metric tensor

Metric tensor

In differential geometry, a metric tensor is a type of tensor that defines the metric (distance) between nearby points in a curved space (a riemannian manifold).
It generalizes the concept of a dot product to curved spaces (like the inner product on euclidian geometry), allowing us to measure lengths and angles locally:

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j$$

Where $ds$ is the infinitesimal distance between two points, $g_{ij}$ are the components of the positive definite metric tensor, and $d{x}^i$, $d{x}^j$ are infinitesimal coordinate differences. The metric tensor varies from point to point on the manifold, encoding its curvature and geometry.