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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 823809, 20 pages
Singularities of Focal Surfaces of Null Cartan Curves in Minkowski 3-Space
1School of Mathematics Science, Harbin Normal University, Harbin 150500, China
2School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
3School of Mathematics, Changchun Normal University, Changchun 130032, China
Received 28 June 2012; Accepted 28 August 2012
Academic Editor: Gabriel Turinici
Copyright © 2012 Zhigang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Singularities of the focal surfaces and the binormal indicatrix associated with a null Cartan curve will be investigated in Minkowski 3-space. The relationships will be revealed between singularities of the above two subjects and differential geometric invariants of null Cartan curves; these invariants are deeply related to the order of contact of null Cartan curves with tangential planar bundle of lightcone. Finally, we give an example to illustrate our findings.
If we imagine a regular curve in , denote , and then imagine the set of all principal normal lines intersecting this curve. Unless is a line, these lines will all meet some locus, in fact, it is the locus of centres of curvatures of , which we call the focal sets. It is obvious that the focal set would be a point or a new curve depending on given curves . Focal sets are useful to the study of certain optical phenomena (namely, scattering, in fact a rainbow is caused by caustics), expressing some geometrical results within fluid mechanics as well as describing many medical anomalies [1–4], and so it is important to study the geometric properties related to the focal curve (i.e., the le we proved that the properties of the vocus of focal set is a curve) of a curve.
It is well known that there exist spacelike curve, timelike curve, and null curve in Minkowski spacetime. For nonnull curve in Minkowski space, many of the classical results from Riemannian geometry have Lorentz counterparts. In fact, spacelike curves or timelike curves can be studied by approaches similar to those taken in positive definite Riemannian geometry. Nonnull curves in Minkowski space, regarding singularity, have been studied extensively by, among others, the second author and by Izumiya et al. [5–10]. The importance of the study of null curves and its presence in the physical theories are clear [4, 11–18]. Nersessian and Ramos  also show us that there exists a geometrical particle model based entirely on the geometry of the null curves, in Minkowskian 4-dimensional spacetime which under quantization yields the wave equations corresponding to massive spinning particles of arbitrary spin. They have also studied the simplest geometrical particle model which is associated with null curves in Minkowski 3-space . However, null curves have many properties which are very different from spacelike and timelike curves [11, 21, 22]. In other words, null curve theory has many results which have no Riemannian analogues. In geometry of null curves difficulties arise because the arc length vanishes, so that it is impossible to normalize the tangent vector in the usual way. Bonner introduces the Cartan frame as the most useful one and he uses this frame to study the behaviors of a null curve . Thus, one can use these fundamental results as the basic tools in researching the geometry of null curves. However, to the best of the authors’ knowledge, the singularities of surfaces and curves as they relate to null Cartan curves (see Section 2) have not been considered in the literature, aside from our studies in de Sitter 3-space [24, 25]. Thus, the current study hopes to serve such a need; in this paper, we study the focal surfaces and the binormal indicatrix associated with a null Cartan curve in Minkowski 3-space from the standpoint of singularity theory.
