Abstract
In this paper, a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves located in nullcone in 4-dimensional semi-Euclidean space with index 2 are discussed. Using the unfolding theory in singularity theory, the singularities of the indefinite hypersurfaces and the indefinite surfaces are classified and the different kinds of singularities are estimated by means of a geometric invariant . Meanwhile, the definition of osculating nullcone is presented; the study shows that the differential geometric invariant of timelike curves measured also the order of the contact between a timelike curve and a osculating nullcone . Finally, some relevant counterexamples are indicated.
1. Introduction
Semi-Euclidean 4-space with index 2 has significant physical background, for example, anti–de Sitter spacetime , as a subspace of semi-Euclidean 4-space with index 2, is a solution of Einstein’s field equations for an empty universe with a negative cosmological constant. Very recently, some new results concerning the non-Euclidean geometry were established by physicists and geometers [1–11]; for example, S. Izumiya et al. investigated the regular curve in the embedded unit spheres in the de Sitter space or the light cone via the theory of Legendrian duality [6]. L. Kong and D. Pei [8] investigated the singularities of lightlike surfaces and focal surfaces of spacelike curves in hyperbolic spacetime sphere, and some geometry properties of the spacelike curves are obtained. The second author and his collaborators presented abundant research results concerning the singularities of submanifolds in semi-Euclidean space [12–15]. In [13], they investigated the null developables of timelike curves that lie on nullcone in 3-dimensional semi-Euclidean space with index 2 and classified the singularities of the null developables of timelike curves. It is shown that the null developable is locally diffeomorphic to the cuspidal edge or swallowtail which are two classes of topologically stable singularities. The study also shows that the differential geometric invariants of timelike curves measured the order of the contact between a timelike curve and a conic . As the generalization and the further exploration on previous study [13], increasing the dimension of semi-Euclidean space from three to four, in the current study we devote ourselves to explore the singularities of a class of indefinite hypersurfaces and a class of indefinite surfaces which are generated by timelike curves in nullcone in 4-dimensional semi-Euclidean space with index 2. In comparison with [13], the current study primarily discusses indefinite hypersurfaces and indefinite surfaces instead of null surfaces and the indefinite hypersurfaces and the indefinite surfaces show more kinds of singularities under certain equivalent conditions, for example, , , . Moreover, the new definition osculating nullcone is presented and the locus of center points of osculating nullcone is exactly the critical value set of the indefinite surface discussed in this paper. This will undoubtedly give the indefinite surface more geometric meanings.
Methods of the unfolding theory in singularity theory which are quite useful for the study of the geometry of hypersurfaces and surfaces are adopted in this paper. To allow a useful study of the singularities of the indefinite hypersurfaces and the indefinite surfaces located in 3-dimensional nullcone, we consider distance squared function denoted by on a nullcone timelike curve . These functions are the unfolding of these singularities in the local neighborhood of , and these functions only depend on the germs that they are unfolding. In this paper, we create these functions by varying a fixed point in the distance squared function to obtain a family of functions. We show that these singularities are -versally unfolded by the family of the distance squared functions. Noticing that the indefinite hypersurface is exactly the discriminant set of the order one of the four-parameter unfolding, the indefinite surface is the discriminant set of the order two of the four-parameter unfolding, while the locus of the center points of the osculating nullcone for timelike curve is the discriminant set of the order three of the four-parameter unfolding. If the singularity of is -type and if the corresponding four-parameter unfolding is -versal, then, applying Bruce’s theory to it (see [16, 17]), we know that the discriminant set (i.e., the indefinite hypersurface) of order one of the four-parameter unfolding is locally diffeomorphic to , , or under certain conditions, the discriminant set (i.e., the indefinite surface) of order two of the four-parameter unfolding is locally diffeomorphic to , , or under certain conditions, and the discriminant set (i.e., the curve) of order three of the four-parameter unfolding is locally diffeomorphic to a line or -cusp under certain conditions. Thus, we have completed the classification of the singularities of the indefinite hypersurface and the indefinite surface .
