The relationship of the Gaussian curvature with the curvature of a Cowen-Douglas operator

It has been recently shown that if $K$ is a sesqui-analytic scalar valued non-negative definite kernel on a domain $\mathrm{\Omega }$ in ${\mathbb{C}}^m$, then the function $(K^2{\mathrm{\partial }}_i{\bar{\mathrm{\partial }}}_j\log K{)}_{i,j=1}^m,$ is also a non-negative definite kernel on $\mathrm{\Omega }$. In this paper, we discuss two consequences of this result. The first one strengthens the curvature inequality for operators in the Cowen-Douglas class $B_1(\mathrm{\Omega })$ while the second one gives a relationship of the reproducing kernel of a submodule of certain Hilbert modules with the curvature of the associated quotient module.