Abstract: In this paper, the manifold geometric structures of nonsingular Hermite matrices <i<H</i< (<em<n</em<) are considered. First, we define a Riemannian metric and introduce the dual <em<α</em<-connections and the <em<α</em<-curvature tensors. Then, the Jacobi fields on manifold <em<H</em< (<em<n</em<) have been considered to investigate the instability of the geodesics in view of differential geometry. Moreover, some examples are given to illustrate our results.