Characterizing geodesics via symmetry or a normal field

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Problem 3.1.15 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 $M$ be a surface in $\mathbb {R}^{3}$ and let $P$ be a plane. Suppose $M$ and $P$ intersect orthogonally. Show that the intersection curve (parameterized by arclength) is a geodesic on $M$ .

Problem 3.2.19 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 3.2 "Differential Geometry of Surfaces"

(a)(b)

Define "geodesics", and characterize (with proof):

(a)

All geodesics on the unit sphere $S^2 \subset \mathbb {R}^{3}$ .

(b)

All geodesics on the surface $\{ (x,y,z) \in \mathbb {R}^{3} : x^2+y^2 \equiv 1\}$ .

Problem 3.2.39 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 3.2 "Differential Geometry of Surfaces"

Let $\Sigma \subset \mathbb {R}^{3}$ be a smooth 2-dimensional submanifold, and $n : \Sigma \to \mathbb {R}^{3}$ a smooth map such that $n(p)$ is a unit length normal to $\Sigma$ at $p$ . Identify the tangent bundle $T\Sigma$ as the subspace of pairs $(p,v) \in \Sigma \times \mathbb {R}^{3}$ such that $\left\langle v, n(p) \right\rangle = 0$ . Suppose now the $t \to p(t)$ is a smoothly parametrized curve in $\mathbb {R}^{3}$ that lies on $\Sigma$ . Prove that this curve is a geodesic if and only if

p''(t) \cdot \bigl(n(p(t)) \times p'(t)\bigr) = 0, \, \forall t.

Here, $p'$ is the derivative of the map $t \to p(t)$ and $p''$ is the second derivative.

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.