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  "body": "\n## Definition\n\nA *Riemannian manifold* is a pair $(M, g)$ where $M$ is a [[smooth manifold|F45VQ47GPV]] and $g$ is a [[Riemannian metric|Q61VFHCH2J]] on $M$.\nAn isometry between Riemannian manifolds $(M, g)$ and $(N, h)$ is a diffeomorphism $\\varphi : M \\to N$ with $\\varphi^* h = g$.\n\n## Remarks\n\nExample: 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.\nIndeed $g_p$ is the restriction of the dot product, so it is symmetric and positive definite, and it depends smoothly on $p$.\nMore 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.\nNon-example: $\\R^2$ with $g = dx^2 - dy^2$ is not a Riemannian manifold, because $g(\\partial_y, \\partial_y) = -1 \u003c 0$; it is a pseudo-Riemannian manifold.\nA 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.\n",
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