Finite type $\xi$-asymptotic lines of plane fields in $\mathbb{R}^3$

TL;DR

This paper demonstrates that finite type curves can serve as $\xi$-asymptotic lines in plane fields in $ ^3$, generalizing Arnold's classical results, and provides explicit examples of such lines.

Contribution

It establishes a connection between finite type curves and $\xi$-asymptotic lines in plane fields, extending Arnold's work to a broader geometric context.

Findings

Finite type curves can be realized as $\xi$-asymptotic lines in plane fields.

Explicit example of a hyperbolic closed finite type $\xi$-asymptotic line provided.

Results generalize Arnold's classical findings to the setting of plane fields.

Abstract

We prove that a finite type curve is an $\xi$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $\xi$-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].

Equations

$$k_n(p,dr)=-\frac{⟨d\xi (p),dr⟩}{⟨dr,dr⟩}.$$

$$k_n=\frac{⟨d^{2}r,\xi ⟩}{⟨dr,dr⟩}=-\frac{⟨dr,d\xi ⟩}{⟨dr,dr⟩}.$$

$$\begin{array}{r}⟨\xi ,dr⟩=0,⟨d\xi ,dr⟩=0.\end{array}$$

$$⟨\tilde{\xi },dr⟩=\phi ⟨\xi ,dr⟩=0,⟨d\tilde{\xi }(dr),dr⟩=d\phi (dr)⟨\xi ,dr⟩+\phi ⟨d\xi (dr),dr⟩=0.$$

$$\alpha (x,y,z)=\gamma (x)+yY(x)+zZ(x).$$

$$\begin{array}{rl}& ⟨\xi ,d\alpha ⟩=adx+bdy+cdz=0,\\ & ⟨d\xi ,d\alpha ⟩=L_{1}d{x}^2+L_{2}dxdy+L_{3}d{y}^2+L_{4}dxdz+L_{5}dydz+L_{6}d{z}^2=0,\end{array}$$

$$\begin{array}{r}a=⟨\xi ,{\alpha }_x⟩,b=⟨\xi ,{\alpha }_y⟩,c=⟨\xi ,{\alpha }_z⟩,\end{array}$$

$$\begin{array}{rl}& L_1=⟨{\xi }_x,{\alpha }_x⟩,L_2=⟨{\xi }_x,{\alpha }_y⟩+⟨{\xi }_y,{\alpha }_x⟩,L_3=⟨{\xi }_y,{\alpha }_y⟩,\\ & L_4=⟨{\xi }_x,{\alpha }_z⟩+⟨{\xi }_z,{\alpha }_x⟩,L_5=⟨{\xi }_y,{\alpha }_z⟩+⟨{\xi }_z,{\alpha }_y⟩,L_6=⟨{\xi }_z,{\alpha }_z⟩.\end{array}$$

$$\begin{array}{r}dz=-\left(\frac{a}{c}\right)dx-\left(\frac{b}{c}\right)dy,ed{x}^2+2fdxdy+gd{y}^2=0,\end{array}$$

$$\begin{array}{r}e=L_1-\frac{a{L}_4}{c}+\frac{a^2{L}_6}{c^2},g=L_3-\frac{b{L}_5}{c}+\frac{b^2{L}_6}{c^2},f=\frac{L_2}{2}-\frac{(a{L}_5+b{L}_4)}{2c}+\frac{ab{L}_6}{2c^2}.\end{array}$$

$$\begin{array}{rl}& \xi (x,y,z)=l_0(x)Y(x)+k_0(x)Z(x)\\ & +\left(y{k}_1(x)+z{l}_1(x)+\left(\frac{y^2}{2}\right){\tilde{k}}_1(x)+yz{\tilde{j}}_1(x)+\left(\frac{z^2}{2}\right){\tilde{l}}_1(x)+A(x,y,z)\right)X(x)\\ & +\left(y{k}_2(x)+z{l}_2(x)+\left(\frac{y^2}{2}\right){\tilde{k}}_2(x)+yz{\tilde{j}}_2(x)+\left(\frac{z^2}{2}\right){\tilde{l}}_2(x)+B(x,y,z)\right)Y(x)\\ & +\left(y{k}_3(x)+z{l}_3(x)+\left(\frac{y^2}{2}\right){\tilde{k}}_3(x)+yz{\tilde{j}}_3(x)+\left(\frac{z^2}{3}\right){\tilde{l}}_3(x)+C(x,y,z)\right)Z(x),\end{array}$$

$$k_0={\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime },l_0=({\gamma }_3^{\prime }{\gamma }_1^{\prime \prime }-{\gamma }_1^{\prime }{\gamma }_3^{\prime \prime }){\gamma }_1^{\prime }+({\gamma }_3^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_3^{\prime \prime }){\gamma }_2^{\prime }.$$

