[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/6911#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/6911","headline":"\u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","name":"\u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","description":"before-content-x4 \u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 \uff08\u77ed\u3044 riemntensor \u3001 \u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 \u307e\u305f \u66f2\u7387 \uff09\u4efb\u610f\u306e\u5bf8\u6cd5\u306e\u90e8\u5c4b\u306e\u66f2\u7387\u3001\u3088\u308a\u6b63\u78ba\u306b\u306friemannscher\u307e\u305f\u306fpseudo-riemann\u306e\u591a\u69d8\u6027\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002\u5f7c\u306f\u6570\u5b66\u8005\u306e\u30d9\u30eb\u30f3\u30cf\u30eb\u30c8\u30fb\u30ea\u30fc\u30de\u30f3\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u3001\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306e\u6700\u3082\u91cd\u8981\u306a\u63f4\u52a9\u306e\u4e00\u3064\u3067\u3059\u3002\u5f7c\u306f\u3001\u4e00\u822c\u7684\u306a\u76f8\u5bfe\u6027\u7406\u8ad6\u306e\u6642\u7a7a\u306e\u66f2\u7387\u306b\u95a2\u9023\u3057\u3066\u5225\u306e\u91cd\u8981\u306a\u7528\u9014\u3092\u898b\u3064\u3051\u307e\u3059\u3002 after-content-x4 Riemann\u306e\u66f2\u7387\u306f\u30ec\u30d9\u30eb4\u306e\u30c6\u30f3\u30bd\u30eb\u3067\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3067\u304d\u307e\u3059\u3002 r \u79c1 k p m {displaystyle r_ {ikp}^{m}} \u793a\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30b5\u30e0\u6761\u7d04\u304c\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3059\u3002","datePublished":"2023-05-28","dateModified":"2023-05-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/655dfa6a880b3c8762df6df1fd86aecf732fe02e","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/655dfa6a880b3c8762df6df1fd86aecf732fe02e","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/6911","wordCount":14542,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4 \u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 \uff08\u77ed\u3044 riemntensor \u3001 \u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 \u307e\u305f \u66f2\u7387 \uff09\u4efb\u610f\u306e\u5bf8\u6cd5\u306e\u90e8\u5c4b\u306e\u66f2\u7387\u3001\u3088\u308a\u6b63\u78ba\u306b\u306friemannscher\u307e\u305f\u306fpseudo-riemann\u306e\u591a\u69d8\u6027\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002\u5f7c\u306f\u6570\u5b66\u8005\u306e\u30d9\u30eb\u30f3\u30cf\u30eb\u30c8\u30fb\u30ea\u30fc\u30de\u30f3\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u3001\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66\u306e\u6700\u3082\u91cd\u8981\u306a\u63f4\u52a9\u306e\u4e00\u3064\u3067\u3059\u3002\u5f7c\u306f\u3001\u4e00\u822c\u7684\u306a\u76f8\u5bfe\u6027\u7406\u8ad6\u306e\u6642\u7a7a\u306e\u66f2\u7387\u306b\u95a2\u9023\u3057\u3066\u5225\u306e\u91cd\u8981\u306a\u7528\u9014\u3092\u898b\u3064\u3051\u307e\u3059\u3002        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Riemann\u306e\u66f2\u7387\u306f\u30ec\u30d9\u30eb4\u306e\u30c6\u30f3\u30bd\u30eb\u3067\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3067\u304d\u307e\u3059\u3002 r \u79c1 k p m {displaystyle r_ {ikp}^{m}} \u793a\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30b5\u30e0\u6761\u7d04\u304c\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3059\u3002 