리만 곡률 텐서

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

 

 

리만 곡률 텐서의 성분

• \({R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})\)
• 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 2형식

• \(R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s\)
  
• \(\,\Omega=d\omega +\frac{1}{2}[\omega,\omega]=d\omega +\omega\wedge \omega\)
  
• \(\Omega^i_{j}=d\omega^i_{j} +\sum_k \omega^i_{k}\wedge\omega^k_{j}\)
  

 

 

 

곡면의 경우

• 제1기본형식이 \(E=e(u,v),F=0,G=g(u,v)\) 로 주어진 경우, 리만 곡률 텐서는 다음과 같다 (이외의 \( R_{jkl}^i\)는 0이다)\[ R_{212}^1 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\]\[R_{112}^2 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\]\[R_{221}^1 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\]\[R_{121}^2 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\]
  

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

 

 

메모

• http://www.math.sunysb.edu/~brweber/401s09/coursefiles/Lecture24.pdf
• http://users-phys.au.dk/fedorov/nucltheo/GTR/09/note6.pdf

 

 

 

관련된 항목들

• 접속 (connection)

 

 

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2ZmMGViMGQtMmI4Ny00MmI3LWE4ZTYtYmQyNjZiYWVhMTc5&sort=name&layout=list&num=50
• http://www.wolframalpha.com/input/?i=

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Riemann_curvature_tensor
• The Online Encyclopaedia of Mathematics[1]

 

 

리뷰논문, 에세이, 강의노트

• http://www.math.csusb.edu/faculty/dunn/lecture1.pdf