Curvature line equation

while reviewing the book on differential geometry by Manfredo do Carmo, I came across the equation of lines of curvature. Do Carmo roughly states the following: A connected regular curve $$C$$ in a coordinate neighborhood $$x$$ is a line of curvature if, and only if for any parametrization $$\alpha(t) = x(u(t), v(t))$$, $$t \in I$$, of $$C$$, the following equation is satisfied: $$N(u,v)(v^{\prime})^{2} + M(u,v) u^{\prime} v^{\prime} + L(u,v) (u^{\prime})^{2} = 0,$$ where $$N(u,v)= gF - fG, M(u,v)= gE - eG, N(u,v)= fE - eF$$, and $$E,F,G$$ are coefficients of the first fundamental form, and $$e,f,g$$ are coefficients of the second fundamental form.

My questions are as follows:

1. If I solve the equation, do I find "directions" in the coordinate plane where, whenever I take derivatives of curves in those directions, their tangent vectors at the derived point are the greatest or smallest? (As, for example, in the case of surfaces of revolution, where I would have $$x(u_{0},v)$$ or $$x(u,v_{0})$$).

2. If I find the points $$(u_{0},v_{0})$$ where the functions $$N=M=L=0$$, am I finding umbilical points? If not, what do I find through this process?

• Directions ( in Euler relation ) associated with max/min normal curvatures. Jul 29, 2023 at 20:14
• Hi @Narasimham. I think the same in the first question, but, in the second question? Jul 30, 2023 at 15:16
• Yes, $\mathrm{II} = c\mathrm I$ iff the shape operator is $c$ times the identity map. Jul 30, 2023 at 21:51
• Hi, @TedShifrin I don't understand your answer, can you explain me? Jul 31, 2023 at 13:45

No, we have proportionality of coefficients in the two fundamental forms. We have in the alternate notation of normal curvature. Let

$$\kappa_n=\frac{Ldu^2+2Mdudv+Ndv^2}{Edu^2+2Fdudv+Gdv^2}$$

$$\frac{E}{L}=\frac{F}{M}=\frac{G}{N} = \mu, \text{a constant that need not be necessarily positive.}$$

It can be shown for surface $$r$$ $$N_1= -\mu \cdot r$$

Differentiate w.r.t $$v$$ and $$u$$ and subtracting,

$$N_{12}= N_{21}, ~\mu_1r_2=\mu_2 r_1;$$ since

$$r_1 \text{ is not parallel to } r_2, \mu s \text { are constant making }$$

$$\mu_1=0,~\mu_2=0$$

If constant $$\mu$$ is non-zero

$$r_1=-N_1/\mu, \text{ it integrates to } r=-N_1/\mu+r_0$$

with $$r_o$$ as a constant of integration

$$(r-r_o)^2 =\frac{N^2}{\mu^2}~=\frac{1}{\mu^2}~$$ Thus $$r$$ locus is a sphere.

If $$\mu$$ is zero, $$N_1=N_2=0,$$ the $$N$$ vector itself is constant and the surface is a plane.

The normal curvature is independent of curvature ratios $$(du,dv),$$ same value holding good for all directions, so all such points are umbilics on the surface.