# the normal curvature for torus

Hello everyone I need little help in differential geometry , I need someone can solve this problem.

Q1 The surface of torus given by $X(U,V)=((a+b\cdot cos(U))\cdot cos(V),(a+b\cdot cos(U))\cdot sin(V),b\cdot sin(U))$ Prove it is asymptotic curve which means that $K_n$(normal curvature)$=0$ this my problem I hope someone give to me the idea how to solve.

Thank you.

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The normal curvature depends on a curve on the surface.$X(u,v)$ is a parametrization of the torus, not a curve. –  jandrew Apr 3 '13 at 1:13
@jandrew Does that mean you can't do a question like this? You need some curve $\gamma(t)$ don't you know to calculate the normal curvature, as you can't use the formula otherwise? –  Kaish Apr 3 '13 at 13:07
@Kaish The normal curvature of a curve on a surface is given by $k_n=k\cdot cos(\theta)$ where $k$ is the curvature of the curve and $\theta$ is the angle between the normal vector of the curve and the normal vector of the surface.You would need to prove that $k=0$ or that the normal vectors are orthogonal. In general I think you can solve a differential equation to find the asymptotic curves on a surface,but this is a much harder procedure because the equation involves coefficients of the second fundamental form. –  jandrew Apr 7 '13 at 16:05