0432\u0438\u0445 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0456\u0432. \u041A\u043E\u0436\u043D\u0456\u0439 \u0442\u043E\u0447\u0446\u0456 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443 \u0432\u043E\u043D\u0430 \u0441\u0442\u0430\u0432\u0438\u0442\u044C \u0443 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u0456\u0441\u0442\u044C \u043E\u0434\u043D\u0435 \u0434\u0456\u0439\u0441\u043D\u0435 \u0447\u0438\u0441\u043B\u043E, \u044F\u043A\u0435 \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u0432\u043D\u0443\u0442\u0440\u0456\u0448\u043D\u044C\u043E\u044E \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0454\u044E \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0430 \u0432 \u043E\u043A\u043E\u043B\u0438\u0446\u0456 \u0446\u0456\u0454\u0457 \u0442\u043E\u0447\u043A\u0438. \u0417\u043E\u043A\u0440\u0435\u043C\u0430, \u0441\u043A\u0430\u043B\u044F\u0440\u043D\u0430 \u043A\u0440\u0438\u0432\u0438\u043D\u0430 \u0432\u0438\u0440\u0430\u0436\u0430\u0454 \u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043E\u0431'\u0454\u043C\u0443 \u043D\u0430 \u044F\u043A\u0438\u0439 \u0432\u0456\u0434\u0440\u0456\u0437\u043D\u044F\u044E\u0442\u044C\u0441\u044F \u0433\u0435\u043E\u0434\u0435\u0437\u0438\u0447\u043D\u0456 \u043A\u0443\u043B\u0456 \u0443 \u0432\u0438\u043A\u0440\u0438\u0432\u043B\u0435\u043D\u043E\u043C\u0443 \u0440\u0456\u043C\u0430\u043D\u043E\u0432\u043E\u043C\u0443 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0456 \u0456 \u0432 \u0435\u0432\u043A\u043B\u0456\u0434\u043E\u0432\u043E\u043C\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0456. \u041E\u0442\u0440\u0438\u043C\u0443\u0454\u0442\u044C\u0441\u044F \u0437\u0433\u043E\u0440\u0442\u043A\u043E\u044E \u0442\u0435\u043D\u0437\u043E\u0440\u0430 \u0420\u0456\u0447\u0447\u0456 \u0437 \u043C\u0435\u0442\u0440\u0438\u0447\u043D\u0438\u043C \u0442\u0435\u043D\u0437\u043E\u0440\u043E\u043C"@uk . . . . . . . . "Section 3.1.5"@en . "Section 3.H.4"@en . "336"^^ . "Gallot"@en . . "1983"^^ . . "\u0421\u043A\u0430\u043B\u044F\u0440\u043D\u0430\u044F \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u0430 \u2014 \u043E\u0434\u0438\u043D \u0438\u0437 \u0438\u043D\u0432\u0430\u0440\u0438\u0430\u043D\u0442\u043E\u0432 \u0440\u0438\u043C\u0430\u043D\u043E\u0432\u0430 \u043C\u043D\u043E\u0433\u043E\u043E\u0431\u0440\u0430\u0437\u0438\u044F, \u043F\u043E\u043B\u0443\u0447\u0430\u0435\u043C\u044B\u0439 \u0441\u0432\u0451\u0440\u0442\u043A\u043E\u0439 \u0442\u0435\u043D\u0437\u043E\u0440\u0430 \u0420\u0438\u0447\u0447\u0438 \u0441 \u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u043C \u0442\u0435\u043D\u0437\u043E\u0440\u043E\u043C.\u041E\u0431\u044B\u0447\u043D\u043E \u043E\u0431\u043E\u0437\u043D\u0430\u0447\u0430\u0435\u0442\u0441\u044F \u0438\u043B\u0438 ."