A Riemannian metric equips each tangent space with an inner product that varies smoothly. It converts topology into quantitative geometry: lengths, angles, volumes, gradients, and distances all arise from this single tensor.
Metrics and their existence
Definition. A Riemannian metric on \(M\) is a smooth symmetric \((0,2)\)-tensor \(g\) such that \(g_p(v,v)>0\) for every \(v\ne0\). In coordinates, \[ g=g_{ij}\,\mathrm{d}x^i\otimes\mathrm{d}x^j, \] where \((g_{ij})\) is positive definite.
Theorem. Every smooth manifold admits a Riemannian metric.
Proof. Choose a locally finite atlas \((U_\alpha,x_\alpha)\) and a subordinate partition of unity \((\rho_\alpha)\). Pull the Euclidean metric back to each \(U_\alpha\) and set \[ g=\sum_\alpha\rho_\alpha\,x_\alpha^*g_{\mathrm{Euc}}. \] The sum is locally finite and smooth. At each point at least one \(\rho_\alpha\) is positive, so \(g(v,v)>0\) for every nonzero \(v\).
Length, energy, and distance
Definition. For a piecewise smooth curve \(\gamma:[a,b]\to M\), \[ L(\gamma)=\int_a^b|\dot\gamma(t)|_g\,\mathrm{d}t,\qquad E(\gamma)=\frac12\int_a^b|\dot\gamma(t)|_g^2\,\mathrm{d}t. \] The Riemannian distance is \[ d_g(p,q)=\inf_{\gamma(a)=p,\gamma(b)=q}L(\gamma). \]
Proposition. On a connected Riemannian manifold, \(d_g\) is a metric and induces the original manifold topology.
Proof outline. Symmetry and the triangle inequality follow by reversing and concatenating curves. In a relatively compact coordinate ball, eigenvalue bounds \[ c|v|^2\le g_x(v,v)\le C|v|^2 \] compare Riemannian and Euclidean lengths. This proves both positivity of \(d_g(p,q)\) for \(p\ne q\) and local equivalence of their topologies.
Cauchy-Schwarz gives \[ L(\gamma)^2\le2(b-a)E(\gamma), \] with equality exactly for constant-speed parametrizations. Thus fixed-endpoint energy minimizers have constant speed and minimize length.
Musical isomorphisms and gradient
The metric defines \[ \flat:T_pM\to T_p^*M,\quad v^\flat=g(v,\cdot), \qquad \sharp=(\flat)^{-1}. \] The gradient of \(f\) is \[ \operatorname{grad}f=(\mathrm{d}f)^\sharp,\qquad g(\operatorname{grad}f,X)=Xf. \] In coordinates, \((\operatorname{grad}f)^i=g^{ij}\partial_jf\).
Volume and Laplacian
On an oriented manifold, \[ \operatorname{vol}_g=\sqrt{\det(g_{ij})}\, \mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n. \] The divergence and Laplace-Beltrami operator are \[ \operatorname{div}X =\frac1{\sqrt{|g|}}\partial_i(\sqrt{|g|}X^i),\qquad \Delta_gf =\frac1{\sqrt{|g|}}\partial_i \left(\sqrt{|g|}g^{ij}\partial_jf\right). \] They satisfy Green's identity \[ \int_M h\Delta_gf\,\mathrm{d}V_g =-\int_Mg(\nabla h,\nabla f)\,\mathrm{d}V_g +\int_{\partial M}h\,\partial_\nu f\,\mathrm{d}A_g. \]
Example (conformal change). If \(\widetilde g=e^{2u}g\) on an \(n\)-manifold, then \[ \mathrm{d}V_{\widetilde g}=e^{nu}\mathrm{d}V_g,\qquad \Delta_{\widetilde g}f =e^{-2u}\bigl(\Delta_gf+(n-2)g(\nabla u,\nabla f)\bigr). \] In dimension two the gradient correction disappears, reflecting conformal invariance of harmonicity.