Surfaces in making constant angle with Killing vector fields
Abstract.
In the present paper we classify curves and surfaces in Euclidean space which make constant angle with a certain Killing vector field. Moreover, we characterize the catenoid and Dini’s surface in terms of constant angle surfaces.
Key words and phrases:
Lancret problem, angle of two curves, Killing vector field, space form, Euclidean space, Dini’s surface2000 Mathematics Subject Classification:
53B251. Introduction
The geometry of curves and surfaces in represented for many years a popular topic in the field of classical differential geometry. Global and local properties were studied in several books and new invariants were defined for curves as well as for surfaces.
An old and important problem in differential geometry of curves and surfaces is that of Bertrand–Lancret–de Saint Venant saying that a curve in the Euclidean space is of constant slope, namely its tangent makes a constant angle with a fixed direction, if and only if the ratio of torsion and curvature is constant. These curves are also called generalized helices. If both and are nonzero constants the curve is called circular helix. Other interesting curves in are slant helices, characterized by the property that the principal normals make a constant angle with a fixed direction.
The problem of Bertrand–Lancret–de Saint Venant was generalized for curves in other 3dimensional manifolds – in particular space forms or Sasakian manifolds. Such a curve has the property that its tangent makes constant angle with a parallel vector field on the manifold or with a Killing vector field respectively. For example, a curve in a dimensional space form is called a general helix if there exists a Killing vector field with constant length along and such that the angle between and is a nonzero constant. See e.g. [4], [17]. Moreover, in [1] it is shown that general helices in a dimensional space form are extremal curves of a functional involving a linear combination of the curvature, the torsion (as functions) and a constant. On the other hand, in a contact manifold, a curve is called a slant curve if the angle between its tangent and the Reeb vector field is constant (see e.g. [8], [2]). General helices, also called Lancret curves are used in many applications (e.g. [5], [13]).
Bertrand studied curves in Euclidean space whose principal normals are also the principal normals of another curve (called Bertrand mate). Such a curve is nowadays called a Bertrand curve and it is characterized by a linear relation between its curvature and torsion. Bertrand mates in are particular examples of offset curves used in computeraided geometric design (see e.g. [23]). Due to previous characterization, one may say that Bertrand curves are strictly related with Weigarten surfaces. Possible applications of Weingarten surfaces are found in [6].
Passing from curves to surfaces, a natural question is to find which surfaces in dimensional space make constant angle with certain vector fields. For example in [22] it is given a new approach to classify surfaces in making a constant angle with a fixed direction. These surfaces were called constant angle surfaces and they appear in the study of liquid crystals and of layered fluids [7]. Recently, the geometry of constant angle surfaces was developed in other dimensional spaces as [10], [11], the Heisenberg group [14], [20], Minkowski space [19], a.s.o. On the other hand, in analogy with the geometry of logarithmic spirals, in [21] there were found all surfaces in whose normals make constant angle with the position vector. Particular surfaces with beautiful shapes were obtained.
The main part of the paper is devoted to the study of surfaces whose normals make constant angle with Killing vectors in . We obtain the complete classification of these surfaces, which includes: halfplanes having axis as boundary, rotational surfaces around axis, right cylinders over logarithmic spirals and Dini’s surfaces. Regarding Dini’s surfaces as a particular kind of helicoidal surfaces, on this broader class of surfaces many studies were made, let us mention for example [3], [12], [24], [25] and references therein. Classification and characterization results were obtained having as starting point the fact that helicoidal surfaces are applicable upon rotational surfaces. For more recent developments and approaches on curves and surfaces, see for example [15].
Different types of helicoidal surfaces were investigated also recently in other ambient spaces as Minkowski space [18], or the product space , [9].
The study of these surfaces did not stop only in the field of classical differential geometry. Also interesting computer graphics results concerning helicoidal surfaces were obtained in e.g. [16]. We conclude the present paper with a new characterization of Dini’s surfaces in terms of constant angle surfaces.
2. Some results on curves in
In the first part of this Section we bring to light new properties of planar and spatial curves involving the angle of two curves. Since the result is local, let us consider two curves and , without selfintersections, parameterized by the same parameter . For an arbitrary , we define the angle (at ) of and as the angle of the tangent vectors and in the corresponding point. Interesting problems appear when the angle is constant.
Let us point our attention first on two planar curves which lie in the same plane. A particular case arises when one of the curves is a straight line. If the other curve makes constant angle with this line, it is also a straight line. A second case occurs when we are looking for planar curves which make a constant angle with the unit circle . Take
(1) 
arc length parameterized and the unit circle parameterized with the same parameter as ,
(2) 
where is a smooth function defined on . Notice that, in general, the arc length parameters for the two curves are not the same.
Denote by and the unit tangent vectors to and to the curve , respectively. The angle between and is and it will be a constant . It follows that
(3) 
Straightforward computations yield the parametrization of the curve
after a translation in .
If we denote , then the parametrization (2) can be rewritten as
In order to give a geometric interpretation of the above formula, let us take a look at Figure 1 where the the angle , the function and .
If the curve (green) makes constant angle with the circle (magenta), then the curve (blue) represents the rotation with angle of in clockwise direction. The line segments represent pairs of tangent vectors to the curve and respectively to the circle .
In the same manner we investigate now the spatial curves. The first question that arises is to find spatial curves which make a constant angle with a straight line and a classical result tells us that is a helix. Without loss of generality the line can be taken to be (parallel with) one of the coordinate axes. But this is an integral curve of a Killing vector vector field in . Motivated by these remarks, a natural question springs up: which curves make a constant angle with a Killing vector field in ?
Let us briefly recall that a vector field in is Killing if and only if it satisfies the Killing equation:
for any vector fields , in , where denotes the Euclidean connection on . The set of solutions of this equation is
and it gives a basis of Killing vector fields in . Here denote the global coordinates on and regarded as a vector space.
As we have mentioned before, spatial curves which make a constant angle with any of the Killing vector fields are helices. Regarding the other three Killing vector fields, the problem is basically the same for each of them and let us choose for example .
We study first planar curves (in plane) defined by (1) which make a constant angle with the Killing vector field .
Since the curve is parameterized by arc length, the unit tangent vector is given by . Since , the condition can be rewritten as
(5) 
A convenient way of handling is to consider its polar parametrization,
and then equation (5) becomes
(6) 
Using now the fact that is parameterized by arc length and combining it with the previous condition, we get after integration
if : and
if : and
where and .
We state now the following theorem:
Theorem 1.
A curve lying in the plane and making constant angle with the Killing vector is

