A question my proof about the line of curvature

Question

I am working on Exercise 2.4.4 of Differential Geometry and Its Application.

The problem statement and my work is available at this link.

At the end of my proof, I claimed that $ S_p(\alpha') $ is both on the plane and the tangent plane, so it must be $ k \alpha' $, at this point I am unsure if that is correct. I think so because $ \alpha' $ must be on the plane and on the tangent plane, but could that two planes be the same?

In your first line, ∇u1U=−Su1(u1)=−k1u1 : is this a typo?, should be $S_p(u_1)$? — JMP, Feb 21 '16 at 15:04

score 1 · Accepted Answer · answered Feb 21 '16 at 19:09

Your proof looks fine.

To justify the last step, refer to Differential Geometry and Its Applications, section 2.2, which defines $T_p(M)$.

Then, $\mathtt{Lemma\; 2.2.10}$ (page 84) states that '$S_p$ is a linear transformation of $T_p(M)$ to itself'.

This is enough to justify that $S_p(\alpha ')$ lies on $P$.

And while you are on page 84, you might find $\mathtt{Example\; 2.2.12}$ useful.

A question my proof about the line of curvature

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