Einsteinian manifold

The Einstein manifold or Einstein manifold is a term from the mathematical branch of differential geometry as well as the general theory of relativity . It is a special case of a (pseudo-) Riemannian manifold and was named after the physicist Albert Einstein .

definition

A pseudo-Riemannian manifold is called an Einstein manifold if there is a real constant such that ${\displaystyle (M,g)}$ ${\displaystyle \lambda }$

${\displaystyle \operatorname {Ric} _{p}(X,Y)=\lambda \,g_{p}(X,Y)}$

applies. Here is the (0,2) - Ricci tensor and for each The pseudo-Riemannian metric is called an Einstein metric under these conditions. ${\displaystyle \operatorname {Ric} _{p}}$ ${\displaystyle X,Y\in T_{p}M}$ ${\displaystyle p\in M.}$ ${\displaystyle g}$

properties

• Einstein's manifolds are only of independent interest for dimensions , since they coincide for and with the spaces with constant scalar curvature or constant sectional curvature . ${\displaystyle n\geq 4}$ ${\displaystyle n=2}$ ${\displaystyle n=3}$

• Let Then an n-dimensional pseudo-Riemannian manifold is Einsteinian if and only if for each there is a constant (depending on ) such that ${\displaystyle n\geq 3.}$ ${\displaystyle p\in M}$ ${\displaystyle \lambda _{p}}$ ${\displaystyle p\,}$

${\displaystyle \operatorname {Ric} _{p}(X,Y)=\lambda _{p}\,g_{p}(X,Y)}$

applies. In contrast to the definition, it depends on the point of the manifold. ${\displaystyle \lambda }$

• The Cartesian product of two Einstein manifolds, which both have the same constant , is again an Einstein manifold with constant . ${\displaystyle \lambda }$ ${\displaystyle \lambda }$

• The definition of the Einstein metric results from the statement that a solution of Einstein's vacuum field equations ${\displaystyle g}$ ${\displaystyle g}$

${\displaystyle \operatorname {Ric} _{p}(X,Y)-{\frac {1}{2}}\,g_{p}(X,Y)\,s_{p}+g_{p}(X,Y)\,\Lambda =0}$

with the cosmological constant and the scalar curvature . By creating a trace in the equation , one obtains ${\displaystyle \Lambda }$ ${\displaystyle s_{p}}$ ${\displaystyle \operatorname {Ric} _{p}(X,Y)=\lambda \,g_{p}(X,Y)}$

${\displaystyle s_{p}=n\lambda ,}$

denotes the dimension of the manifold. ${\displaystyle n\,}$

literature

• Arthur L. Besse : Einstein Manifolds. Reprint of the 1987 edition. Springer, Berlin et al. 2008, ISBN 978-3-540-74120-6 ( Classics in mathematics ).