Editing Metric space (section)

=== Riemannian manifolds ===
{{Main|Riemannian manifold}}
A [[Riemannian manifold]] is a space equipped with a Riemannian [[metric tensor]], which determines lengths of [[tangent space|tangent vectors]] at every point.  This can be thought of defining a notion of distance infinitesimally.  In particular, a differentiable path <math>\gamma:[0, T] \to M</math> in a Riemannian manifold {{mvar|M}} has length defined as the integral of the length of the tangent vector to the path:
<math display="block">L(\gamma)=\int_0^T |\dot\gamma(t)|dt.</math>
On a connected Riemannian manifold, one then defines the distance between two points as the infimum of lengths of smooth paths between them.  This construction generalizes to other kinds of infinitesimal metrics on manifolds, such as [[sub-Riemannian manifold|sub-Riemannian]] and [[Finsler manifold|Finsler metrics]].

The Riemannian metric is uniquely determined by the distance function; this means that in principle, all information about a Riemannian manifold can be recovered from its distance function.  One direction in metric geometry is finding purely metric ([[synthetic geometry|"synthetic"]]) formulations of properties of Riemannian manifolds.  For example, a Riemannian manifold is a [[CAT(k) space|{{math|CAT(''k'')}} space]] (a synthetic condition which depends purely on the metric) if and only if its [[sectional curvature]] is bounded above by {{mvar|k}}.{{sfn|Burago|Burago|Ivanov|2001|p=127}}  Thus {{math|CAT(''k'')}} spaces generalize upper curvature bounds to general metric spaces.