either the circle centered in the origin and of radius

or a straight line passing through the origin

or the logarithmic spiral .
Remark 1.
This is not a surprising fact since the circle is an integral curve for and the logarithmic spiral, known also as the equiangular spiral, is characterized by the property that the angle between its tangent and the radial direction at every point is constant. Moreover, the position vector and the vector are orthogonal.
If is a spatial curve making a constant angle with the Killing vector field , let us consider given in cylindrical coordinates as:
(7) 
where is again arc length parameter.
We have and . Similarly as in previous situation, the property becomes
(8) 
Again, since is parameterized by arc length, we get . Hence, there exists a function such that
Integrating once and combining with (8) we may formulate the following theorem:
Theorem 2.
A curve in the Euclidean space which makes a constant angle with the Killing vector is given, in cylindrical coordinates, up to vertical translations and rotations around axis, by
where and is a smooth function on .
Let us mention some particular cases for function . (We will take different from and .)
3. Classification of surfaces making constant angle with
In this section we are interested to find all surfaces in Euclidean space which make a constant angle with the Killing vector field .
Since the vector field must be nonnull, the surfaces lie in .
A good motivation to study this problem is given by the particular case . Let the surface be parameterized by given by a graph
(9) 
Computing the normal to the surface and imposing for the vector field to be tangent to the surface in this case, namely to be orthogonal to the unit normal , one gets . Hence, the surfaces for which makes a constant angle are given by (9) with , where is an arbitrary real valued function; they are rotational surfaces.
Concerning the other particular case when the constant angle vanishes identically, we obtain halfplanes having axis as boundary.
We focus our attention now on the general case . First, let us fix the notations. Denote by the metric on and by the associated Levi Civita connection and let be the flat connection in the ambient space. The Gauss and Weingarten formulas are:
(G)
(W)
for every tangent to . Here is a symmetric tensor field called the second fundamental form of the surface and is a symmetric tensor field called the shape operator associated to satisfying for any tangent to .
Projecting onto the tangent plane to one has , where is the tangent part with and . At this point we may choose an orthonormal basis on the tangent plane to such that and . Now, we have that .
For an arbitrary vector field in , we have
(10) 
where and stands for the usual cross product in .
If is tangent to , then
Since and are tangent to , let us decompose and in the basis denoting the following angles: , and . By standard calculus one gets up to sign
In these notations, we have
(12) 
Combining (10) and (3), first for and then for , together with (12) we obtain
(13) 
(14) 
As a consequence, the shape operator associated to the second fundamental form is given by
(15) 
where is a smooth function on . Hence and are principal directions on .
In order to study the geometry of the surface , we determine first the Levi Civita connection of :
Thus, the Lie bracket of and is given by
(16) 
and consequently a compatibility condition is found computing in two ways and taking into account (13) and (14):
(17) 
From now on we use cylindrical coordinates, such that the parametrization of the surface may be thought as
(18) 
The Euclidean metric in becomes a warped metric and its Levi Civita connection is given by:
(19)  
The Killing vector field coincides with .
The basis may be expressed in terms of the new coordinates as:
(20)  
In the sequel, since is an involutive system, in other words , one obtains:
(21) 
Remark 2.
When is constant, it follows that since . Hence, from now on we will deal with , while the case will be treated separately.
Writing (21) in an equivalent manner as and taking the derivative along we get
(22) 
Our intention is to find, locally, appropriate coordinates on the surface in order to write explicit embedding equations of in . Let us choose a local coordinate such that . The second coordinate, denote it by , will be defined later. In this framework, equation (22) has the solution , where and . Choosing such that we may consider . Moreover, after a translation in coordinate, one gets
(23) 
Substituting (23) in (21) easily follows
(24) 
Now let us express in terms of and as
(25) 
