The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network buses, and is a nonconvex problem that is generally hard to solve. By leveraging a recent development on the zero duality gap of OPF, we propose a second-order cone programming convex relaxation of the resistive network OPF, and study the uniqueness of the optimal solution using differential topology, especially the Poincare-Hopf Index Theorem. We characterize the global uniqueness for different network topologies, e.g., line, radial, and mesh networks. This serves as a starting point to design distributed local algorithms with global behaviors that have low complexity, are computationally fast, and can run under synchronous and asynchronous settings in practical power grids.