Research Article | Open Access
Singularities of Null Hypersurfaces of Pseudonull Curves
The main goal of this paper is to study the singularities of null hypersurfaces of pseudonull curves. To do this we construct a null frame and a Lorentz distance-squared function of the pseudonull curve. The relations are shown between singularities of the null hypersurfaces and those, of the Lorentz distance-squared function. And we reveal the geometric meanings of the singularities of such hypersurfaces. In addition, we also introduce some properties of the nullcone Gaussian surface of the pseudonull curve.
A pseudonull curve is not a null curve, but its tangent curve is a null curve. Mathematicians have got many good results on the aspect of differential geometry of pseudonull curves [1–7]. Pseudonull helices, pseudonull Mannheim curves, pseudonull rectifying curves, and pseudonull osculating curves are considered, respectively, in , , , and . And the relations are gotten in  between pseudonormal and pseudorectifying curves in Minkowski space-time. Moreover, the involute-evolute of the pseudonull curve is studied in  and they prove that there is no involute of pseudonull curves in Minkowski 3-space.
However, there are few articles about pseudonull curves from the view point of singularities theory. In this paper, we focus on the singularities of null hypersurfaces of pseudonull curves. Here, we investigate the pseudonull curves with . We get the relationship between the singularities of null hypersurfaces of pseudonull curves and the functions and . When , the third normal vector equals . This situation is considered in . The authors, Petrović-Torgašev et al., make a contribution to the research of pseudonull curves. For example, they obtain the Frenet equations of pseudonull curves with only two curvatures and classify all such curves with constant curvatures. Besides curves in semi-Euclidean space, regarding singularity, have been studied extensively by Pei et al. and they make some achievements [8–11].
Here we discuss null hypersurfaces and nullcone Gaussian surfaces of pseudonull curves in . There are some differences between Minkowski space-time (i.e., ) and , for instance, the appearance of nullcone along a timelike pseudonull curve. Thus, we consider a one-parameter family of null Gauss indicatrices. We call it nullcone Gaussian surface. It can locally be a spacelike surface or a Lorentzian surface or a -lightlike surface depending on the characteristic of tangent plane of the surface. Moreover, we point that the nullcone Gaussian surface is also a regular surface. The null hypersurface is a surface bundle along a pseudonull curve whose fibres are the nullcone Gaussian surfaces. To allow a useful study of these singularities, we consider Lorentz distance-squared functions (which are denoted by , ). These functions are the unfolding processes of these singularities in the local neighborhood of , and these functions only depend on the germ that they are unfolding. In this paper, we create these functions by varying a fixed point in the Lorentz distance-squared function , to obtain a family of functions. We show that these singularities are versally unfolded by the family of Lorentz distance-squared functions. If the singularity of is -type and if the corresponding four-parameter unfolding is versal, then, by applying Bruces theory (cf. Bruce and Giblin ), we know that the discriminant set of the four-parameter unfolding processes is locally diffeomorphic to a cusp or a swallowtail or a butterfly under certain conditions. Thus, we have completed the classification of the singularities of a null hypersurface of a pseudonull curve.
The main result is Theorem 2 which is about the geometric meanings of the singularities of null hypersurfaces of pseudonull curves. Our work must be conducive to the research of pseudonull curves and be an exploration to null submanifolds in this space. And we will consider the situation that is not a null vector in our other paper . We assume throughout the paper that all manifolds and maps are unless explicitly stated otherwise.
In this section we introduce some basic notions for semi-Euclidean 4-space with index 2 and pseudonull curves. Let be a 4-dimensional vector space. For and in , the pseudoscalar product of and is defined as . We call (, ) a semi-Euclidean -space with index . We write instead of , . We say that a nonzero vector is spacelike, null, or timelike if , , or , respectively. The norm of the vector is defined as . The signature of a vector is defined as
We call a nullcone with vertex and denote .
For any , we define the vector aswhere is the canonical basis of and . We can easily show that so that is pseudoorthogonal to any .
Let be a smooth regular curve (i.e., ), where is an open interval. For any , the curve is called spacelike, null (lightlike), or timelike if the velocity of the curve is , , or , respectively.
