Convex programming plays a fundamental role in machine learning, data science, and engineering. Disciplined Programming tests and verifies convexity by decomposing a function into basic convex functions (atoms) using convexity-preserving compositions and transformations (rules). We extend disciplined programming to the geodesic setting, allowing for certifying convexity in nonlinear programs on geometric domains. We determine convexity-preserving compositions and transformations for geodesically convex functions on general Cartan-Hadamard manifolds, as well as for the special case of symmetric positive definite matrices, a common setting in matrix-valued optimization. Our paper is accompanied by a Julia package SymbolicAnalysis.jl, which interfaces with manifold optimization software, allowing for directly solving verified geodesically convex programs.