Gauss's lemma (Riemannian geometry)
= Gauss's lemma
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= Gauss's lemma
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= Gauss lemma
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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 coordinates
$$
I=dr^2+G(r,\theta)\,d\theta^2,
\qquad
G=|\phi_\theta|^2,
$$
with $\sqrt{G(r,\theta)}/r\to1$ as $r\to0$.