For pseudosphere $M$ parametrized as in where $X(u,v) =(u-tanhu,sechu\thinspace cosv, sechu\thinspace sinv)$, $0 \leq u$, $0 \leq v \leq 2\pi$ and $M_r$ is the portion defined by $0\leq u \leq r$. I need to calculate the geodesic curvature $\kappa_g $ of the circle $u=u_0$ and $\int_{\partial M_r} \kappa_g ds$.
I have calculated the unit normal $n = (-sech u, -tanhu \thinspace cosv, -tanhu \thinspace sin v)$ and the equation for the geodesic curvature is $\kappa_g = \kappa N(n \times T)$. But I'm confused about how to calculate geodesic curvature of the circle in relation to the pseudosphere as well as the orientations of the two circles.
I attempted to consider the circle with radius r and I get $\alpha(v) = (r-tanhr,sechr\thinspace cosv, sechr\thinspace sinv)$, and parameterize by arclength $\alpha(s) = (r-tanhr,sechr\thinspace cos \frac{s}{r}, sechr\thinspace sin\frac{s}{r})$ and $\kappa = \frac{1}{r}$.
And I got $T = (0, -sin\frac{s}{r}, cos\frac{s}{r})$ and $N = (0, -cos\frac{s}{r}, -sin\frac{s}{r})$
In the end I got that $\kappa_g = \frac{-sechr}{r}$ and $\int_{\partial M_r} \kappa_g ds = -2 \pi sech r$ but it doesn't seem correct. Please help me to proceed this question.