r = a + R r

, where

R^{T} R = I

\partial_{u} r = R \partial_{u} r .

Therefore

g = (R r_{u}) \cdot (R r_{u}) = r_{u} \cdot r_{u} = g .

Since the integration parameter is unchanged,

S (u) = \int_{0}^{u} g (u^{'}) d u^{'} = S (u) .

Both the induced metric and arc length are invariant.

ii

Solution

For an oriented planar curve, the Frenet-Serret formulas are

\partial_{s} (t n) = (0 - k k 0) (t n) .

The frame is orthonormal, so differentiating each scalar product gives

\partial_{s} (e_{i} \cdot e_{j}) = 0.

In matrix form this says that the connection matrix plus its transpose is zero; it must be antisymmetric.

The signed curvature is

k = n \cdot \partial_{s} t .

An orientation-preserving isometry sends

t, n

R t, R n

and leaves

s

unchanged, so

k = (R n) \cdot R (\partial_{s} t) = k .

An orientation-reversing isometry reverses the chosen normal and hence the sign convention for signed curvature, while the geometric curvature

∣ k ∣

remains invariant.

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