Ricci curvature of affine connections (Q957505)

For an affine connection on a smooth manifold \(M\), one can define the Ricci curvature as a contraction of the Riemann curvature tensor. The Ricci curvature can be interpreted as a bilinear form on the tangent bundle \(TM\), which is symmetric provided that the connection preserves some volume form on \(M\). The aim of this article is to construct examples of affine connections in low dimensions for which the Ricci curvature is definite. After reviewing some basic facts about Ricci curvature, the author constructs an affine connection on the torus \(T^n\) for \(n\geq 2\) which is projectively flat, and for which the Ricci curvature is negative definite, divergence free, and has constant determinant. Next, it is shown that any parallelizable manifold of dimension \(n\geq 3\) admits an affine connection with positive definite Ricci curvature. The last part is devoted to the case of dimension two. The author constructs an affine connection with negative definite Ricci curvature on the Klein bottle. Finally it is shown that the method does not extend to the sphere \(S^2\).