The paper is organized as follows. We begin in Section 2 by establishing the theoretical frame of the explicit differential geometry on curves in , defining a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves in nullcone in , calculating the singular sets of the indefinite hypersurfaces and the indefinite surfaces, and presenting the main results in this paper. In Section 3 we present a distance squared function of a timelike curve that lies on nullcone and establish the equivalent relations between -singularity of the distance squared function and geometric invariant which can characterize the types of singularities of the indefinite hypersurfaces and the indefinite surfaces via differential calculations. In Section 4, we give the proof of the main theorem and complete the classifications of singularities of the indefinite hypersurfaces and the indefinite surfaces by using some general results from the singularity theory for families of function germs. In this Section 5, using transversality theorem as the vehicle, we consider generic properties of timelike curves which lie on . To better illustrate our results, we give an example of the indefinite hypersurface and the indefinite surface and draw their graphs in 3-dimensional projection space.
2. Preliminaries
Let denote the 4-dimensional semi-Euclidean space with index 2, that is to say, the manifold with a flat metric of signature (-,-,+,+), for any vectors and in ,We also define the pseudovector product of , , and as follows:where is the canonical basis of We state that a vector is spacelike, null, or timelike if is positive, zero, or negative, respectively. The norm of a vector is defined as , and we call a unit vector if For a vector and a real number , we define the hyperplane with pseudonormal vector by We call a Lorentz hyperplane, a semi-Euclidean hyperplane with index 2, or a null hyperplane if is timelike, spacelike, or null, respectively.
Let be a smooth curve in (i.e., for any ), where is an open interval. For any , the curve is called spacelike curve, null curve, or timelike curve if all its velocities are , , or , respectively. We call the nonnull curve if is a timelike curve or a spacelike curve.
The arc-length of a nonnull curve , measured from , is . Then the parameter s is determined such that for the nonnull curve, where . Therefore, we say that a nonnull curve is parameterized by arc-length; throughout the remainder in this paper, we denote the parameter of as the arc-length parameter.
Let’s define a three-dimensional submanifold in semi-Euclidean 4-spacewhere ; we call a nullcone at the vertex .
There exist spacelike curves, timelike curves, and null curves in ; among them, the null curves had been investigated by the second author [15]; in this paper, we will restrict our attention to the case of timelike curve in ; the case of space curve would be discussed in future work. Let be a timelike regular curve; we consider a basis for all , such that . We choose the , and then there exists a null transversal vector such that . Moreover, defining , we have a Frenet type frame of along satisfyingandIt is easy to see that and have the same positive orientation. In order to study the timelike curve that lies on nullcone, we have the following fundamental formulae to use:and we call (6) the Frenet type equations, whereand and are called cone curvature function of the curve . The Frenet type frame is also called the asymptotic orthonormal frame on along the curve in .
Before stating the main theorem we define several geometric objects involved in this paper. Let be a unit speed timelike curve with ; firstly, we define an indefinite hypersurface in along in :such thatwith . Next, we also define an indefinite surface given bywith . Last but not least, let’s define a curve given byBy applying the formulae expressed in (7), we can calculateAt each of the regular points of ; that is, the formulaholds at ; the tangent hyperplane , , can be Lorentz, semi-Euclidean, or null. Similar calculation shows that the tangent plane of can also be Lorentz, semi-Euclidean, or null. This is the cause why we call an indefinite hypersurface and an indefinite surface.
It is obvious that if , then the four vectors are linearly dependent; this situation is trivial. We shall assume throughout the whole paper that all the maps and manifolds are and unless the contrary is explicitly stated. In this paper, one of our main tasks is to study the singularities of , ; we state the following proposition.
Proposition 1. Let be a regular timelike curve in the nullcone with . One can state the following:
(i) is a singularity of if and only if .
(ii) is a singularity of if and only if and .
Proof. . It has been calculated in (13) and (14) thatThe three vectors above are linearly dependent if and only if that is, if and only if ; thus, assertion is true.