$$\begin{array}{rl}b(x,0,0)& =(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2)[({\gamma }_1^{\prime }{\gamma }_1^{\prime \prime }+{\gamma }_2^{\prime }{\gamma }_2^{\prime \prime }){\gamma }_3^{\prime }-(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2){\gamma }_3^{\prime \prime }],\\ c(x,0,0)& =(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2)(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2+({\gamma }_3^{\prime }{)}^2)({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime }).\end{array}$$

$$\begin{array}{rl}{k}_1& =\frac{(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2{)}^2[({\gamma }_2^{\prime \prime }{\gamma }_3^{\prime \prime \prime }-{\gamma }_3^{\prime \prime }{\gamma }_2^{\prime \prime \prime }){\gamma }_1^{\prime }+({\gamma }_3^{\prime \prime }{\gamma }_1^{\prime \prime \prime }-{\gamma }_1^{\prime \prime }{\gamma }_3^{\prime \prime \prime }){\gamma }_2^{\prime }+({\gamma }_1^{\prime \prime }{\gamma }_2^{\prime \prime \prime }-{\gamma }_2^{\prime \prime }{\gamma }_1^{\prime \prime \prime }){\gamma }_3^{\prime }]}{(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2+({\gamma }_3^{\prime }{)}^2)({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime })}\\ & +\frac{[({\gamma }_1^{\prime }{\gamma }_1^{\prime \prime }+{\gamma }_2^{\prime }{\gamma }_2^{\prime \prime }){\gamma }_3^{\prime }-(({\gamma }_1{)}^2+({\gamma }_2{)}^2){\gamma }_3^{\prime \prime }]l_1+2({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime })}{(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2+({\gamma }_3^{\prime }{)}^2)({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime })}.\end{array}$$

$$\begin{array}{rl}{k}_1& =\frac{(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2{)}^2[({\gamma }_2^{\prime \prime }{\gamma }_3^{\prime \prime \prime }-{\gamma }_3^{\prime \prime }{\gamma }_2^{\prime \prime \prime }){\gamma }_1^{\prime }+({\gamma }_3^{\prime \prime }{\gamma }_1^{\prime \prime \prime }-{\gamma }_1^{\prime \prime }{\gamma }_3^{\prime \prime \prime }){\gamma }_2^{\prime }+({\gamma }_1^{\prime \prime }{\gamma }_2^{\prime \prime \prime }-{\gamma }_2^{\prime \prime }{\gamma }_1^{\prime \prime \prime }){\gamma }_3^{\prime }]}{(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2+({\gamma }_3^{\prime }{)}^2)({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime })}\\ & +\frac{[({\gamma }_1^{\prime }{\gamma }_1^{\prime \prime }+{\gamma }_2^{\prime }{\gamma }_2^{\prime \prime }){\gamma }_3^{\prime }-(({\gamma }_1{)}^2+({\gamma }_2{)}^2){\gamma }_3^{\prime \prime }]l_1+2({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime })H}{(({\gamma }_1^{\prime }{)}^2+({\gamma }_2^{\prime }{)}^2+({\gamma }_3^{\prime }{)}^2)({\gamma }_1^{\prime }{\gamma }_2^{\prime \prime }-{\gamma }_2^{\prime }{\gamma }_1^{\prime \prime })}.\end{array}$$

$$\alpha (x_0+l,y(x_0+l,y_0,z_0),z(x_0+l,y_0,z_0)),$$

$$\begin{array}{rl}& \frac{dz}{dx}=-\frac{a}{c}-\left(\frac{b}{c}\right)\frac{dy}{dx}=A+B\frac{dy}{dx},\\ & e+2f\frac{dy}{dx}+g{\left(\frac{dy}{dx}\right)}^2=0,\\ & (y(x_0,y_0,z_0),z(x_0,y_0,z_0))=(y_0,z_0).\end{array}$$

$$\frac{d}{dx}\left(\mathcal{Q}(x)\right)=\mathcal{M}(x)\mathcal{Q}(x),\mathcal{Q}(0)=\mathcal{I},$$

$$\mathcal{M}(x)=\left(\begin{array}[]{cc}-\frac{e_{y}(x,0,0)}{2f(x,0,0)}&-\frac{e_{z}(x,0,0)}{2f(x,0,0)}\\ \left(A\right)_{y}(x,0,0)&\left(A\right)_{z}(x,0,0)\\ \end{array}\right),\mathcal{Q}(x)=\left(\begin{array}[]{cc}\frac{dy}{dy_{0}}(x,0,0)&\frac{dy}{dz_{0}}(x,0,0)\\ \frac{dz}{dy_{0}}(x,0,0)&\frac{dz}{dz_{0}}(x,0,0)\\ \end{array}\right),$$

$$\begin{array}{r}\frac{d}{dx}\left(\frac{dz}{d{y}_0}\right)=A_y\frac{dy}{d{y}_0}+A_z\frac{dz}{d{y}_0}+B\frac{d}{dx}\left(\frac{dy}{d{y}_0}\right)+\left(B_y\frac{dy}{d{y}_0}+B_z\frac{dz}{d{y}_0}\right)\frac{dy}{dx},\end{array}$$

