Total curvature

In curve theory , a branch of mathematics , the total curvature of a curve is defined as the integral of its curvature , i.e. as ${\displaystyle \varphi \colon [a,b]\to \mathbb {R} ^{n}}$ ${\displaystyle \kappa }$

${\displaystyle \int _{a}^{b}\kappa (s)ds}$.

Curves in the plane

The total curvature of a closed curve in the plane is always an integral multiple of . The integer factor is the tangent revolution number of the curve. ${\displaystyle 2\pi }$

It follows from Whitney-Graustein's theorem that the total curvature of a closed regular curve does not change under regular homotopias .

Space curves

From the Fary-Milnor inequality it follows that the total curvature of a knotted space curve is always greater than . ${\displaystyle 4\pi }$

Higher-dimensional generalization

For higher-dimensional Riemannian manifolds , the integral is called the total scalar curvature (or, in the case of surfaces, also the total curvature ) ${\displaystyle (M,g)}$

${\displaystyle \int _{M}scal\ dvol_{g}}$

the scalar curvature with respect to the volume shape of the Riemannian metric . ${\displaystyle g}$

For surfaces it follows from the Gauss-Bonnet theorem that their total curvature only depends on the Euler characteristic of the surface and not on the Riemannian metric. ${\displaystyle 2\pi \chi (M)}$

literature

• Wolfgang Kühnel : Differential Geometry: Curves - Surfaces - Manifolds , Springer Spectrum 2013, ISBN 978-3-658-00615-0