Restricted Mean Value Property with non-tangential boundary behavior on Riemannian manifolds

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP) for domains in $\mathbb{R}^n$. Recently, the author along with Biswas investigated the problem in the general setting of Riemannian manifolds and obtained results in terms of unrestricted boundary limits of the function on a full measure subset of the boundary. However in the context of classical Fatou-Littlewood type theorems for the boundary behavior of harmonic functions, a genuine query is to replace the condition on unrestricted boundary limits with the more natural notion of non-tangential boundary limits. The aim of this article is to answer this question in the local setup for pre-compact domains with smooth boundary in Riemannian manifolds and in the global setup for non-positively curved Harmonic manifolds of purely exponential volume growth. This extends a classical result of Fenton for the unit disk in $\mathbb{R}^2$.