Einstein metrics and Ricci solitons

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 T_ab gives the matter and energy content of the underlying spacetime.
• In a vacuum (a region of spacetime with no matter) T_ab = 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

• 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.