Surfaces of revolution and ruled surfaces

Show that if $\gamma$ is a geodesic on the cone $z = \sqrt{x^2+y^2}$, $(x,y) \in \mathbb {R}^{2} \setminus \{ (0,0)\}$, then $\gamma$ intersects itself at most a finite number of times.

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.

Let $f : \mathbb {R} \to \mathbb {R}$ be positive and smooth. Let $M$ be the surface in $\mathbb {R}^{3}$ obtained by rotating the graph $\{ (x, f(x)) : x \in \mathbb {R}\}$ of $f$ in the $xz$ plane about the $x$ axis. Characterize in terms of $f$ the set of $x$ such that $\dfrac {1}{f(x)}$ is a principal curvature of $M$ at $(x, 0, f(x))$.

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.

Let $(r(s), 0, z(s))$ be a unit speed curve in $\mathbb {R}^{3}$ with $r(s) > 0$. Consider the surface of revolution $(s,\theta ) \to (r(s)\cos \theta , r(s)\sin \theta , z(s))$. On this surface compute the covariant derivatives $\nabla_{\frac{\partial }{\partial \theta }}\frac{\partial }{\partial s}$ and $\nabla_{\frac{\partial }{\partial \theta }}\frac{\partial }{\partial \theta }$ in terms of $\frac{\partial }{\partial \theta }$ and $\frac{\partial }{\partial s}$.

Consider the torus-of-revolution $T$ obtained by rotating the circle $(x-a)^2+z^2=r^2$ around the $z$-axis:

Parameterize this torus, compute its Gauss curvature function $K$, and verify that $\int_T K \, dA = 0$ by explicit calculation.

By revolving the curve $\gamma$ sketched below around the $x$-axis, we get a surface of revolution $M^2 \subset \mathbb {R}^{3}$. Compute $\int_M K \, dA$, where $K$ is the Gauss curvature and $dA$ the area form, on $M$. (Make sure to justify your answer.)

A surface $\Sigma^2$ immersed in a Riemannian manifold $N$ is said to be ruled if it can be parameterized near any point by a mapping $X : (0,1)^2 \to N$ such that for each fixed $v_0 \in (0,1)$, the $u$-parameter curve $X(u,v_0)$ is a geodesic in $N$.

Let $(f(v)\cos (u), f(v)\sin (u), g(v))$ be a parametrization of a surface of revolution $S \subset \mathbb {R}^{3}$ where $(u,v) \in (0,2\pi )\times (a,b)$. If $S$ is given the induced metric from $\mathbb {R}^{3}$, prove that the following map from $S$ to $\mathbb {R}^{2}$ is locally conformal where $\mathbb {R}^{2}$ is given the standard Euclidean metric:

Let $u \mapsto \rho (u)$ be a smooth curve in $\mathbb {R}^{3}$. Let $S$ be the surface in $\mathbb {R}^{3}$ defined by $(u,v) \mapsto \rho (u) + v\rho '(u)$, where $\rho '(u)$ means the first-order derivative of $\rho (u)$ with respect to $u$. Assume that the two vectors $\rho '(u)$ and $\rho '(u) + v\rho ''(u)$ are $\mathbb {R}$-linearly independent at the point $(u,v) = (u_0,v_0)$, where $\rho ''(u)$ means the second-order derivative of $\rho (u)$ with respect to $u$. Verify directly from the definition of Gaussian curvature that the Gaussian curvature of $S$ is zero at the point $(u,v) = (u_0,v_0)$.

Let $f : [a,b] \to \mathbb {R}^{}_+$ be smooth, and let $X \subset \mathbb {R}^{3}$ be the surface of revolution formed by revolving $z=f(x)$ about $x$-axis.

Let $\gamma$ be a geodesic curve on a regular surface of revolution $S \subset \mathbb {R}^{3}$. Let $\theta (p)$ denote the angle the curve forms with the parallel at a point $p \in \gamma$ and $r(p)$ be the distance to the axis of revolution. Prove Clairaut's relation: $r\cos \theta = \text{const}$.