We calculateThe two vectors above are linearly dependent if and only if , that is, if and only if and . Thus, assertion is proven.
We remark the relationships between , , and as follows.
Remark 2. By Proposition 1, the images of are the critical value sets of , and the images of are the critical value sets of .
Definition 3. Let be a submersion and be a timelike curve with . One says that and have -point contact for if the function satisfies , One also says that and have at least -point contact for if the function satisfies .
In order to characterize the singularities of the indefinite hypersurfaces, the indefinite surfaces, and the curves, we introduce a nullcone invariant defined to beThe main result in the paper is as follows.
Theorem 4. Let be a unit speed timelike curve that lies on nullcone with . For and the nullcone , one has the following:
(1) and have at least 2-point contact for .
(2) and have 3-point contact for if and only ifUnder the condition, the hypersurface at is locally diffeomorphic to , and at is locally diffeomorphic to .
(3) and have 4-point contact for if and only if andUnder the condition, the hypersurface at is locally diffeomorphic to , at is locally diffeomorphic to , and at is locally diffeomorphic to a line.
(4) and have 5-point contact for if and only if , , andUnder the condition, the hypersurface at is locally diffeomorphic to , at is locally diffeomorphic to , and at is locally diffeomorphic to -cusp.
3. Unfoldings of Functions of One-Variable and -Singularity
In this section, let us first recall the following definition (see [16] and references therein): Let be the germ of a function. One calls an r-parameter unfolding of , where . One says that has -singularity at if for all and if ; One also says that has -singularity at if for all .
Furthermore, we define a family of functions on a timelike curves which is useful to the study of geometric invariants of timelike curve. We also study the geometric properties of the indefinite hypersurfaces and the indefinite surfaces of a timelike curve in nullcone. We define the distance squared function of a timelike curve that lies on nullcone bywhere is independent of ; we denote that for any fixed vector in . Using (7) and making a simple calculation, we can state the following facts.
Proposition 5. Suppose that one has a unit speed timelike curve that lies on nullcone with ; then
(i) if and only if there exist some such that and(ii) if and only if there exist and such that(iii) if and only if and such that(iv) if and only if(v) if and only if and(vi) if and only if and
Proof. Because , from , there exist with such that Thus, assertion holds.
Under this condition , if , then we have such that and ; thus, assertion holds.
When , we compute thatimplies that ; hence, and , and, in combination with , it is easy to deduce that if and only if and such that ; that is, assertion holds.
From , we calculate thatIf , then and ; hence, Thus, assertion holds.
Suppose that holds; we notice thatHence, if and only if andThus, assertion holds.
Assume that holds; we calculate thatThus, assertion holds.
Corollary 6. Suppose that we have a unit speed timelike curve that lies on nullcone with ; then
(i) has -singularity at if and only if there exist , , and such that(ii) has -singularity at if and only if , , and such that(iii) has -singularity at if and only if and(iv) has -singularity at if and only if , , and
4. Singularities of Indefinite Hypersurfaces, Indefinite Surfaces, and Curves
In this section, we use some general results from the singularity theory for families of function germs ([16, 17]). Let be an unfolding of , where has 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 conditions as above, is called a -versal unfolding if the matrix of coefficients has rank , where .
We now introduce the following important sets concerning the unfolding:which is called a discriminant set of order . Of course, , and is the set of singular points of . In order to understand the geometric properties of the discriminant set of order , we introduce an equivalence relationship among the unfoldings of functions. Let and be the -parameter unfoldings of and , respectively. We state that and are --equivalent if there exists a diffeomorphism germ of the form such that . Straightforward calculations yield the following proposition.
Proposition 7. Let and be the -parameter unfoldings of and , respectively. If and are --equivalent by a diffeomorphism germ of the form , then are set germs.
Theorem 8. Let be an -parameter unfolding of with -singularity at . Suppose is a versal unfolding of . Then, is -equivalent to one of the following unfoldings:
(a) .
(b) .
(c) .
(d) .
We review the following result [16].