$$\begin{array}{r}\frac{d}{dx}\left(\frac{dz}{d{z}_0}\right)=A_y\frac{dy}{d{z}_0}+A_z\frac{dz}{d{z}_0}+B\frac{d}{dx}\left(\frac{dy}{d{z}_0}\right)+\left(B_y\frac{dy}{d{z}_0}+B_z\frac{dz}{d{z}_0}\right)\frac{dy}{dx}.\end{array}$$

$$\begin{array}{rl}& e_y\frac{dy}{d{y}_0}+e_z\frac{dz}{d{y}_0}+2f\frac{d}{dx}\left(\frac{dy}{d{y}_0}\right)+2\left(f_y\frac{dy}{d{y}_0}+f_z\frac{dz}{d{y}_0}+g\frac{d}{dx}\left(\frac{dy}{d{y}_0}\right)\right)\frac{dy}{dx}\\ & +\left(g_y\frac{dy}{d{y}_0}+g_z\frac{dz}{d{y}_0}\right){\left(\frac{dy}{dx}\right)}^2=0,\end{array}$$

$$\begin{array}{rl}& e_y\frac{dy}{d{z}_0}+e_z\frac{dz}{d{z}_0}+2f\frac{d}{dx}\left(\frac{dy}{d{z}_0}\right)+2\left(f_y\frac{dy}{d{z}_0}+f_z\frac{dz}{d{z}_0}+g\frac{d}{dx}\left(\frac{dy}{d{z}_0}\right)\right)\frac{dy}{dx}\\ & +\left(g_y\frac{dy}{d{z}_0}+g_z\frac{dz}{d{z}_0}\right){\left(\frac{dy}{dx}\right)}^2=0.\end{array}$$

$$\begin{array}{rl}& A_y\frac{dy}{d{y}_0}+A_z\frac{dz}{d{y}_0}=\frac{d}{dx}\left(\frac{dz}{d{y}_0}\right),e_y\frac{dy}{d{y}_0}+e_z\frac{dz}{d{y}_0}+2f\frac{d}{dx}\left(\frac{dy}{d{y}_0}\right)=0,\\ & A_y\frac{dy}{d{z}_0}+A_z\frac{dz}{d{z}_0}=\frac{d}{dx}\left(\frac{dz}{d{z}_0}\right),e_y\frac{dy}{d{z}_0}+e_z\frac{dz}{d{z}_0}+2f\frac{d}{dx}\left(\frac{dy}{d{z}_0}\right)=0.\end{array}$$

$$k_1(x)=\frac{3(3co{s}^2(x)-1)sin(x)l_1(x)+24co{s}^3(x)-18cos(x)-2}{9co{s}^6(x)-18co{s}^4(x)+9co{s}^2(x)+1}.$$

$$\exp \left({\int }_0^{2\pi }-\frac{e_y(x,0,0)}{2f(x,0,0)}dx\right)\mathrm{and}\exp \left({\int }_0^{2\pi }{A}_z(x,0,0)dx\right).$$

$$\begin{array}{rl}{A}_z(x,0,0)& =9sin(x)co{s}^8(x)+54sin(x)co{s}^6(x)-9co{s}^6(x)-117sin(x)co{s}^4(x)\\ & +18co{s}^4(x)+55co{s}^2(x)sin(x)-9co{s}^2(x)-1.\end{array}$$

$${\int }_0^{2\pi }\left(-\frac{e_y(x,0,0)}{2f(x,0,0)}\right)dx=2\pi .$$

$$\alpha (u,v)=\gamma (u)+vN(u)+(k_1(u)v+k_2(u)v^2+k_3(u)v^3+{\mathcal{O}}^4(v))({\gamma }^{\prime }\wedge N)(u).$$

$$N_{\alpha }=\frac{{\alpha }_u\wedge {\alpha }_v}{|{\alpha }_u\wedge {\alpha }_v|}.$$

Full text

MSC(2000): 53C12, 37C27, 34C25.

1. Introduction

A regular plane field in $\mathbb{R}^{3}$ is usually defined by the kernel of a differential form or an unitary vector field $\xi\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$. In this last case $\xi(p)$ is the normal vector to the plane at point $p$. The classical and germinal work about plane fields in $\mathbb{R}^{3}$ is [14].

The normal curvature of a plane field is defined by (see [2] and [5])

[TABLE]

For integrable plane fields the normal curvature is the usual concept of curves on surfaces.

The regular curves $\gamma\colon I\to\mathbb{R}^{3}$ such that $k_{n}(\gamma(t),\gamma^{\prime}(t))=0$ are called $\xi$-asymptotic lines and the directions $dr$ such $k_{n}(p,dr)=0$ are called $\xi$-asymptotic directions.

