A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, a smoothly varying positive definite inner product on each tangent space. An indefinite metric tensor instead defines a pseudo-Riemannian manifold.
On each connected component, the induced distance 
is the infimum of the arc lengths of piecewise smooth curves
joining 
to 
.
This distance is defined without assuming completeness.
If the manifold is without boundary
and its Riemannian metric is complete, the infimum is attained by a minimizing geodesic
(Burtscher 2025, pp. 21-22 and 76-77).
Every connected complete Riemannian manifold without boundary is boundedly compact. This is part of the Hopf-Rinow theorem.