@ru . . "\u30EA\u30FC\u30DE\u30F3\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\uFF08\u3059\u304B\u3089\u30FC\u304D\u3087\u304F\u308A\u3064\u3001\u82F1: Scalar curvature\uFF09\u307E\u305F\u306F\u30EA\u30C3\u30C1\u30B9\u30AB\u30E9\u30FC\uFF08\u82F1: Ricci scalar\uFF09\u306F\u3001\u30EA\u30FC\u30DE\u30F3\u591A\u69D8\u4F53\u306E\u6700\u3082\u5358\u7D14\u306A\u66F2\u7387\u4E0D\u5909\u91CF\u3067\u3042\u308B\u3002\u30EA\u30FC\u30DE\u30F3\u591A\u69D8\u4F53\u306E\u5404\u70B9\u306B\u3001\u305D\u306E\u8FD1\u508D\u306B\u304A\u3051\u308B\u591A\u69D8\u4F53\u306E\u5185\u5728\u7684\u306A\u5F62\u72B6\u304B\u3089\u5B9A\u307E\u308B\u5358\u4E00\u306E\u5B9F\u6570\u3092\u5BFE\u5FDC\u3055\u305B\u308B\u3002 2\u6B21\u5143\u306B\u304A\u3044\u3066\u306F\u3001\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F\u30EA\u30FC\u30DE\u30F3\u591A\u69D8\u4F53\u306E\u66F2\u7387\u3092\u5B8C\u5168\u306B\u7279\u5FB4\u4ED8\u3051\u308B\u3002\u3057\u304B\u3057\u3001\u6B21\u5143\u304C3\u4EE5\u4E0A\u306E\u5834\u5408\u306F\u3001\u66F2\u7387\u306E\u6C7A\u5B9A\u306B\u306F\u3055\u3089\u306B\u60C5\u5831\u304C\u5FC5\u8981\u3067\u3042\u308B\u3002\u8A73\u3057\u3044\u8B70\u8AD6\u306F(en) \u3092\u53C2\u7167\u3002 \u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F\u3057\u3070\u3057\u3070 S (\u305D\u306E\u4ED6\u306E\u8868\u8A18\u3068\u3057\u3066Sc, R)\u3068\u8868\u3055\u308C\u3001\u8A08\u91CF\u30C6\u30F3\u30BD\u30EB g \u306B\u95A2\u3059\u308B\u30EA\u30C3\u30C1\u66F2\u7387 Ric \u306E\u30C8\u30EC\u30FC\u30B9 \u3068\u3057\u3066\u5B9A\u7FA9\u3055\u308C\u308B\u3002\u30EA\u30C3\u30C1\u30C6\u30F3\u30BD\u30EB\u306F (0,2)-\u578B\u30C6\u30F3\u30BD\u30EB\u3067\u3042\u308A\u3001\u30C8\u30EC\u30FC\u30B9\u3092\u3068\u308B\u305F\u3081\u306B\u306F\u6700\u521D\u306E\u6DFB\u5B57\u3092\u4E0A\u3052\u3066 (1,1)-\u578B\u30C6\u30F3\u30BD\u30EB\u3068\u3057\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\u304B\u3089\u3001\u3053\u306E\u30C8\u30EC\u30FC\u30B9\u306F\u8A08\u91CF\u306E\u53D6\u308A\u65B9\u306B\u4F9D\u5B58\u3059\u308B\u3002\u5C40\u6240\u5EA7\u6A19\u7CFB\u3092\u7528\u3044\u3066 \u3068\u66F8\u304D\u8868\u3059\u3053\u3068\u304C\u3067\u304D\u308B\u3002\u305F\u3060\u3057 \u3067\u3042\u308B\u3002\u5EA7\u6A19\u7CFB\u3068\u8A08\u91CF\u30C6\u30F3\u30BD\u30EB\u304C\u4E0E\u3048\u3089\u308C\u305F\u3068\u304D\u3001\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F \u306E\u3088\u3046\u306B\u8868\u793A\u3067\u304D\u308B\u3002\u3053\u3053\u3067 \u0393abc \u306F\u8A08\u91CF\u306E\u30AF\u30EA\u30B9\u30C8\u30C3\u30D5\u30A7\u30EB\u8A18\u53F7\u3067\u3042\u308B\u3002 \u4EFB\u610F\u306E\u30A2\u30D5\u30A3\u30F3\u63A5\u7D9A\u306B\u5BFE\u3057\u3066\u81EA\u7136\u306B\u5B9A\u7FA9\u3055\u308C\u308B\u30EA\u30FC\u30DE\u30F3\u66F2\u7387\u30C6\u30F3\u30BD\u30EB\u3084\u30EA\u30C3\u30C1\u30C6\u30F3\u30BD\u30EB\u3068\u306F\u7570\u306A\u308A\u3001\u30B9\u30AB\u30E9\u30FC\u66F2\u7387\u306F\uFF08\u305D\u306E\u5B9A\u7FA9\u304C\u307E\u3055\u306B\u8A08\u91CF\u3068\u4E0D\u53EF\u5206\u306A\u65B9\u6CD5\u3067\u4E0E\u3048\u3089\u308C\u305F\u3053\u3068\u3092\u601D\u3048\u3070\uFF09\u5B8C\u5168\u306B\u30EA\u30FC\u30DE\u30F3\u5E7E\u4F55\u5B66\u306E\u9818\u57DF\u306B\u7279\u6709\u306E\u6982\u5FF5\u3067\u3042\u308B\u3053\u3068\u304C\u5206\u304B\u308B\u3002"@ja . . . "Section IV.4"@en . . . . .