Abstract
In this paper we introduce and study the properties of a statistical concircular vector field and the Laplacian of a smooth function on a statistical manifold endowed with the dual connection \(\nabla^*\) . Focusing on the compact case, we obtain the fundamental integral formula for a statistical Ricci tensor \(\mathcal Rc\) : \(\int_W\{\mathcal Rc(\nabla^*\psi,\nabla^*\psi)+m(m-1)\mu^2\}\,dv=0.\) We prove that if a compact, connected statistical manifold \((W^m,\nabla^*,h)\) admits a nontrivial statistical concircular vector field \(\nabla^*\psi\) and its statistical Ricci curvature satisfies a specific lower bound condition on the first eigenvalue of the statistical Laplacian \(\Delta^*\) , then the manifold must be isometric to a standard sphere \(S^m(a)\) .
