Riemannian manifold

Remarks

Example: the unit sphere $S^n \subset \R^{n+1}$ with $g_p(u, v) = u \cdot v$ for $u, v \in T_pS^n \subset \R^{n+1}$ is a Riemannian manifold. Indeed $g_p$ is the restriction of the dot product, so it is symmetric and positive definite, and it depends smoothly on $p$. More generally, every immersed submanifold of $\R^N$ is a Riemannian manifold with the induced metric, and by the Nash embedding theorem every Riemannian manifold arises this way. Non-example: $\R^2$ with $g = dx^2 - dy^2$ is not a Riemannian manifold, because $g(\partial_y, \partial_y) = -1 < 0$; it is a pseudo-Riemannian manifold. A connected Riemannian manifold is a metric space, with distance the infimum of the lengths of piecewise smooth curves joining two points, and this metric induces the manifold topology.