Gauss's lemma (Riemannian geometry)

The radial direction is orthogonal to the angular directions under the exponential map. On a surface this makes the first fundamental form in geodesic polar coordinateswith $\sqrt{G(r,\theta )}/r\to 1$ as $r\to 0$.

For $h(r,\theta )=\sqrt{G(r,\theta )}$, the angular Jacobi field equation on a surface becomesThe Riemannian area element is $h(r,\theta )drd\theta$.

If $h^{\prime \prime }+Kh=0$, $h(0)=0$, $h^{\prime }(0)=1$, $h>0$, and $K\le C$ for $C>0$, thenSubtract a smaller multiple of the sine solution and use the monotonicity of its Wronskian with $h$.

If $K\le 0$, then $h_{rr}=-Kh\ge 0$, so $h\ge r$. Every geodesic polar ball of radius $\epsilon$ therefore satisfies

If $K\le C$ with $C>0$, scalar comparison givesbefore $\pi /\sqrt{C}$. Consequently