A singularity is a point (or a function) at which a function (or surface resp.) blows up. It is a point at which a function is at a maximum/minimum or a surface is no longer smooth and regular. Much of the time, these singularities affect a surface not only at a certain point but around it also, and for this reason, we have focused our attention on germs in a local neighbourhood around a fixed point. To allow a useful study of these singularities, we consider volume like distance functions (denoted by ) locally around the point These functions are the unfoldings of these singularities in the local neighbourhood of , and depend only on the germ that they are unfolding. In this paper, we create these functions by varying a fixed point in the volumelike distance function , to get a family of functions. We show that these singularities are versally unfolded by the family of volumelike distance functions. If the singularity of is -type () and the corresponding -parameter unfolding is versal, then applying Bruce’s theory (cf. ), we know that discriminant set of the -parameter unfolding is locally diffeomorphic to cuspidal edge or swallowtail; thus, we finished the classification of singularities of the focal surface (because the discriminant set of the unfolding is precisely the focal surface of a null Cartan curve). Moreover, we see the -singularity () of are closely related to the new geometric invariant . The singular point of the focal surface corresponds to the point of the null Cartan curve which has degenerated contact with a tangential planer bundle of a lightcone. As a consequence, the new Lorentzian invariant describes the contact between the tangential planer bundle of a lightcone and null Cartan curve . It is important that the properties of volumelike distance function (or null Cartan curve ) needed to be generic . Once we proved that the properties of the volumelike distance function were generic, we could deduce that all singularities were stable under small perturbations for our family of volumelike distance function. If transversality is satisfied for the volume like distance function of a null Cartan curve , then the properties of the volumelike distance function are generic. By considering transversality, we prove that these properties given by us are generic. On the other hand, the binormal indicatrix of a null Cartan curve is a curve that lies in lightcone, by defining the lightcone height function and adopting the method similar to those taken in the study of focal surface, we can classify the singularities of the binormal indicatrix the types of these singularities have a direct relation with the other Lorentzian invariant .
A brief description of the organization of this paper is as follows. The main results in this paper are stated in Theorems 2.1 and 6.3. In Section 3, we give volumelike distance functions and light-like height functions of a null Cartan curve, by which we can obtain several geometric invariants of the null Cartan curve. The geometric meaning of Theorem 2.1 is described in Section 4. We give the proof of Theorem 2.1 in Section 5. In Section 7, we give an example to illustrate the results of Theorem 2.1.
Let denote the 3-dimensional Minkowski space, that is to say, the manifold with a flat Lorentz metric of signature , for any vectors and in , we set . We also define a vector where is the canonical basis of . We say that a vector is spacelike, null, or timelike if is positive, zero, or negative, respectively. The norm of a vector is defined by . We call a unit vector if .
Let be a smooth regular curve in (i.e., for any ), parametrized by an open interval . For any , the curve is called a spacelike curve, a null (lightlike) curve, or a timelike curve if all its velocity vectors satisfy or , respectively. We call a non-null curve if is a timelike curve or a spacelike curve.
Let be a null curve in (i.e., for any ). Now suppose that is framed by a null frame. A null frame at a point of is a positively oriented 3-tuple of vectors satisfying The Frenet formula of with respect to is given by The functions , and are called the curvature functions of (cf. ). Employing the usual terminology, the spacelike unit vector filed will be called the principal normal vector filed. The null vector filed is called the binormal vector filed. Null frames of null curves are not uniquely determined. Therefore, the curve and a frame must be given together.
There always exists a parameter of such that in (2.3). This parameter is called a distinguished parameter of , which is uniquely determined for prescribed screen vector bundle (i.e., a complement in to ) up to affine transformation .
Let be a null curve with a distinguished parameter in (i.e., in (2.3)). Moreover, we assume that are linearly independent for all . Then, we consider the basis such that . We choose the , then there exists only one null frame for which is a framed null curve with Frenet equations : where , , , . We call (2.4) the Cartan Frenet equations and their null Cartan curve . We remark that the curvature function is an invariant under Lorentzian transformations.
In case is a null Cartan curve, labeling , then the Frenet formula of with respect to becomes This frame satisfies
Now we define surface by We call the Focal surface of null Cartan curve . We define the -dimensional future lightcone vertex at by When is the null vector , we simply denote by .
Let be a regular null Cartan curve. We define the binormal normal indicatrix of as the map given by and the focal curve of as the map given by Defining the set: for any , we call it the tangential planar bundle of lightcone through . It is obvious that the lightcone is the envelope of the tangential planar bundle.
We give a geometric invariant of a null Cartan curve in by which are related to the geometric meanings of the singularities of the focal surface.
We shall assume throughout the whole paper that all the maps and manifolds are unless the contrary is explicitly stated.
Let be a submersion and be a null Cartan curve. We say that and have k-point contact for if the function satisfies . We also say that and have at least k-point contact for if the function satisfies .
The main result in the paper is as follows.
