the Gauss lemma
/ GOWSS /
Picture firing geodesics in all directions from a point p, and at each, marking the spot you reach after a fixed distance r — that traces out the 'geodesic sphere' of radius r. The Gauss lemma is the elegant fact that these geodesics hit the geodesic spheres at right angles, exactly as the radii of a Euclidean ball meet its concentric spheres perpendicularly. Curvature bends the picture, but this one orthogonality survives untouched.
Precisely: under the exponential map exp_p, geodesics from p (the images of radial rays from 0 in T_p M) are orthogonal to the geodesic spheres exp_p({ |v| = r }). In the language of the differential, for a radial vector v and any vector w tangent to the sphere of radius |v|, the pushforwards satisfy g( d(exp_p)_v (v), d(exp_p)_v (w) ) = g(v, w) restricted appropriately — the radial direction stays length-true and perpendicular to the angular directions. A direct consequence: in geodesic polar coordinates the metric splits as ds^2 = dr^2 + h(r, angles), with no cross term dr·d(angle); radial distance is literally the parameter r.
Why it matters: the Gauss lemma is the technical heart that makes radial geodesics LOCALLY minimizing. Because the radial part contributes dr^2 with no cross terms, any competing curve from p to a nearby point is at least as long as the radial geodesic, with equality only if it is radial — that is the standard local-minimization proof. It is local: the lemma controls geometry inside a geodesic ball before geodesics start to refocus, and says nothing once you pass a cut point or conjugate point, which is the province of global comparison geometry.
In geodesic polar coordinates on a surface, the Gauss lemma gives ds^2 = dr^2 + f(r, theta)^2 dtheta^2, with f(r,theta) -> r and df/dr -> 1 as r -> 0. The way f deviates from the flat value r encodes the Gaussian curvature: f(r) = r - (K/6) r^3 + ..., so a circle of geodesic radius r has slightly less circumference than 2 pi r on a positively curved surface.
The no-cross-term splitting ds^2 = dr^2 + f^2 dtheta^2 is the Gauss lemma; deviations of f from r measure curvature.
The Gauss lemma proves only LOCAL minimization of radial geodesics, valid inside a normal ball. Past the cut locus a radial geodesic can stop being shortest even though the lemma's orthogonality still holds infinitesimally.