Riemannian metric

Remarks

Equivalently, $g$ is a smooth symmetric tensor field of type $(0, 2)$ that is positive definite at every point. In a chart, $g = \sum_{i,j} g_{ij} \, dx^i \otimes dx^j$ where $(g_{ij}(p))$ is a symmetric positive definite matrix depending smoothly on $p$; one often writes $ds^2 = g_{ij} \, dx^i dx^j$. Example: on the upper half-plane $\set{(x, y) : y > 0}$, $g = (dx^2 + dy^2)/y^2$ is a Riemannian metric; its matrix $y^{-2} I$ is symmetric and positive definite because $y > 0$, and its entries are smooth. Non-example: on $\R^2$, $g = x^2 (dx^2 + dy^2)$ fails clause 1 on the line $x = 0$, where $g_p = 0$ is not positive definite. The length of a piecewise smooth curve $\gamma : [a, b] \to M$ is $\int_a^b \sqrt{g(\gamma'(t), \gamma'(t))} \, dt$. Every smooth manifold (Hausdorff and second countable) has a Riemannian metric: glue Euclidean metrics of charts with a partition of unity; positive definiteness survives convex combinations.