is a smooth embedded submanifold of Euclidean space . The graph of a smooth function is a smooth embedded submanifold of with its standard metric. If is not simply connected, there is a covering map , where is the universal cover of . This is an immersion (since it is locally a diffeomorphism), so automatically inherits a Riemannian metric. By the same principle, any smooth covering space of a Riemannian manifold inherits a Riemannian metric.
is a smooth embedded submanifold of Euclidean space . The graph of a smooth function is a smooth embedded submanifold of with its standard metric. If is not simply connected, there is a covering map , where is the universal cover of  This is an immersion (since it is locally a diffeomorphism), so automatically inherits .a Riemannian metric. By the same principle, any smooth covering space of a Riemannian manifold inherits a Riemannian metric.