In this article, we study the following Hamiltonian system: \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} -\varepsilon ^{2}\Delta _{g}u +u & = |v|^{q-1}v,\\ -\varepsilon ^{2}\Delta _{g}v +v & = |u|^{p-1}u, & & \text { in } \mathcal {M}, \\ \quad u,v & >0, & & \text { in } \mathcal {M}, \end{aligned} \end{array}\right. } \end{aligned}\) where \(\mathcal {M}\) is a smooth, compact, and connected Riemannian manifold of dimension \(N\ge 3\) without boundary. The exponents \(p,q>1\) are assumed to lie below the critical hyperbola, ensuring subcritical growth conditions. We investigate a sequence of least energy critical points of the associated dual functional and analyze their concentration behavior as \(\varepsilon \rightarrow 0\) . Our main result shows that the sequence of solutions exhibits point concentration, with the concentration occurring at a point where the scalar curvature of \(\mathcal {M}\) attains its maximum.
