My teacher of Riemannian Geometry told the class about the problem of finding Einstein Manifolds in 5 dimension. My question is, what is the difficulty of this problem? Possessing a first course in Riemmanian geometry can I understand this?
Thanks
My teacher of Riemannian Geometry told the class about the problem of finding Einstein Manifolds in 5 dimension. My question is, what is the difficulty of this problem? Possessing a first course in Riemmanian geometry can I understand this?
Thanks
Answer for the compact homogeneous case: Due to Alekseevsky, Dotti and Ferraris (Homogeneous Ricci positive 5-manifolds, Pacific. J. Math, 175, 1-12, 1996) we know that in the non-symmetric case, the classification of 5-dimensional compact homogeneous Einstein spaces reduces to the classification of Einstein metrics on the space $M_{a, b}=SU(2)\times SU(2)/S^{1}$, with a specific embedding of $S^{1}$. Here, $a\geq b\geq 0$, $a> 0$ are integers with ${\rm gcd}(a, b)>1$.
Wang and Ziller (Einstein metrics on principal torus bundles, J. Differ. Geom., 31, (1990) 215-248), proved that the spaces $M_{a, b}$ for $(a, b)\neq (1, 1)$ admit exactly one (up to isometry) homogeneous Einstein metric and the same result was obtained indepedently by Rodionov. Wang and Ziller also proved that $M_{1, 1}$ admits at least one invariant Einstein metric. Note that the isotropy representation of $M_{1, 1}$ decomposes into three irreducible submodules, two of them are isomorphic to each other, and thus a $SU(2)^{2}$-invariant Riemannian metric is depending on four positive real parameters. Now, $M_{1, 1}$ admits a second homogeneous Einstein metric, which is isometric to a product of symmetric metrics on $S^{3}\times S^{2}$. In fact, the space $M_{1, 1}$ can be viewed as the unit tangent bundle $T_1S^{3}=SO(4)/SO(2)$ of the 3-sphere $S^{3}$, and thus as the Stiefel manifold $V_{2}(\mathbb{R}^{4})$ of 2-frames in $\mathbb{R}^{4}$. Notice also that the dimension $n=5$ and examples like the 5-sphere $S^{5}$, the product $S^{3}\times S^{2}$ and the space $M_{1, 1}$, are very important examples of 5-dimensional homogeneous or cohomogeneity-one Sasaki-Einstein manifolds. Finally, the sphere $S^{5}$ can be considered either as the irreducible symmetric space $SO(6)/SO(5)$, or as $SU(3)/SU(2)$ with a certain invariant metric. The isotropy representation of the second presentation decomposes into the sum of two irreducible summands. The proof of the fact that the canonical metric of $S^{5}=SU(3)/SU(2)$ is the unique $SU(3)$-invariant Einstein metric, was given by Jensen.
Difficulty of constructing general Einstein metrics: It is good to notice that Einstein metrics are relatively rare among Riemannian metrics. Although no topological obstructions are known for big dimensions, it is difficult to construct such metrics because the Einstein equation is a system of second order PDEs and there is no general method known. In recent decades, Einstein manifolds became the subject of many studies (both of mathematicians and physicists). Motivated from classical mechanics and the well known fact that symmetry considerations can simplify the study of geometric problems, the current strategy for constructing examples of Einstein metrics is to assume some symmetry condition for the metric tensor itself, that means that there is a Lie group which acts on the manifold in a smooth way by isometries. Roughly speaking, important progress has been done for homogeneous Riemannian $G$-manifolds and cohomogeneity-one Riemannian $G$-manifolds. The fist case arises from a transitive action of a Lie group $G$ on $(M, g)$, and the second one appears when the Lie group $G$ acts on $(M, g)$ with principal orbits of codimension one. Also, Einstein metrics with special holonomy is a famous topic (with a lot of interest in physics) and there are several results for this kind of Einstein metrics, e.g. Einstein-Sasakian manifolds, 6-dim nearly Kähler manifolds, 7-dim nearly parallel $G_2$-manifolds, etc, and all of them admit real Killing spinors. Also, Ricci-flatness is related with parallel spinors.
Finally: Possessing a first course in Riemmanian geometry and having some basic knowledge of representation theory... it is not difficult to prove for example that $SO(6)/SO(5)$ admits a unique $SO(6)$-invariant Einstein metric. But other problems related with Einstein metrics may need more expertise and studying.
Simply connected 5-manifolds are classified by simple invariants, second homology and second Stiefel-Whitney class. So one can consider the existence problem, i.e. which 5-manifolds admit an Einstein metric. Many are know to have an Eintein metric. See the paper by János Kollár "Einstein metrics on 5-dimensional Seifert bundles".