"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=\ | + | * 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 | * Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons | ||
| − | * Ricci solitons on Finsler spaces | + | * 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== | ||
| + | * In [[general relativity]], [[Einstein field equation]] with a [[cosmological constant]] Λ is | ||
| + | :<math>R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, </math> | ||
| + | written in geometrized units with ''G'' = ''c'' = 1. | ||
| + | * The stress–energy tensor ''T''<sub>''ab''</sub> gives the matter and energy content of the underlying spacetime. | ||
| + | * In a vacuum (a region of spacetime with no matter) ''T''<sub>''ab''</sub> = 0, and one can rewrite Einstein's equation in the form (assuming ''n'' > 2): | ||
| + | :<math>R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}.</math> | ||
| + | * Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with ''k'' proportional to the cosmological constant. | ||
| + | |||
2015년 12월 25일 (금) 02:48 판
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
- In general relativity, Einstein field equation with a cosmological constant Λ is
\[R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, \] 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
- Besse, Arthur L. Einstein Manifolds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://link.springer.com/10.1007/978-3-540-74311-8.
expositions
- Wang, McKenzie Y. “Einstein Metrics from Symmetry and Bundle Constructions: A Sequel.” arXiv:1208.4736 [math], August 23, 2012. http://arxiv.org/abs/1208.4736.
- Cao, Huai-Dong. 2009. “Recent Progress on Ricci Solitons.” arXiv:0908.2006 [math], August. http://arxiv.org/abs/0908.2006.
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.
- 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.