This presentation establishes a theoretical and computational framework for scalar hyperbolic conservation laws (sHCL) on two-dimensional closed, regular manifolds. Central to this approach is a geometry-compatible (GC) flux formulation defined by prescribed flux-directional vectors, ensuring consistency between the surface divergence and the manifold geometry. This construction induces a natural foliation, decomposing the manifold into leaves along which the 2D sHCL reduces to a family of one-dimensional problems. The global flow dynamics emerge as the collective evolution of these leaf-wise solutions. We validate the theoretical analysis using a cp-WENO scheme, combining the Closest Point Method (CPM) embedding with WENO discretization for essentially non-oscillatory capturing of shock and rarefaction waves. Numerical experiments with the inviscid Burgers’ equation on the sphere and torus reveal that rich flow topology is governed by geometric features. On the sphere, the longest leaf acts as an asymptotic separatrix dividing rotational patterns. On the torus, the dynamics depend critically on the flux vector: degenerate cases form invariant barriers separating counter-rotating flows, while generic cases feature isolated singular points that anchor nonlinear wave interactions.