Features for Riemannian N-manifold Classification

Riemannian n-manifolds are topological spaces with a riemannian metric, where Euclidean spaces of dimension n are just special cases. Some straightforward classification rules are available in the topology field by first spotting some core features in any instance of a riemannian n-manifold so as to then assign it to its corresponding homeomorphic counterpart, thus preserving their topological properties, such as connectedness or compactness. Hence, the aim of this study is to present the appropriate topological invariants in order to classify riemannian n-manifolds when n = {1, 2, 3}. Whilst this paper is mainly focused on surface classification, some notes about classifying curves and volumes are also established by touching on their appropriate riemannian n-manifolds.