The Scalar Curvature of a Projectively Invariant Metric Defined by the Kernel Function

Abstract

Considering a projectively invariant metric $\tau$ defined by the kernel function on a strongly convex bounded domain $\Omega\subset\mathbb{R}^n$, we study the asymptotic expansion of the scalar curvature with respect to the distance function, and use the Fubini-Pick invariant to describe the second term in the expansion. This asymptotic expansion implies that if $n\geq 3$ and $(\Omega,\tau )$ has constant scalar curvature, then the convex domain is projectively equivalent to a ball.