This is a homework problem from do Carmo. Given a regular parametrized surface $X(u,v)$ we define the parallel surface $Y(u,v)$ by $$Y(u,v)=X(u,v) + aN(u,v)$$ where $N(u,v)$ is the unit normal on $X$ and $a$ is a constant. I have been asked to compute the Gaussian and mean curvatures $\overline{K}$ and $\overline{H}$ of $Y(u,v)$ in terms of those of X, $K$ and $H$. Now, I know how to do this by brute force: calculate the coefficients of the first and second fundamental forms of $Y$ in terms of those of $X$. However, this is a lengthy and messy calculation. do Carmo says that $$\overline{K}=\frac{K}{1-2Ha+Ka^2}$$ and $$\overline{H}=\frac{H-Ka}{1-2Ha+Ka^2}.$$ The denominator of these fractions is actually something that arose earlier in the problem; I calculated $$Y_u\times Y_v=(1-2Ha+Ka^2)(X_u\times X_v).$$ So, it seems like I should be able to calcuate $\overline{K}$ and $\overline{H}$ from this initial step. Is there something I'm missing? Or, is it actually just a brute force calculation?
Thanks.
This is already implicitly included in the previously posted answers, but if you just want to understand where the formulas for $\overline K$ and $\overline H$ come from, without caring much for rigor, that is actually very easy. Once expressed in terms of the principal radii of curvature, the expressions for $\overline K$ and $\overline H$ are equivalent to \begin{align*} \overline R_1+\overline R_2&=R_1+R_2-2a,\\ \overline R_1\overline R_2&=(R_1-a)(R_2-a). \end{align*} These relations then follow from the intuitively obvious fact that upon translating by $a$ along the normal vector, the principal radii of the surface simply shift by $a$.