diffeyomorphism\u306f\u3001\u5fae\u5206\u53ef\u80fd\u306a\u591a\u69d8\u4f53\u306e\u9593\u306e\u69cb\u9020\u3092\u6d78\u900f\u3055\u305b\u308b\u3082\u306e\u3067\u3042\u308a\u3001\u305d\u308c\u306b\u5fdc\u3058\u3066\uff08\u6ed1\u3089\u304b\u306a\uff09\u30a2\u30a4\u30bd\u30e1\u30c8\u30ea\u30fc\u306f\u3001\u30ea\u30fc\u30de\u30f3\u306e\u30de\u30cb\u30db\u30fc\u30eb\u30c9\u9593\u306e\u69cb\u9020\u753b\u50cf\u3067\u3059\u3002\u30de\u30cb\u30db\u30fc\u30eb\u30c9\u306e\u591a\u69d8\u6027\u3092\u533a\u5225\u3059\u308b\u3053\u3068\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u5730\u57df\u3067\u5c40\u6240\u7684\u306b\u7570\u306a\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u30ea\u30fc\u30de\u30f3\u306e\u591a\u69d8\u6027\u304c\u5c40\u6240\u7684\u306b\u3082\u5bc6\u306b\u306a\u3063\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u306b\u3064\u3044\u3066\u7591\u554f\u304c\u751f\u3058\u307e\u3057\u305f        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4r n {displaystyle mathbb {r} ^{n}} \u305d\u308c\u306f\u3002\u3053\u308c\u306f\u305d\u3046\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387\u304c\u5c0e\u5165\u3055\u308c\u307e\u3057\u305f\u3002 r n {displaystyle mathbb {r} ^{n}} \u306f\u3002\u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387\u306e\u5b9a\u7fa9\u3092\u3088\u308a\u3088\u304f\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u3001\u6b21\u306e\u8003\u616e\u4e8b\u9805 r 2 {displaystyle mathbb {r} ^{2}} \u524d\u306b\u3002 \u591a\u5206        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u3068 \u2208 c \uff08 t r 2 \uff09\uff09 {displaystyle zin gamma\uff08tmathbb {r} ^{2}\uff09} \u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9\u3002\u30e6\u30fc\u30af\u30ea\u30a6\u30ba\u30b9\u3067 r 2 {displaystyle mathbb {r} ^{2}} \u30e6\u30cb\u30c3\u30c8\u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9\u306b\u9069\u7528\u3055\u308c\u307e\u3059 \u2202 \u521d\u3081 \u3001 \u2202 2 {displaystyle partial _ {1}\u3001partial _ {2}} \u5e73\u7b49\u306e\u5ea7\u6a19\u8ef8\u306b\u6cbf\u3063\u3066 \u2207 \u22021\u2207 \u22022\u3068 = \u2207 \u22022\u2207 \u22021\u3068 \u3001 {displaystyle nabla _ {partial _ {1}} urla _ {partial _ {2}} z = ubla _ {partial _ {2}} ubla _ _ partial {1}} of\u3001}} \u9ed2\u306e\u6587\u304c\u78ba\u4fdd\u3059\u308b\u3002\u4e00\u822c\u7684\u306a\u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9\u306e\u5834\u5408 \u30d0\u30c4 \u3001  \u3068 {displaystyle x\u3001y} \u3053\u308c\u306f\u3001\u306b\u3082\u9069\u7528\u3055\u308c\u307e\u3059 r 2 {displaystyle mathbb {r} ^{2}} \u3082\u3046\u9055\u3044\u307e\u3059\u3002\u3082\u3064 \u3068 {displaystyle with} \u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u3092\u8abf\u6574\u3057\u307e\u3059 \u3068 = \u3068 i\u2202 i{dispasSastyle textStyle z = z^{i} partial _ {i}} \u9069\u7528\u3055\u308c\u307e\u3059 \u2207 X\u2207 Y\u3068 = \u2207 X\u3068 \u3068 i\u2202 i= \u30d0\u30c4 \u3068 \u3068 i\u2202 i\u3002 {displaystyle nabla _ {x} nabla _ {y} z = nabla _ {x} yz^{i} partial _ {i} = xyz^{i} partial _ {i}\u3002} \u8868\u73fe \u3068 \u3068 \u79c1 {displaystyle yz^{i}} \u306e\u65b9\u5411\u306e\u65b9\u5411\u3092\u8aac\u660e\u3057\u307e\u3059 \u3068 \u79c1 {displaystyle z^{i}} \u65b9\u5411 \u3068 {displaystyle y} \u3002\u306e\u975e\u5951\u7d04\u3092\u8abf\u3079\u7d9a\u3051\u3066\u3044\u308b\u5834\u5408 \u2207 \u30d0\u30c4 \u2207 \u3068 {displaystyle nabla _ {x} nabla _ {y}} \u3060\u304b\u3089\u3042\u306a\u305f\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u30b9\u30da\u30fc\u30b9\u306b\u5165\u308a\u307e\u3059 \u2207 X\u2207 Y\u3068  –  \u2207 Y\u2207 X\u3068 = \uff08 \u30d0\u30c4 \u3068 \u3068 i –  \u3068 \u30d0\u30c4 \u3068 i\uff09\uff09 \u2202 i= \u2207 [X,Y]\u3068 \u3002 {displaystyle nabla _ {x} nabla _ {y} z-n-nabla _ {y} nabla _ {x} z =\uff08xyz^{i} -yxz^{i}\uff09partial _ {i} = nabla _ {{x\u3001y] z} z\u3002 \u3053\u308c\u306f\u3001\u4e00\u822c\u7684\u306a\u591a\u69d8\u6027\u3067\u306f\u9593\u9055\u3063\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u305f\u3081\u3001\u6b21\u306e\u5b9a\u7fa9\u304c\u884c\u308f\u308c\u307e\u3059\u3002 \u591a\u5206 m {displaystyle m} \u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3068\u306e\u30b9\u30e0\u30fc\u30ba\u306a\u591a\u69d8\u6027 \u2207 {displaystyle nabla} \u3002\u305d\u306e\u5f8c\u3001Riemann\u306e\u66f2\u7387\u306f\u30a4\u30e9\u30b9\u30c8\u3067\u3059 c \u221e\uff08 m \u3001 t m \uff09\uff09 \u00d7 c \u221e\uff08 m \u3001 t m \uff09\uff09 \u00d7 c \u221e\uff08 m \u3001 t m \uff09\uff09 \u2192 c \u221e\uff08 m \u3001 t m \uff09\uff09 \u3001 {displaystyle gamma ^{infty}\uff08m\u3001tm\uff09times gamma ^{infty}\uff08m\u3001tm\uff09times gamma ^{infty}\uff08m\u3001tm\uff09to gamma ^{infty}\uff08m\u3001tm\uff09\u3001} \u305d\u308c\u3092\u901a\u3057\u3066 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3068 = \u2207 X\u2207 Y\u3068  –  \u2207 Y\u2207 X\u3068  –  \u2207 [X,Y]\u3068 {displaystyle r\uff08x\u3001y\uff09z = nabla _ {x} nabla _ {y} z-n-nabla _ {y} nabla _ {x} z-nabla _ {[x\u3001y]} z} \u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u3068 c \u221e \uff08 m \u3001 t m \uff09\uff09 {displaystyle gamma ^{infty}\uff08m\u3001tm\uff09} \u6ed1\u3089\u304b\u306a\u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9\u306e\u7a7a\u9593\u3067\u3042\u308a\u3001 [ \u3002 \u3001 \u3002 ] {displaystyle [\u3002\u3001\u3002]} \u5618\u306e\u30af\u30ea\u30c3\u30d7\u3092\u610f\u5473\u3057\u307e\u3057\u305f\u3002 \u30ed\u30fc\u30ab\u30eb\u5ea7\u6a19\u3067\u306f\u3001Christoffels\u30b7\u30f3\u30dc\u30eb\u3092\u4f7f\u7528\u3057\u3066\u66f2\u7387\u3092\u8868\u793a\u3067\u304d\u307e\u3059\u3002 r ikpm= \u2202 kc ipm –  \u2202 pc ikm+ c ipac akm –  c ikac apm{displaystyle r_ {ikp}^{m} = partial _ {k} gamma _ {ip}^{m} -partial _ {p} gamma _ {ik}^{m}+gamma _ {{ip}^{a}\u30ac\u30f3\u30de_ {ak}^{ak}^{ak} {ak} {ak}\u30ac\u30f3\u30de_ {ap}^{m}} Table of Contents\u6ce8\u91c8 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30c6\u30f3\u30bd\u30eb\u30d5\u30a7\u30eb\u30c9 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u66f2\u7387\u306e\u200b\u200b\u5bfe\u79f0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u610f\u5473 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u66f2\u7387\u3078\u306e\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d1\u30f3\u7c89 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30b9\u30ab\u30e9\u30fc\u66f2\u7387 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6ce8\u91c8 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Do Carmo\u306a\u3069\u306e\u4e00\u90e8\u306e\u8457\u8005 [\u521d\u3081] Oder Gallot\u3001Hulin\u3001Lafontaine\u3001 [2] Riemann\u306e\u66f2\u7387\u3092\u30ea\u30d0\u30fc\u30b9\u30b5\u30a4\u30f3\u3067\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\u3053\u306e\u5146\u5019\u306f\u3001\u4e3b\u8981\u306a\u66f2\u7387\u3068\u30ea\u30c3\u30c1\u306e\u6e7e\u66f2\u306e\u5b9a\u7fa9\u3082\u56de\u8ee2\u3057\u3066\u3044\u308b\u305f\u3081\u3001\u3059\u3079\u3066\u306e\u8457\u8005\u304c\u4e3b\u8981\u306a\u66f2\u7387\u3001\u30ea\u30c3\u30c1\u306e\u66f2\u7387\u3001\u30b9\u30ab\u30e9\u30fc\u66f2\u7387\u306e\u5146\u5019\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002 \u30c6\u30f3\u30bd\u30eb\u30d5\u30a7\u30eb\u30c9 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u66f2\u7387\u306f1\u3064\u3067\u3059 \uff08 \u521d\u3081 \u3001 3 \uff09\uff09 {displaystyle\uff081,3\uff09} -TENSORFELD\u3002 \u66f2\u7387\u306e\u200b\u200b\u5bfe\u79f0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u5dee\u5225\u5316\u3055\u308c\u305f\u591a\u69d8\u6027\u306b\u3064\u3044\u3066 m {displaystyle m} \u3069\u3093\u306a\u6587\u8108\u3067\u3082\u3001\u66f2\u7387\u306f\u6700\u521d\u306e2\u3064\u306e\u30a8\u30f3\u30c8\u30ea\u3067\u66f2\u304c\u3063\u3066\u3044\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u9069\u7528\u3055\u308c\u307e\u3059 \u6700\u521d\u306e\u4ea4\u63db\u5bfe\u79f0 R(X,Y)Z=\u2212R(Y,X)Z{displaystyle r\uff08x\u3001y\uff09z = -r\uff08y\u3001x\uff09zqquad}  Rabcd=\u2212Rabdc\u21d4Rab(cd)=0{displaystyle r_ {abcd} = -r_ {abdc} leftrightarrow r_ {ab\uff08cd\uff09} = 0}  \u30ea\u30fc\u30de\u30f3\u306e\u591a\u69d8\u6027\u306e\u305f\u3081\u306b \uff08 m \u3001 g \uff09\uff09 {displaystyle\uff08m\u3001g\uff09} Levi-Civita\u306e\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3067\u306f\u3001\u4ee5\u4e0b\u3082\u9069\u7528\u3055\u308c\u307e\u3059 2\u756a\u76ee\u306e\u4ea4\u63db\u5bfe\u79f0 g(R(X,Y)Z,T)=\u2212g(R(X,Y)T,Z){displaystyle g\uff08r\uff08x\u3001y\uff09z\u3001t\uff09= -g\uff08r\uff08x\u3001y\uff09t\u3001z\uff09qquad}  Rabcd=\u2212Rbacd\u21d4R(ab)cd=0{displaystyle r_ {abcd} = -r_ {bacd} leftrightarrow r _ {\uff08ab\uff09cd} = 0}  \u30d6\u30ed\u30c3\u30af\u4ea4\u63db\u5bfe\u79f0\u6027 g(R(X,Y)Z,T)=g(R(Z,T)X,Y){displaystyle g\uff08r\uff08x\u3001y\uff09z\u3001t\uff09= g\uff08r\uff08z\u3001t\uff09x\u3001y\uff09}  Rabcd=Rcdab{displaystyle r_ {abcd} = r_ {cdab}}  \u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u306f m {displaystyle m} \u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3068\u306e\u5dee\u5225\u5316\u3055\u308c\u305f\u591a\u69d8\u6027 \u2207 {displaystyle nabla} \u305d\u3057\u3066\u305d\u3046\u3067\u3059 \u306e \u3001 \u30d0\u30c4 \u3001 \u3068 \u3001 \u3068 \u2208 c \u221e \uff08 m \u3001 t m \uff09\uff09 {displaystyle W\u3001X\u3001Y\u3001Zin Gamma ^{infty}\uff08m\u3001tm\uff09} \u30d9\u30af\u30c8\u30eb\u30d5\u30a3\u30fc\u30eb\u30c9\u3001\u6b21\u306b\u3001\u6700\u521d\u306e\u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u304c\u9069\u7528\u3055\u308c\u307e\u3059 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3068 + r \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u30d0\u30c4 + r \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 \u3068 = \uff08 \u2207 Xt \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 + t \uff08 t \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3001 \u3068 \uff09\uff09 + \uff08 \u2207 Yt \uff09\uff09 \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 + t \uff08 t \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u3001 \u30d0\u30c4 \uff09\uff09 + \uff08 \u2207 Zt \uff09\uff09 \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 + t \uff08 t \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 \u3001 \u3068 \uff09\uff09 {displaystyle r\uff08x\u3001y\uff09z+r\uff08y\u3001z\uff09x+r\uff08z\u3001x\uff09y =\uff08nabla _ {x} t\uff09\uff08y\u3001z\uff09+t\uff08t\uff08x\u3001y\uff09\u3001z\uff09+\uff08nabla _ {y} t\uff09\uff08z\u3001x\uff09+t\uff08y\u3001z\uff09\u3001x\uff09 } \u306d\u3058\u308c\u30b9\u30bf\u30ce\u3067 t {displaystylet} \u3068 \uff08 \u2207 \u30d0\u30c4 t \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 = \u2207 \u30d0\u30c4 \uff08 t \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \uff09\uff09  –  t \uff08 \u2207 \u30d0\u30c4 \u3068 \u3001 \u3068 \uff09\uff09  –  t \uff08 \u3068 \u3001 \u2207 \u30d0\u30c4 \u3068 \uff09\uff09 \u3002 {displaStyle\uff08nabla _ {x} t\uff09\uff08y\u3001z\uff09= napla _ {x\uff08y\u3001z\uff09\uff09 –  t\uff08napla _ {x} y\u3001z\uff09-t\uff08y\u3001nah\u3001nabla _ {x} z\uff09\u3002}}  2\u756a\u76ee\u306e\u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u306f\u3067\u3059 \uff08 \u2207 Xr \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 + r \uff08 t \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3001 \u3068 \uff09\uff09 + \uff08 \u2207 Yr \uff09\uff09 \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 + r \uff08 t \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u3001 \u30d0\u30c4 \uff09\uff09 + \uff08 \u2207 Zr \uff09\uff09 \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 + r \uff08 t \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 \u3001 \u3068 \uff09\uff09 = 0 {displaystyle\uff08nabla _ {x} r\uff09\uff08y\u3001z\uff09+r\uff08t\uff08x\u3001y\uff09\u3001z\uff09+\uff08nabla _ {y} r\uff09\uff08z\u3001x\uff09+r\uff08t\uff08y\u200b\u200b\u3001z\uff09\u3001x\uff09+\uff08nabla _ {z} r\uff09\uff08x\u3001y\uff09+r\uff08z\u3001x\uff09\u3001x\uff09\u3001y\uff09= 0} \u3068 \uff08 \u2207 \u30d0\u30c4 r \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u306e = \u2207 \u30d0\u30c4 \uff08 r \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u306e \uff09\uff09  –  r \uff08 \u2207 \u30d0\u30c4 \u3068 \u3001 \u3068 \uff09\uff09 \u306e  –  r \uff08 \u3068 \u3001 \u2207 \u30d0\u30c4 \u3068 \uff09\uff09 \u306e  –  r \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u2207 \u30d0\u30c4 \u306e \u3002 {displaystyle\uff08nabla _ {x} r\uff09\uff08y\u3001z\uff09w = nabla _ {x}\uff08r\uff08y\u3001z\uff09w\uff09-r\uff08nabla _ {x} y\u3001z\uff09w-r\uff08y\u3001nabla _ {x} z\uff09w-r\uff08y\u3001z\uff09nabla _ {x} w\u3002} w\u3002  \u306f \u2207 {displaystyle nabla} \u306d\u3058\u308c – \u30d5\u30ea\u30fc\u3001\u3053\u308c\u306f\u3053\u308c\u3089\u306e\u65b9\u7a0b\u5f0f\u304c\u5358\u7d14\u5316\u3059\u308b\u65b9\u6cd5\u3067\u3059 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3068 + r \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u30d0\u30c4 + r \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 \u3068 = 0 {displaystyle r\uff08x\u3001y\uff09z+r\uff08y\u3001z\uff09x+r\uff08z\u3001x\uff09y = 0} \u3068 \uff08 \u2207 Xr \uff09\uff09 \uff08 \u3068 \u3001 \u3068 \uff09\uff09 + \uff08 \u2207 Yr \uff09\uff09 \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 + \uff08 \u2207 Zr \uff09\uff09 \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 = 0\u3002 {displaystyle\uff08nabla _ {x} r\uff09\uff08y\u3001z\uff09+\uff08nabla _ {y} r\uff09\uff08z\u3001x\uff09+\uff08nabla _ {z} r\uff09\uff08x\u3001y\uff09= 0.} \u306f \uff08 m \u3001 g \uff09\uff09 {displaystyle\uff08m\u3001g\uff09} Levi-Civita\u306e\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3068\u306eRiemann\u306e\u591a\u69d8\u6027 \u2207 {displaystyle nabla} \u3001\u6b21\u306b\u3001\u6700\u521d\u306e\u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u304c\u9069\u7528\u3055\u308c\u307e\u3059 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3068 + r \uff08 \u3068 \u3001 \u3068 \uff09\uff09 \u30d0\u30c4 + r \uff08 \u3068 \u3001 \u30d0\u30c4 \uff09\uff09 \u3068 = 0 {displaystyle r\uff08x\u3001y\uff09z+r\uff08y\u3001z\uff09x+r\uff08z\u3001x\uff09y = 0} \u305d\u3057\u3066\u30012\u756a\u76ee\u306e\u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u306f\u53ef\u80fd\u3067\u3059 \u2207 Wg \uff08 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u3068 \u3001 \u306e \uff09\uff09 + \u2207 Zg \uff08 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u306e \u3001 \u306e \uff09\uff09 + \u2207 Vg \uff08 r \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 \u306e \u3001 \u3068 \uff09\uff09 = 0 {displaystyle nabla _ {w} g\uff08r\uff08x\u3001y\uff09z\u3001v\uff09+nabla _ {z} g\uff08r\uff08x\u3001y\uff09v\u3001w\uff09+nabla _ {v} g\uff08r\uff08x\u3001y\uff09w\u3001z\uff09= 