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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?

Thank you in advance.

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

1 Answer 1

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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.

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