"Einstein metrics and Ricci solitons"의 두 판 사이의 차이

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==introduction==
 
==introduction==
* A Riemannian manifold $(M, g)$ is called Einstein if it has constant Ricci curvature, i.e. $Ric_g=\kappa \cdot g$ for some $kappa\in \mathbb{R}$
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* A Riemannian manifold <math>(M, g)</math> is called Einstein if it has constant Ricci curvature, i.e. <math>Ric_g=\kappa \cdot g</math> for some <math>\kappa\in \mathbb{R}</math>
 
* In local coordinates,
 
* In local coordinates,
$$
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:<math>
 
R_{ab} = \kappa\,g_{ab}
 
R_{ab} = \kappa\,g_{ab}
$$
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</math>
 
* Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons
 
* Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons
 
* Ricci solitons on Finsler spaces are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds.
 
* Ricci solitons on Finsler spaces are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds.
 
  
 
==Einstein equation==
 
==Einstein equation==
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==articles==
 
==articles==
* Arvanitoyeorgos, Andreas, Yusuke Sakane, and Marina Statha. “New Einstein Metrics on the Lie Group $SO(n)$ Which Are Not Naturally Reductive.” arXiv:1511.08849 [math], November 25, 2015. http://arxiv.org/abs/1511.08849.
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* John Lott, Patrick Wilson, Note on asymptotically conical expanding Ricci solitons, arXiv:1605.02128 [math.DG], May 07 2016, http://arxiv.org/abs/1605.02128
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* Grama, Lino, and Ricardo Miranda Martins. “A Numerical Treatment to the Problem of the Quantity of Einstein Metrics on Flag Manifolds.” arXiv:1601.06972 [math-Ph], January 26, 2016. http://arxiv.org/abs/1601.06972.
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* Arvanitoyeorgos, Andreas, Yusuke Sakane, and Marina Statha. “New Einstein Metrics on the Lie Group <math>SO(n)</math> Which Are Not Naturally Reductive.” arXiv:1511.08849 [math], November 25, 2015. http://arxiv.org/abs/1511.08849.
 
* Stolarski, Maxwell. “Steady Ricci Solitons on Complex Line Bundles.” arXiv:1511.04087 [math], November 12, 2015. http://arxiv.org/abs/1511.04087.
 
* Stolarski, Maxwell. “Steady Ricci Solitons on Complex Line Bundles.” arXiv:1511.04087 [math], November 12, 2015. http://arxiv.org/abs/1511.04087.
 
* Bidabad, Behroz, and Mohamad Yar Ahmadi. “On Compact Ricci Solitons in Finsler Geometry.” arXiv:1508.02148 [math], August 10, 2015. http://arxiv.org/abs/1508.02148.
 
* Bidabad, Behroz, and Mohamad Yar Ahmadi. “On Compact Ricci Solitons in Finsler Geometry.” arXiv:1508.02148 [math], August 10, 2015. http://arxiv.org/abs/1508.02148.
37번째 줄: 38번째 줄:
 
* Fernandez-Lopez, Manuel, and Eduardo Garcia-Rio. “On Gradient Ricci Solitons with Constant Scalar Curvature.” arXiv:1409.3359 [math], September 11, 2014. http://arxiv.org/abs/1409.3359.
 
* Fernandez-Lopez, Manuel, and Eduardo Garcia-Rio. “On Gradient Ricci Solitons with Constant Scalar Curvature.” arXiv:1409.3359 [math], September 11, 2014. http://arxiv.org/abs/1409.3359.
 
* Catino, Giovanni, Paolo Mastrolia, Dario D. Monticelli, and Marco Rigoli. 2014. “Conformal Ricci Solitons and Related Integrability Conditions.” arXiv:1405.3169 [math], May. http://arxiv.org/abs/1405.3169.
 
* Catino, Giovanni, Paolo Mastrolia, Dario D. Monticelli, and Marco Rigoli. 2014. “Conformal Ricci Solitons and Related Integrability Conditions.” arXiv:1405.3169 [math], May. http://arxiv.org/abs/1405.3169.
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2020년 11월 13일 (금) 07:22 기준 최신판

introduction

  • A Riemannian manifold \((M, g)\) is called Einstein if it has constant Ricci curvature, i.e. \(Ric_g=\kappa \cdot g\) for some \(\kappa\in \mathbb{R}\)
  • In local coordinates,

\[ R_{ab} = \kappa\,g_{ab} \]

  • Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons
  • Ricci solitons on Finsler spaces are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds.

Einstein equation

\[R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, \] written in geometrized units with G = c = 1.

  • The stress–energy tensor Tab gives the matter and energy content of the underlying spacetime.
  • In a vacuum (a region of spacetime with no matter) Tab = 0, and one can rewrite Einstein's equation in the form (assuming n > 2):

\[R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}.\]

  • Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.


books


expositions


articles

  • John Lott, Patrick Wilson, Note on asymptotically conical expanding Ricci solitons, arXiv:1605.02128 [math.DG], May 07 2016, http://arxiv.org/abs/1605.02128
  • Grama, Lino, and Ricardo Miranda Martins. “A Numerical Treatment to the Problem of the Quantity of Einstein Metrics on Flag Manifolds.” arXiv:1601.06972 [math-Ph], January 26, 2016. http://arxiv.org/abs/1601.06972.
  • Arvanitoyeorgos, Andreas, Yusuke Sakane, and Marina Statha. “New Einstein Metrics on the Lie Group \(SO(n)\) Which Are Not Naturally Reductive.” arXiv:1511.08849 [math], November 25, 2015. http://arxiv.org/abs/1511.08849.
  • Stolarski, Maxwell. “Steady Ricci Solitons on Complex Line Bundles.” arXiv:1511.04087 [math], November 12, 2015. http://arxiv.org/abs/1511.04087.
  • Bidabad, Behroz, and Mohamad Yar Ahmadi. “On Compact Ricci Solitons in Finsler Geometry.” arXiv:1508.02148 [math], August 10, 2015. http://arxiv.org/abs/1508.02148.
  • Nurowski, Pawel, and Matthew Randall. “Generalised Ricci Solitons.” arXiv:1409.4179 [gr-Qc], September 15, 2014. http://arxiv.org/abs/1409.4179.
  • Fernandez-Lopez, Manuel, and Eduardo Garcia-Rio. “On Gradient Ricci Solitons with Constant Scalar Curvature.” arXiv:1409.3359 [math], September 11, 2014. http://arxiv.org/abs/1409.3359.
  • Catino, Giovanni, Paolo Mastrolia, Dario D. Monticelli, and Marco Rigoli. 2014. “Conformal Ricci Solitons and Related Integrability Conditions.” arXiv:1405.3169 [math], May. http://arxiv.org/abs/1405.3169.