"리만 곡률 텐서"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
39번째 줄: 39번째 줄:
 
 
 
 
  
−
<h5 style="margin: 0px; line-height: 2em;">rhr</h5>
+
<h5 style="margin: 0px; line-height: 2em;">곡면의 경우</h5>
 +
 
 +
 
 +
 
 +
<math>\begin{array}{ll}  \begin{array}{ll}  R_{111}^1 & 0 \\  R_{112}^1 & 0 \end{array}  &  \begin{array}{ll}  R_{121}^1 & 0 \\  R_{122}^1 & 0 \end{array}  \\  \begin{array}{ll}  R_{211}^1 & 0 \\  R_{212}^1 & \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)} \end{array}  &  \begin{array}{ll}  R_{221}^1 & -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)} \\  R_{222}^1 & 0 \end{array}  \\  \begin{array}{ll}  R_{111}^2 & 0 \\  R_{112}^2 & -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2} \end{array}  &  \begin{array}{ll}  R_{121}^2 & \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2} \\  R_{122}^2 & 0 \end{array}  \\  \begin{array}{ll}  R_{211}^2 & 0 \\  R_{212}^2 & 0 \end{array}  &  \begin{array}{ll}  R_{221}^2 & 0 \\  R_{222}^2 & 0 \end{array}  \end{array}</math>
 +
 
 +
 
 +
 
 +
 
  
 
 
 
 

2012년 1월 16일 (월) 14:17 판

이 항목의 수학노트 원문주소

 

 

개요
  • 접속 (connection)\(\nabla\)이 정의되어 있다고 하자
  • 세 개의 벡터장 X,Y,Z 가 주어지면, 새로운 벡터장 R(X,Y)Z 를 얻는다
    \(R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\)
  • covariant tensor

 

 

성분
  • \({R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})\)
  • 텐서 표현 
    \(R(\partial_{i},\partial_{j})={R^l}_{kij} dx^{k}\otimes \frac{\partial}{\partial x^{l}}\)
  • 크리스토펠 기호 를 이용한 성분의 계산
    \({R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\)
    \({R^l}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{is}\Gamma^s_{jk} - \Gamma^l_{js}\Gamma^s_{ik}\)
    \(R_{\rho\sigma\mu\nu} = g_{\rho \zeta} {R^\zeta}_{\sigma\mu\nu} .\)

 

 

곡률 2-form

\(R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s\)

\(\Omega_i^j =\frac{1}{2} R_{kli}^j \phi^k \wedge \phi^l \)

\(\Omega_i^j = d\omega_i^j - \omega_i^k \wedge \omega_k^j \)

 

 

곡면의 경우

 

\(\begin{array}{ll} \begin{array}{ll} R_{111}^1 & 0 \\ R_{112}^1 & 0 \end{array} & \begin{array}{ll} R_{121}^1 & 0 \\ R_{122}^1 & 0 \end{array} \\ \begin{array}{ll} R_{211}^1 & 0 \\ R_{212}^1 & \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)} \end{array} & \begin{array}{ll} R_{221}^1 & -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)} \\ R_{222}^1 & 0 \end{array} \\ \begin{array}{ll} R_{111}^2 & 0 \\ R_{112}^2 & -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2} \end{array} & \begin{array}{ll} R_{121}^2 & \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2} \\ R_{122}^2 & 0 \end{array} \\ \begin{array}{ll} R_{211}^2 & 0 \\ R_{212}^2 & 0 \end{array} & \begin{array}{ll} R_{221}^2 & 0 \\ R_{222}^2 & 0 \end{array} \end{array}\)

 

 

 

 

 

Ricci tensor &Ricci scalar

 

 

 

역사

 

 

 

메모

 

 

 

관련된 항목들

 

 

매스매티카 파일 및 계산 리소스

 

 

 

수학용어번역

 

 

 

사전 형태의 자료

 

 

관련논문

 

 

관련도서

 

 

링크
원본 주소 "https://wiki.mathnt.net/index.php?title=리만_곡률_텐서&oldid=8987"