A Mayer optimal control problem with finitely many endpoint constraints is considered. The aim of the paper is the derivation of second order necessary optimality conditions for excluding non optimal critical controls satisfying the Pontryagin maximum principle ( Theorem 2.1). The big challenge of the paper is the assumption that the trajectories belong to a smooth \(n\)-dimensional Riemannian manifold. In the contrary to the Euclidean case the Riemannian curvature tensor applied to the arguments co-state \(p^\ell\), \(X_u\),\(f[t]\) and \(X_u\) occurs in the correlated integral inequality condition of second order (Theorem 2.2). Here \(f[t]\) is the right hand side of the state equation at the found candidates for the control \(\bar u(\cdot)\) and the state \(\bar x(\cdot)\). \(X_u(t)\) is an element of the tangent space at \(\bar x(t)\) of the Riemannian surface satisfying the linearized state equation at \(\bar x(t),\bar u(t)\) with the additional summand \(f[t]-f(t,\bar x(t),u(t))\). Applying some second order needle variation a quasi point-wise condition of second order is proven. Quasi point-wise means that in the condition occurs also some integral term containing the Riemannian curvature tensor. A main role in the proofs plays the construction of convex sets (Lemma 4.1–Lemma 4.4), along critical directions \(X_u\) w.r.t. second order variation for applying separation arguments to get second order necessary conditions. Further, compactness and convexity conditions used in former papers for the control set are not needed here. Some computed examples illustrate the found second order conditions. The main results are proven in detail. For Theorem 2.1. only a sketch of the proof is given. The interested reader should be very familiar with calculus and correlated notations in Riemannian manifolds and the known constructions of second order conditions for control problems over Euclidean spaces. The proofs are mainly self-content.