0} \u66f8\u304f\u3002\u6700\u521d\u306e\u30d3\u30a2\u30f3\u30ad\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u306f\u3001\u4ee3\u6570\u30d3\u30a2\u30f3\u30ad\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u3068\u3082\u547c\u3070\u308c\u30012\u756a\u76ee\u306e\u5fae\u5206\u30d3\u30a2\u30f3\u30ad\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u3068\u3082\u547c\u3070\u308c\u307e\u3059\u3002\u3053\u308c\u3089\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u306f\u3001\u6570\u5b66\u8005\u306e\u30eb\u30a4\u30fc\u30b8\u30fb\u30d3\u30a2\u30f3\u30ad\u306b\u3061\u306a\u3093\u3067\u547d\u540d\u3055\u308c\u3066\u3044\u307e\u3059\u3002 \u610f\u5473 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30ea\u30fc\u30de\u30f3\u306e\u591a\u69d8\u6027 \uff08 m \u3001 g \uff09\uff09 {displaystyle\uff08m\u3001g\uff09} \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306b\u5bfe\u3057\u3066\u5c40\u6240\u7684\u306b\u30a2\u30a4\u30bd\u30e1\u30c8\u30ea\u30fc\u3067\u3042\u308b\u5834\u5408\u3001\u5e73\u3089\u306a\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u3042\u3089\u3086\u308b\u70b9\u306b\u3064\u3044\u3066\u3067\u3059 p \u2208 m {DisplayStyle Pin M} \u74b0\u5883\u304c\u3042\u308a\u307e\u3059 \u306e {displaystyleu} \u305d\u3057\u3066\u30a4\u30e9\u30b9\u30c8 \u03d5 \uff1a \u306e \u2192 \u306e \u2282 r n {displaystyle phi colon uto vsubset mathbb {r} ^{n}} \u3053\u308c\u306f\u30a2\u30a4\u30bd\u30e1\u30c8\u30ea\u30c3\u30af\u306a\u306e\u3067\u3001\u3069\u3061\u3089\u306e\u5834\u5408\u3067\u3059 g \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 = \u03d5 \u2217 g \u00af \uff08 \u30d0\u30c4 \u3001 \u3068 \uff09\uff09 = g \u00af \uff08 \u03d5 \u2217 \u30d0\u30c4 \u3001 \u03d5 \u2217 \u3068 \uff09\uff09 {displaystyle g\uff08x\u3001y\uff09= phi ^{*} {overline {g}}\uff08x\u3001y\uff09= {overline {g}}\uff08phi _ {*} x\u3001phi _ {*}}} \u9069\u7528\u53ef\u80fd\u3067\u3059\u3002\u3053\u3053\u3067\u8aac\u660e\u3057\u307e\u3059 g \u00af {displaystyle {overline {g}}} \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u30b9\u30ab\u30e9\u30fc\u88fd\u54c1\u3068 \u03d5 \u2217 {displaystyle phi _ {*}} \u306e\u30d7\u30c3\u30b7\u30e5\u30d5\u30a9\u30ef\u30fc\u30c9 \u03d5 {displaystylephi} \u3002 \u66f2\u7387\u3078\u306e\u63a5\u7d9a [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Levi-Civita\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3092\u6301\u3064Riemann\u306e\u591a\u69d8\u6027 \u2207 {displaystyle nabla} \u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387\u304c\u30bc\u30ed\u3068\u540c\u4e00\u3067\u3042\u308b\u5834\u5408\u3001\u6b63\u78ba\u306b\u5e73\u3089\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30cf\u30f3\u30c9\u30eb\u30d9\u30a2\u30d6\u30eb\u9818\u57df\u306f\u3001\u30d5\u30e9\u30c3\u30c8\u306a\u591a\u69d8\u6027\u306b\u5bfe\u3059\u308b2\u6b21\u5143\u306e\u985e\u4f3c\u7269\u3067\u3059\u3002 \u30d1\u30f3\u7c89 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Riemann\u5e7e\u4f55\u5b66\u306e\u6700\u3082\u91cd\u8981\u306a\u66f2\u7387\u5909\u6570\u306e1\u3064\u306f\u3001\u4e3b\u8981\u306a\u66f2\u7387\u3067\u3059\u3002\u30ac\u30a6\u30b9\u306e\u901a\u5e38\u306e\u5730\u57df\u306e\u66f2\u7387\u3092\u4e00\u822c\u5316\u3057\u307e\u3059\u3002\u3059\u3079\u3066\u306e\u30ec\u30d9\u30eb\u306f\u3067\u3059 a {displaystyle sigma} \u30ea\u30fc\u30de\u30f3\u306e\u591a\u69d8\u6027\u306e\u6642\u70b9\u3067\u306e\u63a5\u7dda\u90e8\u5c4b\u3067 m {displaystyle m} \u66f2\u7387\u304c\u5272\u308a\u5f53\u3066\u3089\u308c\u307e\u3057\u305f\u3002\u3053\u308c\u306f\u3001\u4e2d\u306e\u30a8\u30ea\u30a2\u306e\u66f2\u7387\u3067\u3059 m {displaystyle m} \u3001 a {displaystyle sigma} \u63a5\u7dda\u30ec\u30d9\u30eb\u3068\u3057\u3066\u3001\u591a\u69d8\u6027\u306e\u4e2d\u3067\u6e7e\u66f2\u3057\u3066\u3044\u307e\u305b\u3093\u3002 a {displaystyle sigma} \u3002\u305f\u3060\u3057\u3001\u5b9a\u7fa9\u306f\u3053\u306e\u9818\u57df\u306e\u52a9\u3051\u3092\u501f\u308a\u3066\u884c\u308f\u308c\u308b\u306e\u3067\u306f\u306a\u304f\u3001Riemann\u306e\u66f2\u7387\u3068\u30ec\u30d9\u30eb\u306e2\u3064\u306e\u30d9\u30af\u30c8\u30eb\u306e\u52a9\u3051\u3092\u501f\u308a\u3066\u5b9f\u884c\u3055\u308c\u307e\u3059\u3002 