Riemannian metrics and length functionals

Let M be a smooth finite-dimensional manifold. A Riemannian metric assigns a positive-definite symmetric bilinear form $g_p$ to each tangent space, with smooth components in every chart. For a vector with coordinate column v, squared norm is $v^{\mathsf T}Gv$, where G is the metric matrix at that point. Off-diagonal entries describe nonorthogonal coordinate directions, not a failure of positive definiteness. Unlike a Lorentzian metric, every nonzero tangent vector has strictly positive squared norm.

Under a coordinate change, vector components and the metric matrix transform together, leaving their pairing unchanged. Thus an identity matrix is a statement about a chosen frame, not a coordinate-independent definition of flatness. One can choose an orthonormal frame at a point for every Riemannian metric, including a curved one. An embedded surface inherits a metric by restricting the ambient inner product to tangent vectors, but intrinsic constructions below need only g and the smooth manifold structure.

For a piecewise smooth curve on a finite parameter interval, length integrates its metric speed. An orientation-preserving regular reparameterization leaves that length unchanged because the speed change cancels the parameter differential. Energy instead integrates half the squared speed and depends on how the same geometric path is traversed. Fixing the parameter duration makes energy a useful variational quantity, but changing that duration changes its numerical value even when the path and length stay the same.

The Cauchy–Schwarz inequality bounds the square of the integrated speed by duration times the integrated squared speed. Equality holds precisely when speed is constant almost everywhere. Thus among parameterizations of a fixed regular path on a fixed duration, constant speed minimizes energy. This statement concerns parameterization of one path; finding a shortest path among different curves is another optimization problem. It also does not imply a globally shortest curve exists between every pair without further geometric assumptions.

On a connected Riemannian manifold, define the distance between two points as the infimum of lengths of piecewise smooth connecting curves. Connected smooth manifolds are path connected, so this infimum is finite. Positive definiteness and local comparison with Euclidean coordinates establish positive distance for distinct points, while reversing and concatenating curves give symmetry and the triangle inequality. The resulting metric induces the original manifold topology, even though its numerical distances depend on g.

An infimum need not be attained on an incomplete manifold: a sequence of routes can approach a missing point and approach a limiting length without realizing it. Later completeness theorems supply conditions guaranteeing minimizing geodesics. Constant rescaling illustrates the distinction between coordinates and geometry: replacing g by c²g for positive c multiplies all lengths and distances by c and energies by c². Merely relabeling coordinates changes no intrinsic measurement when the tensor is transformed correctly.