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C.F.G
C.F.G
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A good paper in this topicdirection is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$, $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$ equality holding if and only if $M$ has constant curvature.

A good paper in this topic is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$, $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$ equality holding if and only if $M$ has constant curvature.

A good paper in this direction is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$, $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$ equality holding if and only if $M$ has constant curvature.

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Source Link
C.F.G
C.F.G
  • 4.2k
  • 6
  • 31
  • 65

A good paper in this topic is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$, $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$ equality holding if and only if $M$ has constant curvature.

A good paper in this topic is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

A good paper in this topic is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$, $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$ equality holding if and only if $M$ has constant curvature.

Source Link
C.F.G
C.F.G
  • 4.2k
  • 6
  • 31
  • 65

A good paper in this topic is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$ whose sectional curvatures are non-negative or nonpositive has non-negative Euler-Poincare characteristic. If the sectional curvatures are always positive or always negative, the Euler-Poincare characteristic is positive.

Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be non-negative.