= Solution
A <connected> <Riemannian manifold> is complete when its <Riemannian distance> is a complete <metric space>, meaning that every <Cauchy sequence> converges in the manifold. The <Hopf-Rinow theorem> says that for a <connected> finite-dimensional <Riemannian manifold>, this is equivalent to <geodesic completeness>, and also to <compactness> of every closed bounded subset. Under these equivalent conditions any two points are joined by a <minimizing geodesic>, and the <Riemannian exponential map> at every point is defined on its entire <tangent space>.
We use the usual connected, boundaryless-manifold convention for the global statements in Q2–Q5. Connectedness is essential, including for the diameter and group-growth conclusions. Without connectedness, the completeness and covering assertions apply to the target components met by the map; an unrelated incomplete component cannot be controlled by a <local isometry> into another component.
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