= Riemannian manifold
{c}
{title2=$(M,g)$}
{wiki}
A Riemannian manifold is a pair $(M,g)$ consisting of a <smooth manifold> $M$ and a <Riemannian metric> $g$. For each point $p$, the <positive-definite> <inner product> $g_p$ on its <tangent space> tells us the size and <angle> of possible instantaneous movements. The <smooth manifold> describes which local coordinates are possible; the <Riemannian metric> adds a choice of measurement.
For example, the plane can carry $ds^2=dx^2+dy^2$ or $ds^2=4dx^2+dy^2$. In the second <Riemannian metric>, a horizontal displacement costs twice as much <length> as the same coordinate displacement in the first. The underlying <smooth manifold> is unchanged. The <coordinate transformation> $X=2x,Y=y$ identifies the second geometry with an ordinary Euclidean plane, illustrating why coordinate appearance alone does not determine <curvature>.
A round <sphere> with its <induced metric> is another <Riemannian manifold>. Its <tangent spaces> each have an ordinary Euclidean <inner product>, although its global geometry is different from a plane's. This is the useful tension: measurements are Euclidean to first order at a point, while how those measurements fit together produces <curvature> and global restrictions on <geodesics>.
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