Criteria for a space curve to lie in a plane or on a line

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Problem 3.1.9 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 $\gamma : [0,1] \to \mathbb {R}^{3}$ be a $C^\infty$ curve with $\left|\gamma '\right| = 1$ and non-vanishing curvature. Assume the torsion $\tau \equiv 0$ .

(a)

Show that $\gamma$ lies in a plane.

(b)

What happens if the curvature is allowed to vanish at a point?

Problem 3.1.18 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 $\alpha : (a,b) \to \mathbb {R}^{3}$ be a smooth curve with non-vanishing curvature.

(a)

Show that if the torsion of $\alpha$ vanishes identically then there is a plane $\pi \subset \mathbb {R}^{3}$ containing $\alpha$ , that is, $\alpha (t) \in \pi$ for all $t \in (a,b)$ . Is $\pi$ unique?

(b)

Is the conclusion of (a) still true if the curvature is allowed to vanish at a single point $c \in (a,b)$ ?

[Of course, the torsion is not defined at $c$ , but is assumed to be zero at all other points.]

Problem 2267 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Using the third Frenet formula, show that a curve whose torsion is identically zero lies in a plane.

Problem 1.24.117 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that if all the osculating planes of a twice continuously differentiable curve pass through a fixed point, then this curve is planar.

Problem 1.24.120 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that the curvature of a curve is identically zero if and only if the curve is a segment of a straight line.

Problem 1.24.130 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.24.2 "Problems"

Prove that if all the normal planes of a twice continuously differentiable curve without singular points and with nonvanishing curvature are parallel to a constant vector, then this curve is planar.

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.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.