Recall that asymptotic lines on surfaces are regular curves $\gamma$ such that $k_{n}(\gamma(t),\gamma^{\prime}(t))=0$. Also, asymptotic lines are the curves $\gamma$ such that the osculating plane of $\gamma$ coincides with the tangent plane of the surface along it, so asymptotic lines are of extrinsic nature.

The local study, and singular aspects of asymptotic lines on surfaces in $\mathbb{R}^{3}$, near parabolic points, is a very classical subject, see [6, 7, 8], [9] and references therein.

The study of closed asymptotic lines of surfaces in $\mathbb{R}^{3}$ under the viewpoint of qualitative theory of differential equations is more recent, see [6, 7, 8]. It is worth to mention that existence of closed asymptotic lines on the tubes of “T-surfaces”  is still an open problem. See [1, page 107] and [11].

Also, it is not known if there is a surface in $\mathbb{R}^{3}$ having a cylinder region foliated by closed asymptotic lines (see [13, page 110]). In $\mathbb{S}^{3}$ all asymptotic lines of the Clifford torus are globally defined and they are the Villarceau circles.

V. Arnold in [4] studied the topology of asymptotic lines being curves of type $(t,t^{m},t^{n})$ near $t=0$, which are called of finite type. Also it was shown in [4] that the projection of a closed asymptotic line of an hyperbolic surface of graph type $(x,y,h(x,y))$ in the horizontal plane $(x,y)$ cannot be a starlike curve.

The main results of this work are the following.

The Theorem 3.1 states that any finite type curve is an $\xi$-asymptotic line of a suitable plane field in $\mathbb{R}^{3}$.

The Theorem 4.3 gives an example of a hyperbolic closed finite type $\xi$-asymptotic line of a plane field in $\mathbb{R}^{3}$.

2. Preliminaries and Previous Results

In this paper, the space $\mathbb{R}^{3}$ is endowed with the Euclidean norm $|\cdot|=\langle\cdot,\cdot\rangle^{\frac{1}{2}}$.

Definition 2.1** ([10, Definition 5.15]).**

A subset $\Omega\subset\mathbb{R}^{2}$ is called a starlike convex set if there is a point $p\in\Omega$, called the star point, such that, for every $q\in\Omega$, the segment $\overline{pq}$ lies in $\Omega$. The boundary of a starlike convex set is called a starlike curve.

Theorem 2.2** (D. Panov, see [4]).**

The projection of a closed asymptotic line of a surface $z=\varphi(x,y)$ to the plane $\{z=0\}$ cannot be a starlike curve (in particular, this projection cannot be a convex curve).

Definition 2.3** ([4]).**

A smoothly immersed curve $\gamma:I\to\mathbb{R}^{3}$ is said to be of finite type at a point $x$, if $\{\gamma^{\prime}(x),\gamma^{\prime\prime}(x),\ldots,\gamma^{(k)}(x)\}$ generate all the tangent space $T_{\gamma(x)}\mathbb{R}^{3}$ for some $k\in\mathbb{N}.$ Here $\gamma^{(k)}(x)$ denotes the derivative of order $k$ of $\gamma$. In a neighborhood of this point, the curve is parametrized locally by $\gamma(x)=(x,a_{m}x^{m}+\mathcal{O}^{m+1}(x),b_{n}x^{n}+\mathcal{O}^{n+1}(x))$, where $m,n\in\mathbb{N}$, $a_{m}b_{n}\neq 0$ and $1<m<n$.

The set $\{1,m,n\}$, $(1<m<n)$, of the degrees of $\gamma$ is called the symbol of the point. If $n=m+1$, then $\gamma$ is said to be of rotating type at the point.

If a curve is of finite type (resp. rotating type) at every point, then it is called of finite type curve (resp. rotating type curve).

A finite type curve $\gamma$ can be have inflection points, i.e., points where the curvature of $\gamma$ vanishes.

Arnold’s Theorem** (See [4]).**

An asymptotic curve of finite type on a hyperbolic surface is a rotating curve.

Every rotating space curve of finite type is an asymptotic line on a suitable hyperbolic surface.

A new proof of Arnold’s Theorem will be given in Appendix Finite type $\xi$-asymptotic lines of plane fields in $\mathbb{R}^{3}$.

2.1. Plane fields in $\mathbb{R}^{3}$

Let $\xi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ be a vector field of class $C^{k}$, where $k\geq 3$.

Definition 2.4**.**

A plane field $\xi$ in $\mathbb{R}^{3}$, orthogonal to the vector field $\xi$, is defined by the 1-form $\langle\xi,dr\rangle=0$, where $dr$ is a direction in $\mathbb{R}^{3}$. See Fig. 1.

Theorem 2.5** ([2, Jacobi Theorem, p.2]).**

There exists a family of surfaces orthogonal to $\xi$ if, and only if, $\langle\xi,curl(\xi)\rangle\equiv 0$.

A plane field $\xi$ is said to be completely integrable if $\langle\xi,curl(\xi)\rangle\equiv 0$. A surface of the family of surfaces orthogonal to $\xi$ is called an integral surface.

