Riemannian Metrics

Google Maps computes routes on the WGS-84 ellipsoid: the Moscow-New York geodesic is 418 km shorter than the Mercator flat-map path.

• **GPS:** WGS-84 with specific g_{ij} is the global standard for all GPS devices. Routes are geodesics on this manifold.
• **GR:** Schwarzschild metric describes gravity. Riemannian geometry is the language of Einstein's equations.
• **Riemannian ML:** Poincare embeddings (Facebook AI, 2017) use the hyperbolic space metric for hierarchical data.

The Metric Tensor

Google Maps uses a Riemannian metric on the WGS-84 ellipsoid (semi-major axis a=6,378,137 m) to compute shortest routes: the geodesic between Moscow and New York is 418 km shorter than the Mercator flat-map path.

Isometries and Symmetry Groups

Einstein's general relativity uses the pseudo-Riemannian Minkowski metric ds^2=-c^2 dt^2+dx^2+dy^2+dz^2. The isometry group of this metric is the Lorentz group, which describes the relativity of simultaneity at speeds ~c.

Key ideas

• **Metric tensor g_{ij}:** symmetric (0,2)-tensor. Inner product <u,v>_p=g_{ij}u^iv^j in T_pM.
• **Volume form:** dvol=sqrt(det g) dx^1 wedge...wedge dx^n. For S^2: dvol=sin(theta) dtheta wedge dphi.
• **Isometries:** f*g=g (isometry), L_X g=0 (Killing field). dim Isom(M)<=n(n+1)/2.