Gauss lemma

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 = d r^{2} + G (r, θ) d θ^{2}, G = ∣ ϕ_{θ} ∣^{2},

with

G (r, θ) / r \to 1

r \to 0

Jacobi equation in geodesic polar coordinates ( $h_{rr} + K h = 0$ )

For

h (r, θ) = G (r, θ)

h_{rr} + K (r, θ) h = 0, h (0, θ) = 0, h_{r} (0, θ) = 1.

h (r, θ) d r d θ

h^{''} + K h = 0

h (0) = 0

h^{'} (0) = 1

h > 0

, and

K \leq C

for

C > 0

, then

h (r) \geq \frac{s i n ( C r )}{C} (0 \leq r < π / C) .

Subtract a smaller multiple of the sine solution and use the monotonicity of its Wronskian with

h

K \leq 0

, then

h_{rr} = - K h \geq 0

, so

h \geq r

. Every geodesic polar ball of radius

ε

therefore satisfies

Area B (p, ε) \geq π ε^{2} .

K \leq C

with

C > 0

, scalar comparison gives

h (r, θ) \geq \frac{s i n ( C r )}{C}

before

π / C

. Consequently

Area B (p, ε) \geq \frac{2 π}{C} (1 - cos (C ε)) = π ε^{2} (1 + O (ε^{2})) .

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