2.2. Normal curvature of a plane field

Definition 2.6** ([2, p. 8]).**

The normal curvature $k_{n}$ of a plane field in the direction $dr$ orthogonal to $\xi$ is defined by

[TABLE]

This definition agrees with the classical one given by L. Euler, see [5].

In the plane $\pi(p_{0},dr)$ generated by $\xi(p_{0})$ and $dr$ (direction orthogonal to $\xi(p_{0})$) we have a line field $\ell(p)$ orthogonal to vector $\bar{\xi}(p)\in\pi(p_{0},dr)$ obtained projecting $\xi(p)$ in the plane $\pi(p_{0},dr)$, with $p\in\pi(p_{0},dr)$. The integral curves $\varphi_{p}(t)$ of the line field $\ell$ are regular curves and $k_{n}(p_{0},dr)$ is the plane curvature of $\varphi_{p_{0}}(t)$ at $t=0$. See Fig. 2.

2.3. $\xi$-asymptotic lines and parabolic points of a plane field

The $\xi$-asymptotic directions of a plane field $\xi$ are defined by the following implicit differential equation

[TABLE]

and will referred as the implicit differential equation of the $\xi$-asymptotic lines.

A solution $dr$ of equation (2.2) is called an $\xi$-asymptotic direction. A curve $\gamma$ in $\mathbb{R}^{3}$ is an $\xi$-asymptotic line if $\gamma$ is an integral curve of equation (2.2). Analogously to the case of asymptotic lines on surfaces, for plane fields the osculating plane of an $\xi-$asymptotic line coincides with the plane of the distribution of planes passing through the point of the curve. See also [3, page 29].

Definition 2.7**.**

If at a point $r$ there exists two real distinct $\xi$-asymptotic directions (resp. two complex $\xi$-asymptotic directions), then $r$ is called a hyperbolic point (resp. elliptic point).

Definition 2.8**.**

If at $r$ the two $\xi$-asymptotic directions coincide or all the directions are $\xi$-asymptotic directions then $r$ is called a parabolic point.

Example 2.9*.*

The circle in $\mathbb{R}^{3}$ given by $x^{2}+y^{2}=1$, $z=0$, is an $\xi$-asymptotic line without parabolic points of the plane field $\xi$ defined by the orthogonal vector field $\xi=(\rho,\varrho,\sigma)$, where $\rho=x^{2}yz+y^{3}z-x^{2}y-y^{3}+xz-2yz+y$, $\varrho=x^{3}-x^{3}z-xy^{2}z+xy^{2}+2xz+yz-x$ and $\sigma=-x^{2}-y^{2}$. The plane field $\xi$ is not completely integrable. By the Theorem 2.2, this circle cannot be an asymptotic line of a regular surface $z=\varphi(x,y)$.

Proposition 2.10**.**

Given a plane field $\xi$, let $\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}$ be a differentiable nonvanishing function. Then a curve $\gamma$ is an $\xi$-asymptotic line if, and only if, $\gamma$ is an $\xi$-asymptotic line of the plane field $\widetilde{\xi}$ orthogonal to the vector field $\widetilde{\xi}=\varphi\xi$.

Proof.

The implicit differential equation of $\xi$-asymptotic lines of $\widetilde{\xi}$ is given by

[TABLE]

Then $\gamma$ is an $\xi$-asymptotic line of $\xi$ if, and only if, $\gamma$ is an $\xi$-asymptotic line of the plane field $\widetilde{\xi}$. ∎

2.4. Tubular neighborhood of a integral curve of a plane field

Let $\xi$ be a plane field orthogonal to a vector field $\xi(x,y,z)$. Then $d\xi=\xi_{x}dx+\xi_{y}dy+\xi_{z}dz$. Let $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))$ be a curve such that $(\gamma_{1}^{\prime}(x),\gamma_{2}^{\prime}(x))\neq(0,0)$ for all $x$. Set $X(x)=\gamma^{\prime}(x)$, $Y(x)=(\gamma_{2}^{\prime}(x),-\gamma^{\prime}(x),0)$, $Z(x)=(X\wedge Y)(x)$ and $\alpha:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$,

[TABLE]

The map (2.4) is a parametrization of a tubular neighborhood of $\gamma$. At this neighborhood, the position point is given by $r=\alpha(x,y,z)$ and then $dr=d\alpha=\alpha_{x}dx+\alpha_{y}dy+\alpha_{z}dz$. It follows that the implicit differential equation (2.2) of the $\xi$-asymptotic lines, at this neighborhood, is given by

[TABLE]

where,

[TABLE]

and

[TABLE]

Proposition 2.11**.**

Let $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))$ be a curve such that, for all $x$, $(\gamma_{1}^{\prime}(x),\gamma_{2}^{\prime}(x))\neq(0,0)$. Consider the tubular neighborhood (2.4). If $\xi$ is a plane field such that $\frac{a}{c}$ and $\frac{b}{c}$ are well defined in a neighborhood of $\gamma$, where $a,b,c$ are given by (2.5), then the implicit differential equation of the $\xi$-asymptotic lines, in this neighborhood, is given by

[TABLE]

where,

[TABLE]

Furthermore, in this neighborhood, the parabolic set of $\xi$ is given by $eg-f^{2}=0$.

