This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and more general type of family of subspaces in manifolds that we call barycen-tric subspaces. They are implicitly defined as the locus of points which are weighted means of k + 1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces. Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear sub-spaces of increasing dimension). We show that the Euclidean PCA minimizes the sum of the unexplained variance by all the subspaces of the flag, also called the Area-Under-the-Curve (AUC) criterion. Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUC criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds, that we call Barycentric Subspaces Analysis (BSA).