Obstructions to Einstein metrics in high dimensions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:17:50Z http://mathoverflow.net/feeds/question/69427 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69427/obstructions-to-einstein-metrics-in-high-dimensions Obstructions to Einstein metrics in high dimensions Viktor Bundle 2011-07-03T23:40:20Z 2012-01-29T19:32:19Z <p>It is well known that there exists three and four manifolds that do not admit an Einstein metric, but I wonder if this question is still open for manifolds of dimension higher than four. That is, does anyone know of a compact n-manifold, $n>4$, that does not admit an Einstein metric?</p> http://mathoverflow.net/questions/69427/obstructions-to-einstein-metrics-in-high-dimensions/69473#69473 Answer by Dmitri for Obstructions to Einstein metrics in high dimensions Dmitri 2011-07-04T15:23:42Z 2011-07-04T15:23:42Z <p>No one knows this. Here is a citation from Gromov's beautiful article: <a href="http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf" rel="nofollow">http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf</a></p> <p>Page 19.</p> <p><em>Following Alex we (I speak for myself) are lead to the pessimistic conclusion that there is no chance for a distinguished</em> $g_{best}$ <em>for</em> $n\ge 5$ <em>and "natural" metrics, e.g. Einstein $G$, $Ri(g)=\lambda g$ for $\lambda&lt;0$, must be chaotically scattered in the vastness of $G$ with no meaningful link between geometry and topology (This does not preclude, but rather predicts, the existence of such metrics, e.g. Einstein, on all $V$ if dimension $\ge 5$: the problem is there may be too many of them)</em></p> http://mathoverflow.net/questions/69427/obstructions-to-einstein-metrics-in-high-dimensions/86442#86442 Answer by math3.14159 for Obstructions to Einstein metrics in high dimensions math3.14159 2012-01-23T13:12:53Z 2012-01-29T19:32:19Z <p>In general, for dimensions $n>5$ we dont have topological obstructions to the existence of Einstein metrics (like as Hitchin-Thorpe inequality when $n=4$).</p> <p>However, there exist <strong>compact homogeneous Riemannian manifolds</strong> with <strong>no $G$-invariant Einstein metrics.</strong> In this case we work on cosets $G/K$ of a compact Lie group $G$ and we consider the Einstein equation $Ric = c \cdot g$ for a $G$-invariant Riemannian metric on $M=G/K$. For such a metric the Einstein equation reduces to a polynomial system and positive real solutions correspond to invariant Einstein metrics.</p> <p>Moreover, if $\frak{g}=\frak{k}\oplus\frak{m}$ is a reductive decomposition for $G/K$, and we assume that tangent space $\frak{m}=T_{o}G/K$ decomposes into $q$ pairwise inequivalent isotropy summands (i.e., irreducible $Ad(K)$-submodules)</p> <p>$\frak{m}= \frak{m}_1 + \frak{m}_2 + ..... + \frak{m}_q,$</p> <p>then any $G$-invariant Riemannian metric is given by </p> <p>$g = &lt; , > = x_1\cdot (-B)|_{\frak{m}_1}+\cdots+x_s\cdot (-B)|_{\frak{m}_q}$</p> <p>for some positive real numbers $x_1, x_2, ... x_q$. Here $-B$ is the negative of the Killing form. (the induced inner product on the tangent space).</p> <p>Homogeneous Einstein metrics are real soulutions of the system ${r_1-r_2=0, ...., r_{q-1}-r_{q}=0}$ where $r_{i}$ are the components of the invariant Ricci tensor on any isotropy summand. given by </p> <p>$$\label{ricc} r_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{i, j}\frac{x_{k}}{x_{i}x_{j}}[ijk]-\frac{1}{2d_{k}}\sum_{i, j}\frac{x_{j}}{x_{k}x_{i}}[kij], \qquad (k=1, \ldots, q).$$ Here $[ijk]$ are the <strong>structure constants</strong> of $G/K$. Their computation is usually non-trivial. For particular classes of homogeneous spaces (e.g. isotropy irreducible), they are related with the Casimir operator of $\frak{g}$. They are non-zero only for these triples $\frak{m}_i$, $\frak{m}_j$, $\frak{m}_k$, for which </p> <p>$B([\frak{m}_i, \frak{m}_j], \frak{m}_k)\neq 0$. </p> <p>Positive real solutions of the above system may exist or not. Therefore, there are homogeneous spaces with no invariant Einstein metrics. For example, <strong>Wang and Ziller</strong>, by applying the variational approach of homogeneous Einstein metrics on compact homogeneous spaces, they proved that the <strong>12-dimensional</strong> space $SU(4)/SU(2)$ does not admit any homogeneous Einstein metric.</p> <p>see: M. Wang and W. Ziller: Existence and non-excistence of homogeneous Einstein metrics}, Invent.~Math.~84 (1986) 177--194.</p> <p>This space seems to be the lowest dimensional example of a compact homogeneous manifold with no invariant Einstein metrics. In particular, from a recent work of B\"ohm and Kerr we know that :</p> <p><strong>Theorem:</strong> Any simply connected compact homogeneous Einstein manifod admts at least an invariant Einstein metric.</p> <p>B\"ohm and M. Kerr: Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc. 358 (4) (2005) 1455--1468. </p> <p>Other interesting examples of compact homogeneosu spaces with no invariant Einstein metrics were given in the latter article, but also in a work of Sakane and Park: </p> <p>J-S. Park and Y. Sakane: Invariant Einstein metrics on certain homogeneous spaces, Tokyo J. Math. 20 (1) (1997) 51--61. </p> <p>More general arguments about the existence of homogeneous EInstein metrics (which are based on the topology of compact homogeneous spaces and applications of variational analysis), are presented in the articles:</p> <p>1) C. B\"ohm, M. Wang and W. Ziller: A variational approach for homogeneous Einstein metrics}, Geom. Funct. Anal. 14 (2004) (4) 681-733.</p> <p>2) C. LeBrun and M. Wang (editors): Surveys in Differential Geometry} Volume VI Essays on Einstein Manifolds, International Press, 1999.</p> <p>3) C. B\"ohm: Homogeneous Einstein metrics and simplicial complexes, J. Diff. Geom. 67 (2004) 79-165.</p> <p>We mention that for the problem of <strong>non-homogeneous Einstein metrics</strong> on homogeneous spaces, <strong>less are knwon</strong> (see the work of <strong>Page, or Bohm</strong> for the existence of non homogeneous Einstein metrics.)</p> <p>For non compact homogeneous manifolds (solvmanifolds, nilmanifolds, etc) we refer the reader to Heber's work and the refernces therin, although a lot of progress has been made in the last decade in this case too (see the works of Laurent, Tamaru, amd others).</p> <p>J. Heber: Noncompact homogeneous Einstein space, Invent.~ Math.~133 (1998) 279-352.</p>