Global Differential Geometry of Curves in Three-Dimensional Heisenberg Group and CR Sphere

Abstract

In this paper, we study some global properties of curves in the Heisenberg group $H_{1}$. In particular, we obtain Fenchel-type theorem and Fáry–Milnor type theorem, together with Bray–Jauregui type theorem. We also prove the fundamental theorem of curves in the standard CR 3-sphere $S^{3}$. As an application, we classify some horizontally regular curves in $S^{3}$.

Appendix

In this appendix, we prove that the number of extremal points of a continuous function on the unit circle must be even. It is probably well known. But we include it here for the sake of completeness.

First we have the following notations. We say that p is a relative maximum (relative minimum, respectively) of f if there exists a neighborhood U of p such that

$$\begin{aligned} f(p)\ge f(x) ~~(f(p)\le f(x) \text{ respectively })~~ \text{ for } \text{ all } x\in U. \end{aligned}$$

We say that p is an extremal point of f if p is a relative maximum of f or a relative minimum of f.

Lemma 9.1

Suppose $f:C\rightarrow \mathbb {R}$ is a real-valued continuous function defined on the unit circle C. If the number of extremal points of f are finite, then the number of extremal points of f must be even.

Proof

It suffices to prove the corresponding statement for the continuous function $f:[0,2\pi ]\rightarrow \mathbb {R}$ with $f(0)=f(2\pi )$. We have the following: $\square$

Claim 1

Each relative maximum p of f must be strict maximum, i.e., there exists a neighborhood U of p such that

$$\begin{aligned} f(p)>f(x) \text{ for } \text{ all } x\in U-\{p\}. \end{aligned}$$

Suppose this is not true. Then we can find a sequence of distinct points $\{x_i\}_{i=1}^\infty$ such that $x_i\ne p$ and $x_i\rightarrow p$ as $i\rightarrow \infty$ with

$$\begin{aligned} f(p)\le f(x_i). \end{aligned}$$

This implies that $x_i$ are all relative maximum of f, which contradicts to the assumption that the number of extremal points of f are finite. This proves the claim.

Similarly, we have the following:

Claim 2

Each relative minimum p of f must be strict maximum, i.e., there exists a neighborhood U of p such that

$$\begin{aligned} f(p)<f(x) \text{ for } \text{ all } x\in U-\{p\}. \end{aligned}$$

Now we are ready to prove Lemma 9.1. Since the number of extremal points of f are finite, we can assume that $\{a_1<\cdots <a_m\}$ and $\{b_1<\cdots <b_M\}$ are, respectively, the relative minimum of f and relative maximum of f. Since f is a continuous function defined on the compact set $[0,2\pi ]$, f must have a relative maximum and a relative minimum, i.e., $m, M\ge 1$. On the other hand, in view of Claims 1 and 2, $a_i\ne b_j$ for all i, j. Then we have the following:

Claim 3

$m=M$.

Claim 3 implies Lemma 9.1, since the number of extremal points is equal to $M=m$. To prove claim 3, suppose on the contrary, $m\ne M$. Without loss of generality, we can assume that $m<M$. Then there exists $i=1,\ldots ,M$ such that the interval $[b_i,b_{i+1}]$ does not contain any $a_j$, i.e., f does not have any relative minimum in the interval $[b_i,b_{i+1}]$. (If $i=M$, then we identify $b_{M+1}$ as $b_1$.) Since f is continuous on the compact interval $[b_i,b_{i+1}]$, there exists an absolute minimum $x_0\in [b_i,b_{i+1}]$, i.e.,

$$\begin{aligned} f(x_0)\le f(x)~~ \text{ for } \text{ all } x\in [b_i,b_{i+1}]. \end{aligned}$$

In view of Claim 1, $x_0\ne b_i, b_{i+1}$. As a result, $x_0$ is a relative minimum of f, which is a contradiction. This proves Claim 3. $\square$