Proof.

In a neighborhood of $\gamma$, solve the first equation of (2.5) for $dz$ to get the first equation of (2.8). Replace this $dz$ in the second equation of (2.5) to get the second equation of (2.8).

If $eg-f^{2}<0$ at a point (resp. $eg-f^{2}>0$), then the equations (2.8) defines two distinct $\xi$-asymptotic directions at this point (resp. two complex $\xi$-asymptotic directions).

If $eg-f^{2}=0$ at a point, then at it the $\xi$-asymptotic directions coincide or, if $e=g=f=0$, all directions are $\xi$-asymptotic directions. ∎

Definition 2.12** ([2, p. 11]).**

Let $\xi$ be a plane field satisfying the assumptions of Lemma 2.11. The function defined by $\mathcal{K}=eg-f^{2}$ is called the Gaussian curvature of $\xi$.

Lemma 2.13**.**

Let $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))$ be an $\xi$-asymptotic line of a plane field $\xi$, such that $(\gamma_{1}^{\prime}(x),\gamma_{2}^{\prime}(x))\neq(0,0)$ for all $x$. Consider the tubular neighborhood (2.4). Then, in a neighborhood of $\gamma$, the vector field $\xi$ is given by

[TABLE]

where $X(x)=\gamma^{\prime}(x)$, $Y(x)=(\gamma_{2}^{\prime}(x),-\gamma_{1}^{\prime}(x),0)$, $Z(x)=(X\wedge Y)(x)$, $A(x,0,0)=B(x,0,0)=C(x,0,0)=0$ and

[TABLE]

Furthermore, if $\gamma_{1}^{\prime}(x)\gamma_{2}^{\prime\prime}(x)-\gamma_{2}^{\prime}(x)\gamma_{1}^{\prime\prime}(x)\neq 0$ for all $x$, then the implicit differential equation of the $\xi$-asymptotic lines is given by (2.8).

Proof.

The expression (2.10) holds, since $\gamma$ is an integral curve of the plane field defined by $\xi$. Also, as $\gamma$ is an $\xi$-asymptotic line, $\langle\xi(x),\gamma^{\prime\prime}(x)\rangle=0$ for all $x$, which gives the expressions (2.11).

If $\gamma_{1}^{\prime}(x)\gamma_{2}^{\prime\prime}(x)-\gamma_{2}^{\prime}(x)\gamma_{1}^{\prime\prime}(x)\neq 0$, then $c(x,0,0)\neq 0$. The conclusion then follows from Proposition 2.11. ∎

3. Finite type $\xi$-asymptotic lines of plane fields

In this section the following result is established.

Theorem 3.1**.**

Any finite type curve is an $\xi$-asymptotic line without parabolic points of a suitable plane field.

Proof.

Let $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))=(x,a_{m}x^{m}+\mathcal{O}^{m+1}(x),a_{n}x^{n}+\mathcal{O}^{n+1}(x))$

be a finite type curve. Consider the tubular neighborhood $\alpha$ given by (2.4) and the vector field $\xi$ given by (2.10), with $k_{0}(x)$, $l_{0}(x)$ given by (2.11). Then $\gamma$ is a $\xi$-asymptotic line of the plane field orthogonal to $\xi$.

We have that $a(x,0,0)=0$ and

[TABLE]

We can factor the term $x^{m-2}$ from $b(x,0,0)$ and $c(x,0,0)$, $b(x,0,0)=x^{m-2}B(x,0,0)$ and $c(x,0,0)=x^{m-2}C(x,0,0)$ where $B(0,0,0)=0$ and $C(0,0,0)=a_{m}m(m-1)$.

Then, after factoring $x^{m-2}$ from the first equation of (2.5), the equation (2.5) will become $adx+bdy+cdz=0$, where $a(x,0,0)=0$, $b(0,0,0)=0$ and $c(0,0,0)=a_{m}m(m-1)\neq 0$. By Proposition 2.11, in a neighborhood of $(0,0,0)$, the equation of $\xi$-asymptotic lines are given by (2.8).

Let $k_{1}(x)$ be defined by

[TABLE]

Then $\mathcal{K}(x,0,0)=-1$. ∎

4. Hyperbolic closed finite type $\xi$-asymptotic line

Examples of hyperbolic asymptotic lines on surfaces are given in [6, 7, 8].

In this section it will be given an example of a hyperbolic closed $\xi$-asymptotic line of finite type for a suitable plane field.