Let $F(x,y,z)$ be a smooth homogenous function of degree $k$, i.e., $F(\lambda x, \lambda y, \lambda z) = \lambda^k F(x,y,z)$. Prove that away from the origin, the induced metric on the conical surface $\Sigma = \{ (x,y,z) \mid F(x,y,z) = 0\}$ has Gaussian curvature equal to $0$.

A line parallel to the plane $xOy$ slides along the axis $Oz$ and along the curve

Find the equation of the resulting surface.

A line parallel to the plane $x+y+z = 0$ slides along the axis $Oz$ and along the circle $x = b$, $y^{2}+z^{2} = a^{2}$. Find the resulting surface.

The curve

lies on the surface of revolution $x^{2}+y^{2}=a^{2}\ch^{2}\dfrac {z}{a}$, called a catenoid. Prove that at every point of the curve the binormal coincides with the normal to the surface.

Prove that the equation of a surface of revolution about the axis $Oz$ has the form $f(x^{2}+y^{2}, z)=0$.

Prove that the equation of a surface of revolution about the line

has the form

where

Find the equation of the surface obtained by rotating the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ about the axis $Ox$.

Find the equation of the surface obtained by rotating the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1$ about the axis $Oy$.

Find the equation of the surface obtained by rotating the parabola $y^{2}=2px$ about the axis $Ox$.

Find the equation of the surface obtained by rotating the circle $(x-a)^{2}+y^{2}=R^{2}$ about the axis $Oy$.

Find the equation of the surface obtained by rotating the parabola $y^{2}=2px$ about the axis $Oy$.

Find the equation of the surface obtained by rotating the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ about $Ox$.

Find the equation of the surface obtained by rotating the same curve (the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$, see problem 2328) about the axis $Oy$.

The surface obtained by rotating the sinusoid

about the axis $Oz$ is illuminated by parallel rays making an angle $\dfrac {\pi }{4}$ with $Oz$. Find the shape of the shadow it casts on the plane $xOy$.

Find the equation of the tangent plane to the skew helicoid

Find the total and mean curvatures of the surface of revolution $z=f(\rho )$, where $\rho =\sqrt{x^{2}+y^{2}}$.

Prove that the surface of revolution of the tractrix

about the axis $Ox$ has constant negative curvature, equal to $-\dfrac {1}{a^{2}}$.

Through the vertex of the paraboloid of revolution $y^{2}+z^{2}=2px$, curves are drawn on its surface. Find the locus of their centers of curvature.

Prove that for the helicoid of problem 2385, along each line of curvature one of the quantities $\ln \left(u+\sqrt{u^{2}+m^{2}}\right)\pm v$ remains constant.

Show that on the surface

the points of a geodesic satisfy the condition

where $c=\mathrm{const}$.

Find the analogous condition for the geodesics of the surface of revolution

Prove that the curves of intersection of the cylinders $y^{2}+z^{2} = b^{2}$ with the surface $xy = az$ intersect, at a right angle, all the generators of this surface belonging to one family.

Prove that the curve $x = a \operatorname {tg} t, y = b \cos t, z = b \sin t$ lies on the surface of a paraboloid and intersects, at a right angle, all its generators of one family.

Let the function $f(x)$ be defined in the interval $a \leq x \leq b$ and satisfy the following conditions:

1. $f(x)$ is positive in the interior and $=0$ at the end-points of the interval $a$, $b$;

2. $f(x)$ is twice continuously differentiable for $a < x < b$;

3. $f'(x)$ converges to $+\infty$ as $x \to a$, and to $-\infty$ as $x \to b$. Under these conditions the function

cannot be monotonic in the whole interval $a < x < b$, except in the case that

On a surface of revolution with continuous curvature there are always two different parallel circles with the same Gaussian curvature.