Lines of Principal Curvature on Canal Surfaces in R^3

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Riccati equations, it is established that canal surfaces have at most two isolated periodic principal lines. Examples of canal surfaces with two simple and one double periodic principal lines are given.


Introduction
The study of principal curvature lines, along which a surface in R 3 bends extremely and their umbilic singularities was founded by Monge, Dupin and Darboux. See (Gray 1998) and (Sotomayor 2003) for references. For the basic facts about principal curvature lines on surfaces the reader is addressed to (do Carmo 1976), (Spivak 1979) and (Struik 1988).
As a consequence of the work of Monge and Dupin the lines of curvature on quadrics and toroidal -Dupin Cyclides-surfaces were determined. See (Fischer 1986, Chap. 3), for an outline of the theory and for a collection of remarkable illustrations.
In (Gutierrez and Sotomayor 1982, 1991, 1998 ideas originating in the Qualitative Theory of Differential Equations and Dynamical Systems, such as Structural Stability and Genericity, were incorporated into the subject; see also (Garcia and Sotomayor 2002). Historical comments, going up to some recent developments on principal curvature lines, can be found in (Sotomayor 2003).
The global dynamic complexity of principal curvature in simple smooth surfaces was illustrated in Sotomayor 1991, 1998). This includes examples of recurrent (dense) principal curvature lines on spheroidal surfaces, which are small perturbations of ellipsoids of revolution and canal surfaces with constant radial functions. The methods established in these works show how to make hyperbolic a principal cycle by means of a small smooth perturbation. This leads to smooth toroidal immersions with arbitrary large number of principal isolated principal cycles.
In this paper is improved and reproved a result that goes back to Vessiot, establishing that principal curvature lines on canal surfaces immersed in R 3 (see Section 2, Definition 2) verify a Riccati equation (see Section 3, Remark 7). In fact, (Vessiot 1919) is the first source known by the authors for the connection between Riccati equations and principal curvature lines on canal surfaces. The improvement consists in the formulation of precise conditions for canal surfaces be regular immersions and also to have umbilic points (see Section 3, Theorem 4 and Remark 5). A consequence of the Riccati structure for principal curvature lines on canal immersed surfaces implies that the maximal number of isolated periodic principal lines is 2. Examples of canal surfaces with two (simple i.e. hyperbolic) and one (double i.e. semi-stable) principal periodic lines, absent in (Vessiot 1919) and also in later references, are given in this paper (see Section 4, Proposition 8).

Regular Canal Surfaces
Consider the space R 3 endowed with the Euclidean inner product < , > and norm | | =< , > 1/2 as well as with a canonical orientation. The wedge product ∧ of vectors is defined relative to this orientation.
The expression for E is as follows: Additional calculation and simplification using equations (9), with the tangency condition (5) imposed, and (10) gives This expression, which is equal to |∂α/∂s ∧ ∂α/∂ϕ| 2 , vanishes if and only if Condition (6) states that the absolute value of the right-hand member of (12) is larger than 1, which by (11) implies the linear independence of α s and α ϕ at every point (s, ϕ), namely that α is an immersion.
Definition 2. A mapping such as α, of T 2 into R 3 , satisfying conditions (5) and (6) will be called an immersed canal surface with center along c(s) and radial function r(s). When r is constant, it is called an immersed tube. Due to the tangency condition (5), the immersed canal surface α is the envelope of the family of spheres of radius r(s) whose centers range along the curve c(s).

