# Gaussian curvature of the pseudosphere

I am asked to show that the pseudosphere has Gaussian Curvature $$-1$$ at all points. So I parametrized the tractrix as: $$$$\alpha(t) = (\sin{t},\cos{t}+\log{\tan{\frac{t}{2}}})$$$$ So the surface of revolution would be: $$$$x(\theta,t)=(\sin{t}\cos{\theta},\sin{t}\sin{\theta},\cos{t}+\log{\tan{\frac{t}{2}}})$$$$ By calculating the first and second fundamental forms we have that $$F=f=0$$ and $$E=(\sin{t})^2,G=(\cot{t})^2,e=(\cos{t})^2$$ and $$g=-(\cot{t})^2$$. So the Gaussian curvature is: $$$$K = \frac{eg}{EG} = -(\cot{t})^2$$$$ What am I doing wrong?

• Since you haven't shown the details of your computations, I can't tell you what you're doing wrong. But your $e$ and $g$ are most definitely wrong. Did you remember to take the unit normal vector? – Ted Shifrin Oct 16 '18 at 18:29

We have \begin{align} x_\theta &= (-\sin t\sin\theta, \sin t \cos \theta, 0) = \sin t(-\sin\theta, \cos\theta, 0) \\ x_t &= (\cos t \cos\theta, \cos t \cos\theta, -\sin t + \frac{\sec^2(t/2)}{2\tan(t/2)}) = \cos t(\cos\theta, \sin\theta, \cot t), \end{align} so \begin{align} E&= \langle x_\theta, x_\theta \rangle = \sin^2 t\\ F&= 0 \quad\text{[surface of revolution]}\\ G&= \langle x_t, x_t \rangle = \cos^2 t (1+\cot^2t) = \cot^2t. \end{align} So far so good. Now $$x_\theta \times x_t$$ is parallel to $$(-\sin\theta, \cos\theta, 0) \times(\cos\theta,\sin\theta,\cot t) = (\cos\theta\cot t, \sin\theta\cot t, -1)$$ so we can (after normalizing to make it a unit vector, i.e. multiplying by $$\sin t$$) choose the Gauss map to be $$N = (\cos\theta\cos t, \sin\theta\cos t, -\sin t).$$ This gives \begin{align} N_\theta &= (-\sin\theta\cos t, \cos\theta\cos t, 0)\\ N_t &= (-\cos\theta\sin t, -\sin\theta\sin t, -\cos t), \end{align} so that \begin{align} e &= - \langle N_\theta, x_\theta \rangle = -\sin t \cos t\\ f &= 0 \quad\text{[surface of revolution]}\\ g &= - \langle N_t, x_t \rangle = -\cos t (-\sin t - \cos t \cot t) = \cot t, \end{align} which gives $$K = \frac{eg-f^2}{EG-F^2} = \frac{-\cos^2t}{\sin^2t\cot^2t} = -1.$$