Theorem 2.1. Let be a regular null Cartan curve with . For and the tangential planar bundle of lightcone, one has the following. (1)The null Crtan curve and have at least 2-point contact for . (2)The null Crtan curve and have 3-point contact for if and only if
Under this condition, the germ of Image at is locally diffeomorphic to the cuspidal edge (cf., Figure 1). (3) The null Crtan curve and have 4-point contact for if and only if
Under this condition, the germ of Image at is locally diffeomorphic to the swallowtail (cf., Figure 2).
Let : be a null Cartan curve. If is a point of the binormal indicatrix of at , then locally at , (1)the binormal indicatrix is diffeomorphic to a line at if , (2)the binormal indicatrix is diffeomorphic to the ordinary cusp at if and .
Here, is the cuspidal edge and is the swallowtail.
3. Volume-Like Distance Function and Lightcone Height Function of Null Cartan Curve
The purpose of this section is to obtain one geometric invariants of null Cartan curves by constructing a family of functions of the null Cartan curve.
Let be a regular null Cartan curve with . We define a three-parameter family of smooth functions by . Here, denotes the determinant of matrix . We call the volumelike distance function of null Cartan curve . We denote that for any fixed vector in . Using (2.5) and making a simple calculation, we can state the following facts.
Proposition 3.1. Suppose is a regular null Cartan curve with and . Then (1) if and only if there exist real numbers such that ,(2) if and only if ,(3) if and only if ,(4) if and only if and ,(5) if and only if and .
Proof. (1) Let . We have
The assertion (1) follows. (2) By (1), we have . Using (3), we obtain
It follows that if and only if . (3) Under the assumption that , we will compute Hence, the assertion (3) holds. (4) When , the assertion (4) follows from the fact that (5) Under the condition that , this derivative is computed as follows: which implies that is equivalent to . Moreover, in combination with , it follows that if and only if and .
Let be a regular null Cartan curve. We define a two-parameter family of functions by . We call the lightcone height functions of null Cartan curve . We denote that for any fixed vector in . We have the following proposition.
Proposition 3.2. Suppose is a regular null Cartan curve and . Then (1) if and only if there exist real numbers such that and .(2) if and only if .(3) if and only if and . (4) if and only if and .
Proof. Let in , where are real numbers. (1) If then . Moreover, in combination with in , which means . It follows that if and only if and .(2) When , the second derivative then if and only if .(3) When , the assertion (3) follows from the fact that (4) Under the condition that , this derivative is computed as follows: The assertion (4) follows.
4. Geometric Meanings of Invariant of a Null Cartan Curve
The purpose of this section is to study the geometric properties of the focal surface of a null Cartan curve in . Through these properties, one finds that the functions have special meanings. These properties will be stated below.
Proposition 4.1. Let be a regular null Cartan curve with . Then (1)the singularities of are the set , (2)if is a constant vector, then is in for any in and .
Proof. (1) A straightforward computation shows that The two equalities above imply that and are linearly dependent if and only if . This completes the proof of assertion (1).
(2) For a smooth function , define
5. Unfoldings of Functions of One Variable
In this section, we use some general results on the singularity theory for families of function germs .
Let be a function germ. We call an r-parameter unfolding of , where . We say that has -singularity at if for all and . We also say that has -singularity at if for all . Let be an unfolding of and has -singularity () at . We denote the -jet of the partial derivative at by , for . Then is called a versal unfolding if the matrix of coefficients has rank . Under the same as the above, is called a versal unfolding if the matrix of coefficients has rank , where .
We now introduce several important sets concerning the unfolding. The singular set of is the set The bifurcation set of is the critical value set of the restriction to of the canonical projection and is given by The discriminant set of is the set Then we have the following well-known result .
Theorem 5.1. Let be an -parameter unfolding of which has the singularity at . (1) Suppose that is a versal unfolding. (a)If , then is the fold point of and is locally diffeomorphic to .(b)If , then is locally diffeomorphic to .(2)Suppose that is a versal unfolding. (a)If , then is locally diffeomorphic to .(b)If , then is locally diffeomorphic to .(c)If , then is locally diffeomorphic to .
