Riemannian Metrics and Geodesics

A Riemannian metric measures lengths and angles on a manifold, and geodesics are the curves that locally minimize length — the curved-space analogues of straight lines.

Definition

A Riemannian metric assigns to each tangent space a positive-definite inner product depending smoothly on the point; a geodesic is a curve that locally minimizes length, equivalently one whose velocity is parallel along itself.

Scope

This topic defines the Riemannian metric as a smoothly varying inner product on tangent spaces, the resulting notions of arc length, angle, and Riemannian volume, and the distance function making a connected Riemannian manifold a metric space. It develops geodesics both as length-minimizing curves and as solutions of the geodesic equation, the exponential map and normal coordinates, geodesic completeness, and the Hopf-Rinow theorem relating completeness to the existence of minimizing geodesics. Isometries and the variational characterization of geodesics are included.

Core questions

• How does a metric turn a smooth manifold into a metric space with a well-defined distance?
• In what sense are geodesics the straightest and locally shortest curves?
• How does the exponential map provide canonical coordinates around a point?
• When does geodesic completeness guarantee minimizing geodesics between any two points (Hopf-Rinow)?

Key concepts

• Riemannian metric, arc length, and volume
• Riemannian distance function and isometries
• Geodesic equation and length minimization
• Exponential map and normal coordinates
• Geodesic completeness and the Hopf-Rinow theorem

Clinical relevance

Geodesics model free particle motion and light paths in relativity, optimal paths in shape spaces and robotics, and the shortest routes on curved surfaces; the metric structure makes a manifold a genuine geometric and metric-space object.

History

Riemann introduced the metric in 1854; the variational study of geodesics matured in the late 19th and early 20th centuries, and the Hopf-Rinow theorem (1931) clarified the equivalence of metric and geodesic completeness, completing the foundational picture taught today.

Key figures

• Bernhard Riemann
• Heinz Hopf
• Willi Rinow

Seminal works

• lee1997
• docarmo1992

Frequently asked questions

Are geodesics always the shortest paths?

Only locally. A geodesic minimizes length between sufficiently close points, but globally a geodesic between two distant points may not be the shortest — for example, a great-circle arc going the long way around a sphere.

What does the Hopf-Rinow theorem guarantee?

On a connected Riemannian manifold, geodesic completeness, metric completeness, and the property that closed bounded sets are compact are all equivalent, and any of them ensures every pair of points is joined by a minimizing geodesic.