Affine differential geometry

Remarks

For a regular curve $\gamma \colon I \to \R^2$ with $\det(\gamma', \gamma'') > 0$, the affine arc length is $$\sigma(t) = \int_{t_0}^{t} \det\big(\gamma'(u), \gamma''(u)\big)^{1/3} \, du.$$ It is invariant: if $\Phi(x) = Ax + b$ with $\det A = 1$, then $\det(A\gamma', A\gamma'') = \det(\gamma', \gamma'')$. For the parabola $\gamma(t) = (t, t^2/2)$, one has $\gamma' = (1, t)$ and $\gamma'' = (0, 1)$, so the determinant is $1$ and the affine arc length from $0$ is $t$.

Euclidean arc length is not an affine invariant. The shear $(x, y) \mapsto (x + y, y)$ has determinant $1$, and it maps the segment from $(0,0)$ to $(0,1)$, of length $1$, to the segment from $(0,0)$ to $(1,1)$, of length $\sqrt{2}$.

For a nondegenerate hypersurface, the basic invariants are the affine normal vector field and the induced Blaschke metric. Some authors use the full affine group, with any invertible $A$; then volume is no longer preserved and the invariants change.