Here, is swallowtail and is the ordinary cusp. We also say that a point is a fold point of a map germ if there exist diffeomorphism germs and such that .
In the following propositions, the range of the index is used unless otherwise stated.
For the volumelike distance function and the lightcone height function , we can consider the following propositions.
Proposition 5.2. Let be the volumelike distance function on a null Cartan curve with . If has -singularity at , then is a versal unfolding of .
Proof. Let in , and in .
Under this notation the same as above proposition, we obtain Let be the 2-jet of at and so We denote that Thus, which implies that the rank of is 3, which finishes the proof.
Proposition 5.3. Let be the lightcone height function on a null Cartan curve . If has -singularity at , then is a versal unfolding of .
Proof. Consider a null Cartan curve , and let in , we have
Let be the 2-jet of at , then we can show that
We denote that
We require rank , which is verified from the fact that This completes the proof.
Proof of Theorem 2.1. Let be a null Cartan curve with . For , we define a function by . Then we have . Since and 0 is a regular value of , has the -singularity at if and only if and have -point contact for .
On the other hand, we can obtain from Proposition 3.1 that the discriminant set of is the assertion (A) of Theorem 2.1 follows from Propositions 3.1 and 5.2 and Theorem 5.1.
The bifurcation set of is The assertion (B) of Theorem 2.1 follows from Propositions 3.2 and 5.3 and Theorem 5.1.
6. Generic Properties of Null Cartan Curves
In this section, we consider generic properties of null Cartan curves in . The main tool is transversality theorem. Let be the space of null embedding equipped with Whitney -topology. We also consider the functions defined by . We claim that is a submersion for any in , where . For any in , we have . We also have the -jet extension defined by . We consider the trivialization . For any submanifold , we denote that . It is evident that both is a submersions and is an immersed submanifold of . Then is transversal to . We have the following proposition as a corollary of Lemma 6 in Wassermann .
Proposition 6.1. Let be submanifolds of . Then the set is residual subset of . If is closed subset, then is open.
Let be a function germ which has an -singularity at . It is well known that there exists a diffeomorphism germ such that . This is the classification of -singularities. For any in , we have the orbit given by the action of the Lie group of -jet diffeomorphism germs. If has an -singularity, then the codimension of the orbit is . There is another characterization of versal unfoldings as follows .
Proposition 6.2. Let be an -parameter unfolding of which has an -singularity at . Then is a versal unfolding if and only if is transversal to the orbit for .
Here, is the -jet extension of given by .
The generic classification theorem is given as follows.
Theorem 6.3. There exists an open and dense subset such that for any , then the focal surface of is locally diffeomorphic to the cuspidal edge or the swallowtail at any singular point.
Proof. For , we consider the decomposition of the jet space into orbits. We now define a semialgebraic set by Then the codimension of is 4. Therefore, the codimension of is 5. We have the orbit decomposition of into where is the orbit through an -singularity. Thus, the codimension of is . We consider the -jet extension of the volumelike distant function . By Proposition 6.1, there exists an open and dense subset such that is transversal to and the orbit decomposition of . This means that and is a versal unfolding of at any point . By Theorem 5.1, the discriminant set of (i.e., the focal surface of ) is locally diffeomorphic to cuspidal edge or swallowtail if the point is singular.
In this section, we give an example to illustrate the idea of Theorem 2.1.
Let be a null Cartan curve of defined by with respect to a distinguished parameter (Figure 3). The Cartan Frenet frame as follows: Thus, using the Cartan Frenet equations (2.5), we obtain We give, respectively, the vector parametric equations of the focal curve (Figure 4), the focal surface (Figure 5), and the singular locus of the focal surface (Figure 6) We can calculate the geometric invariant We see that gives two real roots and gives one real root , and two complex roots . Hence, we have is locally diffeomorphic to the cuspidal edge at any singularity , where . Moreover, is locally diffeomorphic to the swallowtail at , where or .
This work was supported by the Project of Science and Technology of Heilongjiang Provincial Education Department of China no. 12521151.
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