a {displaystyle sigma} \u7559\u3081\u91d1\u3002 \u30ea\u30fc\u30de\u30f3\u306e\u591a\u69d8\u6027\u304c\u3042\u308a\u307e\u3059 m {displaystyle m} Riemannscher Metrik\u3068 g {displaystyle g} \u30011\u3064\u306e\u70b9 p {displaystyle p} \u306e m {displaystyle m} \u30682\u6b21\u5143\u30b5\u30d6\u30b9\u30da\u30fc\u30b9\uff08\u30ec\u30d9\u30eb\uff09 a \u2282 t p m {displaystyle sigma\u30b5\u30d6\u30bb\u30c3\u30c8t_ {p} m} \u63a5\u7dda\u90e8\u5c4b\u306e t p m {displaystyle t_ {p} m} \u304b\u3089 m {displaystyle m} \u30dd\u30a4\u30f3\u30c8\u3067 p {displaystyle p} \u3002\u306a\u308c \u306e {displaystyle v} \u3068 \u306e {displaystyle in} \u3053\u306e\u30ec\u30d9\u30eb\u3092\u56fa\u5b9a\u3059\u308b2\u3064\u306e\u63a5\u7dda\u30d9\u30af\u30c8\u30eb\u3002\u3068 | \u306e \u2227 \u306e | = g(v,v)g(w,w)\u2212g(v,w)2{displaystyle | vwedge w | = {sqrt {g\uff08v\u3001v\uff09g\uff08w\u3001w\uff09-g\uff08v\u3001w\uff09^{2}}}}} \u306e\u9818\u57df\u306e\u5834\u5408 \u306e {displaystyle v} \u3068 \u306e {displaystyle in} \u7dca\u5f35\u3057\u305f\u5e73\u884c\u56db\u8fba\u5f62\u3002\u305d\u306e\u5f8c\u3001\u30b5\u30a4\u30ba\u304c\u639b\u3051\u3089\u308c\u307e\u3059 k \uff08 \u306e \u3001 \u306e \uff09\uff09 = g(R(v,w)w,v)|v\u2227w|2= g(R(v,w)w,v)g(v,v)g(w,w)\u2212g(v,w)2{displaystyle k\uff08v\u3001w\uff09= {frac {g\uff08r\uff08v\u3001w\uff09w\u3001v\uff09} {| vwedge w |^{2}}}} = {frac {g\uff08r\uff08v\u3001w\uff09w\u3001v\uff09} {g\uff08v\u3001v\uff09g\uff08w\u3001w\uff09-g\uff08v\u3001w\uff09^{2}}}}}} \u30ec\u30d9\u30eb\u304b\u3089\u306e\u307f a {displaystyle sigma} \u30aa\u30d5\u3067\u3059\u304c\u3001\u30d9\u30af\u30c8\u30eb\u306e\u9078\u629e\u304b\u3089\u3067\u306f\u3042\u308a\u307e\u305b\u3093 \u306e {displaystyle v} \u3068 \u306e {displaystyle in} \u3002\u3060\u304b\u3089\u3042\u306a\u305f\u306f\u66f8\u304f k \uff08 \u306e \u3001 \u306e \uff09\uff09 {displaystyle k\uff08v\u3001w\uff09} \u307e\u305f k \uff08 a \uff09\uff09 {displaystyle k\uff08sigma\uff09} \u3053\u308c\u3092\u547c\u3073\u307e\u3059 \u30d1\u30f3\u7c89 \u304b\u3089 a {displaystyle sigma} \u3002 \u306f m {displaystyle m} 2\u6b21\u5143\u3067\u3001\u3059\u3079\u3066\u306e\u30dd\u30a4\u30f3\u30c8\u306b\u3042\u308a\u307e\u3059 p {displaystyle p} \u304b\u3089 m {displaystyle m} \u63a5\u7dda\u90e8\u5c4b\u306e\u305d\u306e\u3088\u3046\u306a2\u6b21\u5143\u306e\u90e8\u5206\u7a7a\u9593\u3001\u3064\u307e\u308a\u63a5\u7dda\u90e8\u5c4b\u81ea\u4f53\u3060\u3051\u3001\u305d\u3057\u3066 k \uff08 a \uff09\uff09 {displaystyle k\uff08sigma\uff09} \u305d\u306e\u5f8c\u3001\u30ac\u30a6\u30b9\u30eb\u30df\u30f3\u30b0\u3067\u3059 m {displaystyle m} \u30dd\u30a4\u30f3\u30c8\u3067 p {displaystyle p}  \u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u30d5\u30a3\u30fc\u30eb\u30c9\u65b9\u7a0b\u5f0f\u3067\u306f\u3001 \u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb  r m n {displaystyle r_ {mu nu}} \uff08Gregorio Ricci-Curbastro\u306b\u3088\u308b\u3068\uff09\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u30c6\u30f3\u30bd\u30eb\u306e\u82e5\u8fd4\u308a\u306e\u66f2\u7387\u306b\u8d77\u56e0\u3057\u307e\u3059\u3002 r \u03bc\u03bd= \u00b1 r \u03bc\u03bb\u03bd\u03bb{displaystyle r_ {mu nu} = pm r_ {mu lambda nu}^{lambda}}} Einstein Sum\u6761\u7d04\u306b\u3088\u308b\u3068\u3001\u767a\u751f\u3059\u308b\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u304c\u8ffd\u52a0\u3055\u308c\u3066\u304a\u308a\u3001\u305d\u306e\u3046\u3061\u306e1\u3064\u306f\u4e0a\u90e8\u306b\u3042\u308a\u3001\u3082\u30461\u3064\u306f\u4ee5\u4e0b\u3067\u3059\u3002\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u306e\u5f62\u6210\u3001\u30a4\u30f3\u30c7\u30c3\u30af\u30b9 