측지선

수학노트
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개요

  • n차원 다양체 M의 coordinate chart 에서 \(\alpha(t)=(\alpha_1(t),\alpha_2(t),\cdots, \alpha_n(t))\) 로 표현되는 곡선이 측지선이 될 조건은 크리스토펠 기호를 사용하여 다음 미분방정식으로 쓸 수 있다

\[\frac{d^2\alpha_k }{dt^2} + \sum_{i,j}\Gamma^{k}_{~i j }\frac{d\alpha_i }{dt}\frac{d\alpha_j }{dt} = 0,\quad k=1,2,\cdots, n\]
또는\[\ddot{\alpha_k } + \sum_{i,j}\Gamma^{k}_{~i j }\dot{\alpha_i}\dot{\alpha_j }= 0,\quad k=1,2,\cdots, n\]

 

 

곡면의 측지선

  • 곡선 (\((x(t),y(t))\) 가 다음의 미분방정식을 만족해야 한다\[x''(t)+\Gamma _{1,1}{}^1 x'(t)^2+\Gamma _{1,2}{}^1 x'(t) y'(t)+\Gamma _{2,1}{}^1 x'(t) y'(t)+\Gamma _{2,2}{}^1 y'(t)^2=0\]\[y''(t)+\Gamma _{1,1}{}^2 x'(t)^2+\Gamma _{1,2}{}^2 x'(t) y'(t)+\Gamma _{2,1}{}^2 x'(t) y'(t)+\Gamma _{2,2}{}^2 y'(t)^2=0\]

 

 

 

 

역사

 

 

 

메모

 

 

관련된 항목들

 

 

 

 

사전 형태의 자료

 

   

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관련논문

  • Christian Lange, On metrics on 2-orbifolds all of whose geodesics are closed, arXiv:1603.08455[math.DG], March 28 2016, http://arxiv.org/abs/1603.08455v1
  • Radeschi, Marco, and Burkhard Wilking. “On the Berger Conjecture for Manifolds All of Whose Geodesics Are Closed.” arXiv:1511.07852 [math], November 24, 2015. http://arxiv.org/abs/1511.07852.
  • Erlandsson, Viveka, and Juan Souto. “Counting Curves in Hyperbolic Surfaces.” arXiv:1508.02265 [math], August 10, 2015. http://arxiv.org/abs/1508.02265.
  • Kennard, Lee, and Jordan Rainone. “Characterizations of the Round Two-Dimensional Sphere in Terms of Closed Geodesics.” arXiv:1507.00414 [math], July 1, 2015. http://arxiv.org/abs/1507.00414.
  • Sapir, Jenya. ‘Lower Bound on the Number of Non-Simple Closed Geodesics on Surfaces’. arXiv:1505.06805 [math], 26 May 2015. http://arxiv.org/abs/1505.06805.