Let be a unit speed timelike curve, parameterized by the arc length parameter ; that is, . For any , the curve is called a pseudonull curve if is a null vector (see ).
One can define the null frame of which is positively oriented -tuple of vectors satisfyingFor the sake of the simple forms and the convenience of our research objects, we take . The Frenet formula of with respect to the frame is as follows:When , which is the situation that the authors consider in . We focus on . And we construct the null frame of in a different way. In , they also investigate W-curves, that is, the pseudonull curves with constant curvatures, and obtain the classifications of W-curves. Here, we do not discuss W-curves. We focus on the fact that is not a constant. In particular ; otherwise , are all null vectors with the same direction. It means is a lightlike straight line.
Without loss of generality, we suppose that and are not null vectors. And we have under the condition that is not a constant. Thus, we take which are appropriate for conditions (4). Thus, we have
We define the family of null Gauss indicatrices of by , where . It is also called the nullcone Gaussian surface of . By a straightforward calculation, we have The above two vectors are definitely linearly independent, because . Therefore, the nullcone Gaussian surface is a regular surface. Obviously, is a spacelike vector. Let We have and the vector might be spacelike, timelike, or null. Therefore, the nullcone Gaussian surface might locally be a spacelike surface or a Lorentzian surface or a -lightlike surface depending on the characteristic of tangent plane of the surface. In , the authors introduce the notion of -lightlike surface. As a lightlike submanifold, its null space is not empty and its nullity degree is just . In other words, there will always be some vectors simultaneously pseudoorthogonal to the tangent space as well as the normal space. And we will give some results associated with 1-lightlike nullcone Gaussian surface in our other paper . On the other hand, when the nullcone Gaussian surface degenerates onto or , it can be constructed by the sum or minus of the spacelike vector and the timelike vector . We can use similar method in , which we do not consider here.
Remark 1. If we take , we can have the same discussion. For the sake of the continuity of the surface, we only consider one situation.
We define the null hypersurface along by . If we fix , the null hypersurface turns into a null ruled surface along .
We also define new invariants of the pseudonull curve in by
Let be a submersion and a timelike curve. We say that and have -point contact for if the function satisfies , . We also say that and have at least -point contact for if the function satisfies . For any fixed , we have a model surface . It is a light cone with vertex . We now consider the following conditions.(A 1)The number of points of where the model surface at having five-point contact with the curve is finite.(A 2)There is no point of where the model surface at has greater than or equal to six-point contact with the curve .
Here, we present the main results in this paper.
Theorem 2. Let be a timelike pseudonull curve. Let ; one has the following.
(1) and have at least 2-point contact at .
(2) and have 3-point contact at if and only if there exists such that where . Under this condition, the null hypersurface at is locally diffeomorphic to and the null focal set is nonsingular.
(3) and have 4-point contact at if and only if there exists such that where . Under this condition, the null hypersurface at is locally diffeomorphic to , the null focal set is locally diffeomorphic to , and the singular value set of is a regular curve.
(4) and have 5-point contact at if and only if there exists such that where . Under this condition, the null hypersurface at is locally diffeomorphic to , the null focal set is locally diffeomorphic to , and the singular value set of is locally diffeomorphic to the -cusp.
3. A Family of Lorentz Distance-Squared Functions of Pseudonull Curve γ
In this section we introduce one very useful family of functions on a pseudonull curve. For a pseudonull curve , we define the function: This function is called the Lorentz distance-squared function of the pseudonull curve . We use the notation for any fixed vector in . They describe the contact between and a nullcone. As we study this family of functions, it will become clear how singularities and the corresponding catastrophes arise.
Proposition 3. Let be a timelike pseudonull curve. Supposing that , then one has the following:(1) if and only if there exist and such that ;(2) if and only if there exists such that ;(3) if and only if there exists such that and , so one can write ;(4) if and only if there exists such that , , and ;(5) if and only if there exists such that and .
The above proposition also states that the discriminant set of the Lorentz distance-squared function is given by which is the image of the null hypersurface along . Therefore, a singular point of the null hypersurface is the point , where .
We define as we call it the null focal set of . By definition, the null focal set is the singular value set of the null hypersurface .