l {displaystyle lambda} \u8ffd\u52a0\u3057\u305f\u3002\u30b5\u30a4\u30f3\u306f\u6163\u7fd2\u306b\u3088\u3063\u3066\u8a2d\u5b9a\u3055\u308c\u3066\u304a\u308a\u3001\u539f\u5247\u3068\u3057\u3066\u81ea\u7531\u306b\u9078\u629e\u3067\u304d\u307e\u3059\u3002 \u30b9\u30ab\u30e9\u30fc\u66f2\u7387 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30ea\u30c3\u30c1\u30c6\u30bd\u30eb\u306e\u30c6\u30f3\u30bd\u30eb\u306e\u82e5\u8fd4\u308a\u307e\u305f\u306f\u53ce\u7e2e\u306f \u66f2\u7387 \uff08\u307e\u305f ricci-sklar \u307e\u305f \u30b9\u30ab\u30e9\u30fc\u66f2\u7387 \uff09\u3002\u305d\u306e\u5f62\u72b6\u3092\u8aac\u660e\u3059\u308b\u305f\u3081\u306b\u3001\u8868\u73fe\u306f\u6700\u521d\u3067\u3059 r k l {displaystyle r_ {kappa}^{lambda}} ricci-tesor\u304b\u3089\u6d3e\u751f\u3057\u305f\uff1a r \u03ba\u03bb= g \u03bc\u03bbr \u03bc\u03ba\u3002 {displaystyle r_ {kappa} ^ {lambda} = g ^ {in lambda} r_ {in cappa}\u3002} \u3042\u308b g m l {displaystyle g^{mu lambda}} \u77db\u76fe\u3057\u305f\u30e1\u30fc\u30c8\u30eb\u30c6\u30f3\u30bd\u30eb\u3002\u66f2\u7387\u306e\u200b\u200b\u30b9\u30b1\u30fc\u30eb\u306f\u53ce\u7e2e\u306b\u8d77\u56e0\u3057\u3001\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u3092\u8d85\u3048\u3066\u3044\u307e\u3059 l {displaystyle lambda} \u8ffd\u52a0\u3057\u305f\u3002 r = r \u03bb\u03bb{displaystyle r = r_ {lambda}^{lambda}} \u66f2\u7387\u30b9\u30b1\u30fc\u30eb\u306f\u3001\u30ea\u30c3\u30c1\u30c6\u30f3\u30bd\u30eb\u304b\u3089\u3082\u76f4\u63a5\u8ca9\u58f2\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059 r m r {displaystyle r_ {mu rho}} \u52dd\u3064\uff1a r = g \u03bc\u03c1r \u03bc\u03c1{displaystyle r = g^{mu rho} r_ {mu rho}} \u30a4\u30f3\u30c7\u30c3\u30af\u30b9 m {displaystyle mu} \u3068 r {displaystyle rho} \u8ffd\u52a0\u3057\u305f\u3002 \u76f8\u5bfe\u6027\u306e\u4e00\u822c\u7406\u8ad6\u3067\u306f\u3001\u66f2\u7387\u306e\u30b9\u30b1\u30fc\u30eb\u306f\u30a2\u30a4\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u8981\u56e0\u306b\u304b\u304b\u3063\u3066\u3044\u307e\u3059 k {displaystyle kappa} \u3068\u3068\u3082\u306b Laue\u30b9\u30b1\u30fc\u30eb  t {displaystylet} \u4e00\u7dd2\u306b\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u885d\u52d5\u304b\u3089\u306e\u53ce\u7e2e\u306b\u3088\u3063\u3066 t n m {displaystyle t_ {nu}^{mu}} \u5f62\u6210\u3055\u308c\u307e\u3059\uff1a t = t \u03bb\u03bb= r \/ k {displaystyle t = t_ {lambda}^{lambda} = r\/kappa} \u2191  ManfredoPerdig\u00e3odoCarmo\uff1a Riemannian Geometry\u3002 1992\u3001S\u300289  \u2191  Sylvestre Gallot\u3001Dominique Hulin\u3001Jacques Lafontaine\uff1a Riemannian Geometry\u3002 \u7b2c2\u7248\u200b\u200b1990\u3001p\u3002107  ManfredoPerdig\u00e3odoCarmo\uff1a Riemannian Geometry\u3002 Birkh\u00e4user\u3001Boston 1992\u3001ISBN 0-8176-3490-8\u3002 Sylvestre Gallot\u3001Dominique Hulin\u3001Jacques Lafontaine\uff1a Riemannian Geometry\u3002 \u7b2c2\u7248\u200b\u200b\u3002 Springer-Verlag\u3001\u30d9\u30eb\u30ea\u30f3 \/\u30cf\u30a4\u30c7\u30eb\u30d9\u30eb\u30af1990\u3001ISBN 3-540-52401-0\u3002 \u30b8\u30e7\u30f3\u30fbM\u30fb\u30ea\u30fc\uff1a Riemannian\u30de\u30cb\u30db\u30fc\u30eb\u30c9\u3002\u66f2\u7387\u306e\u200b\u200b\u7d39\u4ecb\u3002 Springer\u3001New York 1997\u3001ISBN 0387983228\u3002 \u30d4\u30fc\u30bf\u30fc\u30fbW\u30fb\u30df\u30b7\u30e5\u30fc\u30eb\uff1a \u5fae\u5206\u5f62\u72b6\u306e\u30c8\u30d4\u30c3\u30af\u3002 AMS\u3001\u30d7\u30ed\u30d3\u30c7\u30f3\u30b9\u30012008\u5e74\u3001ISBN 978-0-8218-2003-2\u3002      (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/6911#breadcrumbitem","name":"\u30ea\u30fc\u30de\u30f3\u306e\u66f2\u7387 – \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2"}}]}]