Geodesic curvature of a curve on a surface

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Problem 3.1.8 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

Let $M \subset \mathbb {R}^{3}$ be the torus obtained by rotating the circle $\{ (0, y, z) : (y-2)^2 + z^2 = 1\}$ around the $z$ -axis, and let $c(t) = (2\cos t, 2 \sin t, 1)$ ("top circle"). Is this curve a geodesic on $M$ ? Explain without long computations.

Problem 3.1.12 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

(a)(b)

Let $M^2 \subset \mathbb {R}^{3}$ be the cylinder $x^2+y^2=1$ . Suppose the curve $\alpha (s) \subset M^2$ is parameterized by arclength.

(a)

If $k(s) > 0$ and $\tau (s) = 0$ , show that $\alpha (s)$ is a closed curve. (Here $k, \tau$ are curvature and torsion of $\alpha$ in $\mathbb {R}^{3}$ .)

(b)

If $k_g(s) = 1$ , show that $\alpha (s)$ is a closed curve ( $k_g$ is the geodesic curvature in $M$ ).

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.