Proposition 4.1**.**

Let $\gamma$, $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))$, be a curve such that $(\gamma_{1}^{\prime}(x),\gamma_{2}^{\prime}(x))\neq(0,0)$, $\gamma_{1}^{\prime}(x)\gamma_{2}^{\prime\prime}(x)-\gamma_{2}^{\prime}(x)\gamma_{1}^{\prime\prime}(x)\neq 0$ for all $x$. Consider the tubular neighborhood $\alpha$ given by (2.4) and the vector field $\xi$ given by (2.10), with $k_{0}(x)$, $l_{0}(x)$ given by (2.11). Let $H(x)$ be a nonvanishing function and define $k_{1}(x)$ by

[TABLE]

Then, $\gamma$ is a $\xi$-asymptotic line, without parabolic points, of the plane field orthogonal to the vector field $\xi$.

Furthermore, $\mathcal{K}(x,0,0)=-(H(x))^{2}$.

Proof.

By direct calculations, we can see that $\gamma$ is a $\xi$-asymptotic line. The implicit differential equation of the $\xi$-asymptotic lines are given by (2.8) and $e(x,0,0)=0$, $f(x,0,0)=H(x)$. Since $e(x,0,0)=0$, then $\mathcal{K}(x,0,0)=-(H(x))^{2}$ for all $x$. ∎

4.1. Poincaré map associated to a closed $\xi$-asymptotic line

Let $\gamma:[0,l]\rightarrow\mathbb{R}^{3}$, $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))$, be a closed $\xi$-asymptotic line, without parabolic points, of a plane field $\xi$, such that $\gamma(0)=\gamma(l)$, $(\gamma_{1}^{\prime}(x),\gamma_{2}^{\prime}(x))\neq(0,0)$, $\gamma_{1}^{\prime}(x)\gamma_{2}^{\prime\prime}(x)-\gamma_{2}^{\prime}(x)\gamma_{1}^{\prime\prime}(x)\neq 0$ for all $x$, and consider the tubular neighborhood $\alpha$ given by (2.4).

This means that $\gamma$ is a regular curve having a projection in a plane which is a strictly locally convex curve.

By the Proposition 2.13, $\xi$ is given by (2.10) and the implicit differential equations of the $\xi$-asymptotic lines is given by (2.8).

Let $\Sigma_{x_{0}}=\{(x_{0},y,z)\}$ be a transversal section. Then $\alpha(\Sigma_{x_{0}})$ is the plane spanned by $Y(x_{0})$ and $Z(x_{0})$. By Lemma 2.13, in a neighborhood of $\gamma$, the $\xi$-asymptotic line passing through $\alpha(x_{0},y_{0},z_{0})$ intersects $\alpha(\Sigma_{x_{0}})$ again at the point

[TABLE]

where $(y(x,y_{0},z_{0}),z(x,y_{0},z_{0}))$ is solution of the following Cauchy problem

[TABLE]

The Poincaré map $\mathcal{P}$, also called first return map, associated to $\gamma$ is defined by $\mathcal{P}:\mathcal{U}\subset\Sigma\rightarrow\Sigma$, $\mathcal{P}(y_{0},z_{0})=(y(l,y_{0},z_{0}),z(l,y_{0},z_{0}))$. See Fig. 3.

A closed $\xi$-asymptotic line $\gamma$ is said to be hyperbolic if the eigenvalues of $d\mathcal{P}_{(0,0)}$ does not belongs to $\mathbb{S}^{1}$. See [12] for the generic properties of the Poincaré map associated to closed orbits of vector fields.

We will denote by $d\mathcal{P}_{(0,0)}$ the matrix of the first derivative of the Poincaré map evaluated at $(y_{0},z_{0})=(0,0)$.

Proposition 4.2**.**

Let $\gamma:[0,l]\rightarrow\mathbb{R}^{3}$, $\gamma(x)=(\gamma_{1}(x),\gamma_{2}(x),\gamma_{3}(x))$, be a closed $\xi$-asymptotic line, having a projection in a plane which is locally strictly convex curve. Let $\mathcal{P}$ be the Poincaré map associated to $\gamma$. Then $d\mathcal{P}_{(0,0)}=\mathcal{Q}(l)$, where $\mathcal{Q}(x)$ is solution of the following Cauchy problem:

[TABLE]

where $\mathcal{I}$ is the identity matrix, and $\mathcal{M}(x)$, $\mathcal{Q}(x)$ are the matrices given by

[TABLE]

where $A=-\frac{a}{c}$.

Proof.

To fix the notation suppose that $\gamma(0)=\gamma(l)$, $(\gamma_{1}^{\prime}(x),\gamma_{2}^{\prime}(x))\neq(0,0)$ and $\gamma_{1}^{\prime}(x)\gamma_{2}^{\prime\prime}(x)-\gamma_{2}^{\prime}(x)\gamma_{1}^{\prime\prime}(x)\neq 0$ for all $x$.