Thus, . On the other hand, replacing the expression of given by (23) in (14) we get that . Hence,
(26) 
Exploiting the compatibility condition (17) and combining it with (26) we get
(27) 
Consequently, . In terms of and , from (16), one obtains . After a homothetic transformation of the coordinate, one can take and thus
(28) 
Replacing now (26) and (28) in (25) the expression of is explicitly found
(29) 
The Levi Civita connection may also be written in terms of the coordinates and in the following manner:
(\theparentequation.a)  
(\theparentequation.b)  
At this point, using the expression of the shape operator (15), and the formulas (23), (24) and respectively (27) we obtain the expression of the second fundamental form in terms of and as:
(\theparentequation.a)  
(\theparentequation.b)  
(\theparentequation.c) 
Since the surface is given in cylindrical coordinates by the isometric immersion (18), and using the Euclidean connection (19) one has
(32) 
On the other hand, using Gauss formula (G) one computes
Comparing the two previous expressions (32) and (3) we have the following PDEs:
(\theparentequation.a)  
(\theparentequation.b)  
(\theparentequation.c) 
Using the same technique also for the expressions of and , we get in addition:
(\theparentequation.a)  
(\theparentequation.b)  
(\theparentequation.c) 
respectively
(\theparentequation.a)  
(\theparentequation.b)  
(\theparentequation.c) 
First, let us note that the (cylindrical) coordinate has the same meaning as the function , namely is given by (24). As a consequence, . Moreover, from the choice of and , together with (3), we immediately obtain the third component of the parametrization
(37) 
after a translation along axis. Furthermore, straightforward computations yield
(38) 
Hence, when , combining (24), (37) and (38), we get the following parametrization in cylindrical coordinates:
(39) 
where is a nonzero real constant.
Let us return now to the case . Following the same steps as in the general case, we get the next parametrization in cylindrical coordinates
(40) 
These surfaces are right cylinders over logarithmic spirals.
At this point one can state the main result of this section.
Theorem 3.
Let be a surface isometrically immersed in and let be a Killing vector field. Then makes a constant angle with if and only if it is one of the following surfaces:
Proof.
The direct implication is the conclusion of the previous computations, so we briefly sketch it.
We analyze different cases for the constant angle :

: the halfplanes passing through axis as in case (i) are obtained;

: we get rotational surfaces, case (ii);
The converse part follows by direct computations showing that the surfaces in each of the cases (i)–(iv) make a constant angle with the Killing vector field . ∎
Willing to give more information about the geometry of Dini’s surface described in case (iv) of Theorem 3, we note the following
Remark 3.
The surface given by (39) is a helicoidal surface with axis , namely it can be parametrized as
where . The pitch is .
Recall a classical construction [15, §15.7]: A helicoidal surface, known also as a generalized helicoid, is always related to a rotational surface (when the pitch vanishes identically). Twisting the pseudosphere of radius , one obtains the Dini’s surfaces (39).
The next result makes a connection with the previous Section.
Proposition 1.
The parametric curves of surfaces parameterized by (39) are circular helices and spherical curves.
Proof.
The parameter curves () are circular helices with the same pitch and with radius , depending on . On the other hand, the parameter curves () are unit speed spherical curves and they lie on 2spheres centered in and with the same radius . ∎
Remark 4.
Looking backward to Section 2, the parameter curves make the constant angle with the Killing vector field and the affine function is given by .
We conclude this paper with the following proposition which shows that surfaces making constant angle with define a particular class of Weingarten surfaces.
Proposition 2.
Let be a surface isometrically immersed in which makes a constant angle with the Killing vector field . Then,

is totally geodesic if and only if it is a vertical plane with the boundary axis;

is minimal not totally geodesic if and only if it is a catenoid around axis;

is flat if and only if it is a vertical plane with the boundary axis, a flat rotational surface or a right cylinder over a logarithmic spiral;

the Dini’s surface from case (iv) of the classification theorem has constant negative Gaussian curvature, .
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