Principal Lines on Regular Canal Surfaces
In this work the positive orientation on the torus T 2 = S 1 × S 1 is defined by the ordered tangent frame {∂/∂s, ∂/∂ϕ}. Therefore the positive unit normal -or Gaussian map-N α of the immersion α is defined by |∂α/∂s ∧ ∂α/∂ϕ|N α = ∂α/∂s∧∂α/∂ϕ. By the tangency condition (5) this unit vector is given by (8), which can be written as follows: It points inwards the toroidal surface defined by α.
Below will be studied the global behavior of the principal curvature lines of α.
Theorem 4. Let α : S 1 × S 1 → R 3 be a smooth immersion expressed by (4). Assume the regularity conditions (5) and (6) as in Proposition 1 and also that The maximal principal curvature lines are the circles tangent to ∂/∂ϕ. The maximal principal curvature is k 2 (s) = 1/r(s).
The minimal principal curvature lines are the curves tangent to The expression is negative, and the minimal principal curvature is given by . (17) There are no umbilic points for α: k 1 (s, ϕ) < k 2 (s).
Proof. Direct calculation gives: By Rodrigues equation, see (do Carmo 1976) and (Struik 1988), the circles s = constant are principal curvature lines of α, with principal curvature k 2 (s).
Denoting the metric of α by Eds 2 + 2F dsdϕ + Gdϕ 2 , it follows that the direction orthogonal to ∂/∂ϕ, giving the other principal direction, is defined by the vector field which by equation (9) is collinear with V (s, ϕ) in (15).
Remark 5 (Umbilic Points in Canal Immersions). The calculation of k 1 and the condition for the appearance of umbilic points for regular canal surfaces has not been considered in previous works on the subject. The discussion leading to (21) leads to the following equation for umbilic points: (14) is not imposed. This curve however consists of removable singularities for the principal line fields which have smooth extensions to the whole torus, given by ∂/∂ϕ and V . Under generic conditions on the radial function, r(s), and curvature, κ(s), functions it will be expected to appear curves of umbilic points as those studied in Proposition 2 of (Garcia and Sotomayor 2005).

It gives a non-empty curve if condition
Remark 6 (Minimal Principal Foliation in terms of Differential Forms). Written as a differential form, the vector field (15) becomes: When it is used another parameter t, say T -periodic, related with the arc length s of c in the form s = s(t), withṡ =<ċ,ċ > 1/2 , the form in (22) writes as: Remark 7 (Vessiot). By the change of coordinates tan(ϕ/2) = z, equation (22) is transformed into the L-periodic Riccati equation Therefore, the solutions of equation (15) can be obtained from the solutions of equation (24)

Canal Surfaces with one and two Principal Cycles
In this section is carried out a discussion on the qualitative properties of equation (15), or of its equivalent form (23), absent in (Vessiot 1919).
Being equivalent to a periodic Riccati equation, this equation has a Möebius transformation as return map (see Hille 1976) and therefore can have either: (a) all its solutions periodic, (b) all its solutions dense, (c) two hyperbolic (simple) periodic solutions or (d) one semi-hyperbolic (double or semi-stable) periodic solution. An example of situation (a) is exhibited by the standard torus of revolution. An example of (b) is given in Sotomayor 1991, 1998), for a canal surface of constant radius (a tube) around a curve that is not bi-regular. Below will be given examples of cases (c) and (d). This will also provide examples of case (b) for bi-regular curves. The cost of this is heavier calculation which, nevertheless, is easy to corroborate with Computer Algebra.
Therefore, the surface of 2π-periodic orbits, in the (ν, µ, ϕ)-space consists on a plane µ = 0 crossing transversally a regular sheet P at the sinusoidal curve of equation ν + sin ϕ 0 = 0. The critical values of the projection on the (ν, µ)−plane of the surface P, coming from double (semi-stable) periodic solutions, the fold curve, must cross transversally the ν-axis at points corresponding to ν = ±1, which are the critical values of the projection of the sinusoidal curve. Going back to the coordinates (ε, µ) gives the conclusion formulated in the proposition.
Remark 9 (Arnold Tongues). Due to equations (31) and (27), for ρ small, the return map on the tube S ε,ρ,0 is a rotation with its rotation number changing monotonically with ε (taken small). Therefore it takes irrational values and the tube presents case (b). It also takes rational values, q/p, which are the vertices of the Arnold Tongues (in the (ε, µ)-plane), corresponding to canal surfaces with periodic closed principal lines winding p-times around the parallels (t-circles) and q-times around the meridians (ϕ-circles) of the torus. The sector with vertex at (0, 0) established in Proposition 8 is the principal tongue, with rotation number 0. See (Chicone 1999, page 372