Network coding encourages information mixing at the intermediate nodes within a network. The multiple-unicast conjecture proposed by Li and Li  in 2004 is one of the most well-known unsolved problems in network coding field. The conjecture asserts that, for multiple independent unicast transmissions in an undirected network, network coding has no advantage over traditional routing. In this thesis, we study the conjecture by embedding graphs into Riemannian manifolds using a geometric framework developed by Xiahou el al. . We prove that isometric embedding of graphs into a Riemannian manifold is impossible. Then, interestingly, we construct an embedding that achieves an infinitesimally small distortion. We show that if the multiple-unicast network coding conjecture is true on Riemannian manifolds, it is also true for undirected networks. Our hope is to develop a Riemannian geometry approach for making new progresses against the long-time open conjecture.