An Einstein manifold is a type of Riemannian manifold where the Ricci curvature is proportional to the metric tensor. This means that the way the manifold curves is uniform in a specific sense, making it a key object of study in differential geometry. Einstein manifolds are important in both mathematics and theoretical physics, particularly in the context of general relativity. These manifolds can be classified based on their scalar curvature, which is a measure of how the manifold bends in space. Examples of Einstein manifolds include spheres and toroids, which exhibit constant curvature. Understanding these structures helps in exploring the geometric properties of spaces and their implications in various scientific fields.