Liouville's theorem (differential geometry)

The Liouville's theorem is a result of classical differential geometry . This was named after the mathematician Joseph Liouville . The result provides a formula for calculating the geodetic curvature of surface curves. Sometimes this result is also called the Liouville formula .

statement

Let be an oriented differentiable surface and be a neighborhood of with an orthogonal parametric representation . Is also one after the arc length parameterized representation of a regular curve and become the angle between and referred to. Then applies ${\displaystyle S}$ ${\displaystyle U\subset S}$ ${\displaystyle p\in M}$ ${\displaystyle x(u,v)}$ ${\displaystyle c}$ ${\displaystyle \phi (s)}$ ${\displaystyle \textstyle {\frac {\partial x(u,v)}{\partial x}}}$ ${\displaystyle {\tfrac {\mathrm {d} c(t)}{\mathrm {d} t}}}$

${\displaystyle \kappa _{g}=\kappa _{g,u}\cos(\varphi )+\kappa _{g,v}\sin(\varphi )+{\frac {\mathrm {d} \varphi }{\mathrm {d} s}}.}$

And denote the geodetic curvatures with respect to the coordinate lines. This means that at is the curvature of , where constant is chosen and the curve therefore only depends on. The analog is meant by. ${\displaystyle \kappa _{g,u}}$ ${\displaystyle \kappa _{g,v}}$ ${\displaystyle \kappa _{g,u}}$ ${\displaystyle x(u,v)}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle \kappa _{g,v}}$

literature

• Manfredo Perdigão do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs NJ 1976, ISBN 0-13-212589-7 .