One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.

Proposition. If a closed manifold $M^4$ admits an Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.

Proof. Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$. This implies $\chi(M)=1/32\pi^2 \int|Rm|^2dV\ge 0$ with equality iff $Rm=0$. $\square$

Combining with another formula for the signature of a 4-manifold $$ \sigma(M)=\frac{1}{12\pi^2}\int_M(|W_+|^2-|W_-|^2)dV, $$ where $W_{\pm}$ are compontens of the Weyl tensor, one can prove a stronger inequality: for a closed Einstein manifold $M^4$ $$ \chi(M)\ge 3/2|\sigma(M)|. $$

The equality case was studied by Nigel Hitchin. He proved that if $(M^4, g)$ is an Einstein manifold with $\chi(M)= 3/2|\sigma(M)|$, then $M$ is Ricci flat and either universal cover of $M$ is K3-surface, or $M$ is flat.

These are important results, since topological obstructions for the existence of Einstein metrics are very scarce. In particular in dimensions $\ge 5$ there are no known such obstructions.

One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.

Proposition. If a closed manifold $M^4$ admits an Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.

Proof. Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$. This implies $\chi(M)=1/32\pi^2 \int|Rm|^2dV\ge 0$ with equality iff $Rm=0$. $\square$

Combining with another formula for the signature of a 4-manifold $$ \sigma(M)=\frac{1}{12\pi^2}\int_M(|W_+|^2-|W_-|^2)dV, $$ where $W_{\pm}$ are compontens of the Weyl tensor, one can prove a stronger inequality: for a closed Einstein manifold $M^4$ $$ \chi(M)\ge 3/2|\sigma(M)|. $$

The equality case was studied by Nigel Hitchin, Compact four-dimensional Einstein manifolds, (1974). He proved that if $(M^4, g)$ is an Einstein manifold with $\chi(M)= 3/2|\sigma(M)|$, then $M$ is Ricci flat and either universal cover of $M$ is K3-surface, or $M$ is flat.

These are important results, since topological obstructions for the existence of Einstein metrics are very scarce. In particular in dimensions $\ge 5$ there are no known such obstructions.

One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.

Proposition. If a closed manifold $M^4$ admits Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.

Proof. Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$, so $\chi(M)=1/32\pi^2 \int|Rm|^2dV\ge 0$ $\square$.

Combining with another formula for the signature of a 4-manifold $$ \sigma(M)=\frac{1}{12\pi^2}\int_M(|W_+|^2-|W_-|^2)dV, $$ where $W_{\pm}$ are compontens of the Weyl tensor, one can prove a stronger inequality: for a closed Einstein manifold $M^4$ $$ \chi(M)\ge 3/2|\sigma(M)|. $$

The equality case was studied but Nigel Hitchin. He proved that if $(M^4, g)$ is an Einstein manifold with $\chi(M)= 3/2|\sigma(M)|$, then $M$ is Ricci flat and either universal cover of $M$ is K3-surface, or $M$ is flat.

These are important results, since in higher dimensions there are no known topological obstructions for the existence of Einstein metrics.

One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.

Proposition. If a closed manifold $M^4$ admits an Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.

Proof. Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$. This implies $\chi(M)=1/32\pi^2 \int|Rm|^2dV\ge 0$ with equality iff $Rm=0$. $\square$

Combining with another formula for the signature of a 4-manifold $$ \sigma(M)=\frac{1}{12\pi^2}\int_M(|W_+|^2-|W_-|^2)dV, $$ where $W_{\pm}$ are compontens of the Weyl tensor, one can prove a stronger inequality: for a closed Einstein manifold $M^4$ $$ \chi(M)\ge 3/2|\sigma(M)|. $$

The equality case was studied by Nigel Hitchin. He proved that if $(M^4, g)$ is an Einstein manifold with $\chi(M)= 3/2|\sigma(M)|$, then $M$ is Ricci flat and either universal cover of $M$ is K3-surface, or $M$ is flat.

These are important results, since topological obstructions for the existence of Einstein metrics are very scarce. In particular in dimensions $\ge 5$ there are no known such obstructions.