Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $S\subset \mathbb{R}^3$ be a connected smooth surface. Suppose that every point of $S$ is an umbilic point. Prove that $S$ is a subset of either a plane or a sphere in $\mathbb{R}^3$. Here's a HW problem. I wonder how to prove it.

share|cite|improve this question

Since this is homework and you haven't showed us your work, I'll give you the start and let you take it from there.

Let $\sigma(u, v)$ be a regular surface patch and $\mathbf{N}$ be the normal vector.

Since the surface is umbilic at every point, we have:

\begin{align*} \mathcal{W}(\sigma_{u}) &= -\mathbf{N}_u = \kappa \ \sigma_{u} \\ \mathcal{W}(\sigma_{v}) &= -\mathbf{N}_v = \kappa \ \sigma_{v} \end{align*}

Where $\mathcal{W}$ is the shape operator (Weingarten map), and $\kappa$ is the principal curvature.

By differentiating the first equation with respect to $v$ and the second with respect to $u$, we have:

\begin{align*} - \mathbf{N}_{uv} &= \kappa_{v} \ \sigma_{u} + \kappa \ \sigma_{uv} \\ &= \kappa_{u} \ \sigma_{v} + \kappa \ \sigma_{uv} \end{align*}

From that we get:

$$ \kappa_{v} \ \sigma_{u} = \kappa_{u} \ \sigma_{v} $$

Since $\sigma_{u}$ and $\sigma_{v}$ are linearly independent, $\kappa_{u}$ and $\kappa_{v}$ must be 0. Hence, $\kappa$ is constant everywhere.

Can you take it from there? Use the equations in the beginning of my answer to find a relationship containing $\sigma$ and $\mathbf{N}$.

Rest of the proof (to reference in another question):

We have:

\begin{align*} -\mathbf{N}_u &= \kappa \ \sigma_{u} \\ -\mathbf{N}_v &= \kappa \ \sigma_{v} \end{align*}

$\kappa$ is constant. Hence:

$$ -\mathbf{N} = \kappa \ \sigma + v $$

For a constant vector $v$.

Either $\kappa = 0$, so $\mathbf{N}$ is constant and the surface is (part of) a plane. Or $\kappa \ne 0$ and:

$$ \|\sigma + \frac{1}{\kappa} v \| = \|-\frac{1}{\kappa} N\| = \frac{1}{|\kappa|} $$

And $\sigma$ is (part of) a sphere with radius $\frac{1}{|\kappa|}$.

share|cite|improve this answer
can u give details? I cant get it – Shu May 7 '12 at 12:43
@Shu What don't you get exactly so I can elaborate? Can you show us your work so far? – Ayman Hourieh May 7 '12 at 12:45
Well, I got your ideas after reading the textbook. Thx a lot. – Shu May 11 '12 at 7:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.