9c6c8fc5bd-p-73.pdf - 7.1 METRICAS RIEMANNIANAS Y...

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7.1. M ´ ETRICAS RIEMANNIANAS Y LORENTZIANAS 133 (1) riemanniana si g p es un producto escalar eucl´ ıdeo de T p Q para todo p Q . (2) lorentziana si g p es una producto escalar lorentziano de T p Q para todo p Q . (3) semi-riemanniana si g p es un producto escalar de T p Q con ´ ındice s constante para todo p Q 1 . Seg´un el caso, llamaremos al par ( Q, g ) variedad riemanniana , lorentziana o semi-riemanniana , respectivamente. En adelante todas las m´ etricas que consideremos ser´an semi-riemannianas, salvo especificaci´on contraria. Ejemplos: (1) La m´ etrica riemanniana usual de R n es g 0 = dx 1 dx 1 + · · · + dx n dx n n i =1 ( dx i ) 2 . An´alogamente, la m´ etrica lorentziana usual se define como g L = - dx 1 dx 1 + dx 2 dx 2 + · · · + dx n dx n ≡ - ( dx 1 ) 2 + n i =2 ( dx i ) 2 . (2) Sea S una subvariedad de R n . Como T p S es un subespacio de T p R n para todo p S , la m´ etrica usual g 0 de R n puede restrin- girse a T p S para proporcionar un producto escalar eucl´ ıdeo g S p sobre cada T
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  • Winter '15
  • Producto escalar, Variedad de Riemann, Tp S, m´etrica usual

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