4. Classifications of Singularities
In this section we classify singularities of the null hypersurface along as an application of the unfolding theory of functions. Detailed descriptions could be found in . Let be a function germ. We call an r-parameter unfolding of f, if . We say has -singularity at , if for all and . Let be an -parameter unfolding of and has -singularity at . We denote the -jet of the partial derivative at byIf the rank of matrix is , then is called a versal unfolding of , where .
Inspired by the proposition in the previous section, we havewhich is called a discriminant set with order . Therefore, we have the following proposition.
Proposition 4. For a pseudonull curve , one has
Then we have the following classification theorem as a Corollary 6.6 in .
Theorem 5. Let be an r-parameter unfolding of f with -singularity at . Suppose is a versal unfolding of f; then one has the following assertions.(a)If , then is locally diffeomorphic to and .(b)If , then is locally diffeomorphic to and is locally diffeomorphic to and .(c)If , then is locally diffeomorphic to , is locally diffeomorphic to and is locally diffeomorphic to and .(d)If , then is locally diffeomorphic to , is locally diffeomorphic to , is locally diffeomorphic to , and is locally diffeomorphic to and .
For the proof of Theorem 2 we have the following fundamental proposition in this paper.
Proposition 6. If has -singularity at , then is a versal unfolding of .
Proof. By definition, we have where and .
Thus, we get For a fixed , the - of at isThe condition for versatility can be checked as follows.
(1) By Proposition 3, has -singularity at if and only if there exist and such that and . When has -singularity at , we require the matrix to have rank , which it always does since .
(2) It also follows from Proposition 3 that has -singularity at if and only if there exists such that and . We require matrixto have rank . Otherwise, if rank, it means that and are linearly dependent. This contradicts with the fact that is the pseudoorthogonal frame of .
(3) By Proposition 3, has -singularity at if and only if there exists such that , , and . We require matrixto have rank . Otherwise, if rank, it means that can be generated by and . Through a straightforward calculation, we can easily show that it is a contradiction.
(4) By Proposition 3, has -singularity at if and only if there exists such that , , and . We require matrixto have rank .
In fact, Therefore, rank.
In summary, is a versal unfolding of . This completes the proof.
We now give the proof of Theorem 2.
Proof of Theorem 2. Let be a pseudonull curve. As , we give a function , by ; then we assume that . Because and 0 is a regular value of , and have -point contact for if and only if has -singularity at . Thus and have at least -point contact for if and only if . By Proposition 4, we have By combining Proposition 3, Theorem 5, and Proposition 6, we get the results.
5. Generic Properties
In this section we consider the generic properties of pseudonull curves in . The main tool is transversality theorem. Assume that is the space of timelike embedding equipped with Whitney -topology. We also consider the function defined by . We claim that is a submersion for any , where . For any , we have . We also have the -jet extensiondefined by ; we consider the trivialization . For any submanifold , we denote that . It is obvious that is a submersion and is an immersed submanifold of . Then is transversal to . We have the following proposition as a corollary of Lemma 6 in .
Proposition 7. Let Q be a submanifold of . Then the setis a residual subset of . If Q is a closed subset, then is open.
Let be a function germ which has -singularity at 0. It is well known that there exists a diffeomorphism germ such that . This is the classification of -singularities. For any , we have the orbit given by the action of the Lie group of -jet diffeomorphism germs. If has -singularity, then the codimension of the orbit is . There is another characterization of versal unfolding as follows .
Proposition 8. Assume that is an r-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 9. There exists a dense and open subset such that, for any , the lightlike hypersurface of is locally diffeomorphic to the cuspidal edge, the swallowtail, or the butterfly at any singular point.
Proof. For , we discuss the decomposition of the jet space into orbits; we now define a semialgebraic set byThen the codimension of is . Therefore, the codimension of is . Then the orbit decomposition of is where is the orbit going through the -singularity. Thus, the codimension of is . We consider the -jet extension of the function . By Proposition 7, there exists a dense and open 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, the discriminant set of (i.e., the lightlike hypersurface of ) is locally diffeomorphic to cuspidal edge, swallowtail, or butterfly if the point is singular.
Conflict of Interests
The authors declare that there is no conflict of interests in this work.
The second author Donghe Pei was partially supported by NSF of China no. 11271063 and NCET of China no. 05-0319.
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