Let $(y(x,y_{0},z_{0}),z(x,y_{0},z_{0}))$ be solution of the Cauchy problem given by equation (4.2). Then, at $(y,z)=(0,0)$, $\frac{dy}{dx}(x,0,0)=\frac{dz}{dx}(x,0,0)=0$.

Differentiating the first equation of (4.2) with respect to $y_{0}$ (resp. $z_{0}$), it results that:

[TABLE]

respectively,

[TABLE]

Differentiating the second equation of (4.2) with respect to $y_{0}$ (resp. $z_{0}$), it results that:

[TABLE]

respectively,

[TABLE]

Evaluating (4.5), (4.6), (4.7), (4.8) at $(y,z)=(0,0)$, it follows that:

[TABLE]

Then $\frac{d}{dx}\left(\mathcal{Q}(x)\right)=\mathcal{M}(x)\mathcal{Q}(x)$. Since $(y(0,y_{0},z_{0}),z(0,y_{0},z_{0}))=(y_{0},z_{0})$, it follows that $\mathcal{Q}(0)=\mathcal{I}$.

Since $\mathcal{P}(y_{0},z_{0})=(y(l,y_{0},z_{0}),z(l,y_{0},z_{0}))$, the first derivative $d\mathcal{P}_{(0,0)}$ is given by $\mathcal{Q}(l)$. ∎

4.2. Example of a hyperbolic closed finite type $\xi$-asymptotic line

An explicit example of a hyperbolic closed $\xi$-asymptotic line is given in the next result.

Theorem 4.3**.**

Let $\gamma:[0,2\pi]\rightarrow\mathbb{R}^{3}$, $\gamma(x)=(sin(x),cos(x),sin^{3}(x))$, see Fig. 4. Then it is a hyperbolic finite type $\xi$-asymptotic line of a suitable plane field.

Proof.

Let $\xi$ be a plane field orthogonal to the vector field $\xi$ given by (2.10), where $k_{0}(x)$ and $l_{0}(x)$ are given by (2.11). Let $k_{1}(x)$ given by (4.1), with $H(x)\equiv 1$. Then

[TABLE]

By Proposition 4.1, $\gamma$ is a $\xi$-asymptotic line without parabolic points and $\mathcal{K}(x,0,0)=-1$. Performing the calculations, $e_{z}(x,0,0)=\mathcal{E}(x)+l_{2}(x)$. Solve $e_{z}(x,0,0)=0$ for $l_{2}(x)$. This vanishes the entry $\left(-\frac{e_{z}(x,0,0)}{2f(x,0,0)}\right)$ of $\mathcal{M}(x)$ given by Theorem 4.2. From (4.3), it follows that the eigenvalues of $d\mathcal{P}_{(0,0)}$ are given by

[TABLE]

Set $l_{1}(x)=cos(x)$. Then

[TABLE]

It follows that $\int_{0}^{2\pi}A_{z}(x,0,0)dx=-\frac{25\pi}{8}$. Let $k_{3}(x)=0$ and $k_{2}(x)$ a solution of the equation $e_{y}(x,0,0)+2f(x,0,0)=0$. It follows that

[TABLE]

∎

Acknowledgments

The second author is fellow of CNPq. This work was partially supported by Pronex FAPEG/CNPq.

Appendix A A new proof of the Arnold’s Theorem

Here it will be given a geometric proof of Arnold’s Theorem.

Proof.

Let $\gamma$ be an asymptotic line of finite type $(u,u^{m},u^{n})$, $n\geq m$. Set $N(u)=(\gamma_{2}^{\prime}(u),-\gamma_{1}^{\prime}(u),0)$. Let

[TABLE]

Let

[TABLE]

The implicit differential equations of the asymptotic lines of $\alpha$ is given by

[TABLE]

where $e=\langle\alpha_{uu},N_{\alpha}\rangle$, $f=\langle\alpha_{uv},N_{\alpha}\rangle$ and $g=\langle\alpha_{vv},N_{\alpha}\rangle$.

Since $\gamma$ is an asymptotic line of $\alpha$, and parametrized by $v=0$, we have that $e(u,0)=0$. Then by equation (4.1) it follows that

[TABLE]

Direct calculations shows that

[TABLE]

It follows that $f(0,0)\neq 0$ if, and only if, $n=m+1$.

If $\gamma$ is a rotating space curve of finite type $(u,u^{m},u^{m+1})$, $m\geq 2$, set $N(u)=(\gamma_{2}^{\prime}(u),-\gamma_{1}^{\prime}(u),0)$ and let

[TABLE]

where $k_{1}(u)$ is given by (A.4) with $n=m+1$. Therefore, $e(u,0)=[\beta_{u},\beta_{v},\beta_{uu}](0,0)=0$ and $f(0,0)=[\beta_{u},\beta_{v},\beta_{uv}](0,0)=\frac{m+1}{m-1}\neq 0$. Then $\gamma$ is an asymptotic line, without parabolic points, of the surface parametrized by $\beta$ in a neighborhood of $(u,v)=(0,0)$. ∎