Riemannian metrics and geodesics

A Riemannian metric is a smooth symmetric positive-definite bilinear form g at each point. Positive definiteness makes the squared length of every nonzero tangent vector strictly positive. In a chart it is represented by a symmetric matrix, but that matrix must transform with the coordinates. Rescaling a coordinate changes its metric coefficient rather than changing physical length. An indefinite Lorentzian metric is a different structure with different causal and variational interpretations, so the Riemannian shortest-length discussion does not transfer unchanged.

For a piecewise smooth curve, integrate the norm of its velocity to obtain length. The length is unchanged by an orientation-preserving smooth reparametrization of the same route. The metric also identifies covectors with vectors: if df has components ∂ᵢf, the gradient has components $g^{ij}\partial_j f$, where the superscript matrix is the inverse metric. Thus changing the metric changes what steepest ascent means, even when the function and its differential remain fixed.

The Levi-Civita connection associated with g defines covariant acceleration. A geodesic with an affine parameter has zero covariant acceleration and satisfies a second-order differential equation involving Christoffel coefficients. It need not have constant coordinate components. A straight line in the Euclidean plane looks curved in polar coordinates, while the plane remains intrinsically flat. On a round sphere, great circles are geodesics even though their embeddings curve visibly in three-dimensional space. Intrinsic and extrinsic curvature answer different questions.

Given a point and an initial tangent velocity, smoothness gives a unique local geodesic. A sufficiently short geodesic segment minimizes length between its endpoints in an appropriate normal neighborhood. A long segment need not minimize: following a great circle more than halfway around the unit sphere is longer than going the other way. The geodesic equation encodes a local stationary-path condition, not a promise that every solution is the unique global winner among all routes.

Removing a point from the Euclidean plane leaves a smooth Riemannian manifold, but a straight trajectory aimed at the missing point can end after finite parameter time. This manifold is geodesically incomplete. For a connected finite-dimensional Riemannian manifold without boundary, the Hopf-Rinow theorem relates metric completeness to geodesic completeness and guarantees a length-minimizing geodesic between any two points when the manifold is complete. The finite-dimensional Riemannian hypotheses are part of the statement.

Completeness does not imply uniqueness of a minimizing geodesic. Antipodal points on a sphere are joined by many equal-length semicircles. Nor does completeness force zero curvature or compactness; ordinary Euclidean space is complete and noncompact. When applying geometry to optimization or physical configuration spaces, specify whether a theorem is local, whether the domain has excluded boundaries, and whether a numerical path finder is seeking any stationary route or a global minimum.