  Regression on manifolds, and, more broadly, statistics on manifolds, has
garnered significant importance in recent years due to the vast number of
applications for this type of data. Circular data is a classic example, but so
is data in the space of covariance matrices, data on the Grassmannian manifold
obtained as a result of principal component analysis, among many others. In
this work we investigate prediction sets for regression scenarios when the
response variable, denoted by $Y$, resides in a manifold, and the covariable,
denoted by X, lies in Euclidean space. This extends the concepts delineated in
[Lei and Wasserman, 2014] to this novel context. Aligning with traditional
principles in conformal inference, these prediction sets are distribution-free,
indicating that no specific assumptions are imposed on the joint distribution
of $(X, Y)$, and they maintain a non-parametric character. We prove the
asymptotic almost sure convergence of the empirical version of these regions on
the manifold to their population counterparts. The efficiency of this method is
shown through a comprehensive simulation study and an analysis involving
real-world data.