Theorem 9. Let be an -parameter unfolding of , which has singularity at . Suppose that is a versal unfolding.
(i) If , then is locally diffeomorphic to and .
(ii) If , then is locally diffeomorphic to , is locally diffeomorphic to , .
(iii) If , then is locally diffeomorphic to , is locally diffeomorphic to , is locally diffeomorphic to , and .
(iv) If , then is locally diffeomorphic to , is locally diffeomorphic to , is locally diffeomorphic to , is locally diffeomorphic to , and .
We remark that all of diffeomorphisms in the above assertions are diffeomorphism germs. We, respectively, call a -cusp (see Figure 1), a -cusp (see Figure 2), a -cusp (see Figure 3), a cuspidal edge (see Figure 4), a swallowtail (see Figure 5), a -cusp (see Figure 5), a butterfly, and a -butterfly (i.e., the critical value set of the butterfly) (see Figure 6). We have the following key proposition for .
Proposition 10. For a unit speed timelike curve with and is the distance squared function of with . If has -singularity at ,), then is a -versal unfolding of .
Proof. LetBy the definition of distance squared function, we obtain thatFor a fixed , let be the 3-jet of at () such thatWhen has -singularity at , we haveIt is easy to see that the rank of is 1 since .
When has -singularity at , we getBecause is a null vector and is a timelike vector for each , is independent of implying that the rank of is 2.
Assume that has -singularity at (; we want to verify the rank of the matrixis four. In fact, we can use an elementary transformation such that matrix becomes matrix , whereIt is obvious that the rank of is equal to the rank of . In Corollary 6, it has been shown that when has -singularity at , ; in combination with formulae (6), we calculate the determinant of Noticing that , the rank of matrix is . Thus, we can conclude that when has -singularity at , is a -versal unfolding of . This completes the proof.
Just as the definition of osculating sphere in Euclidean 3-space, we define the osculating nullcone as follows.
Definition 11. Let be a timelike curve in . Then the nullcone having at least 4-point contact with is called the osculating nullcone of . One denotes the osculating nullcone with vertex as .
As a corollary of Proposition 5, we give the following statements.
Corollary 12. Let be a regular timelike curve with in . The center points (or vertex points) of the osculating nullcone at a point areIn other words, the critical value set of is the locus of center points (or vertex point) of the osculating nullcone.
Remark 13. Based on Definition 11 and Corollary 12, we can rewrite the asserts (3) and (4) of Theorem 4 as follows:
and osculating nullcone have 4-point contact for if and only if andUnder the condition, the indefinite hypersurface at is locally diffeomorphic to , the indefinite surface at is locally diffeomorphic to , and the locus of the center points of the osculating nullcone at is locally diffeomorphic to a line.
and osculating nullcone have 5-point contact for if and only if , , andUnder the condition, the indefinite hypersurface at is locally diffeomorphic to , the indefinite surface at is locally diffeomorphic to , and the locus of the center points of the osculating nullcone at is locally diffeomorphic to -cusp.
Proof of Theorem 4. In Corollary 6, some equivalent conditions are presented when has -singularity (). Moreover, it is noticed that the indefinite hypersurface is exactly the discriminant set of the order one of the four-parameter unfolding, the indefinite surface is the discriminant set of the order two of the four-parameter unfolding, and the locus of the center points of the osculating nullcone for timelike curve is the discriminant set of the order three of the four-parameter unfolding. Proposition 10 shows that when has -singularity (), is a -versal unfolding of . Applying Theorem 9, it follows that under equivalent conditions corresponding to the assertions that has -singularity (), the indefinite hypersurface , the indefinite surface , and the locus of the center points of the osculating nullcone exhibit different characteristics of singularities; that is, the assertions in Theorem 4 are supported.
5. Generic Properties of Timelike Curves
In this section we consider generic properties of timelike curves which lie on The main tool is transversality theorem. Let be 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 We have the following proposition as a corollary of Lemma 6 in Wassermann [18].
Proposition 14. Let Q be submanifolds of Then the setis a residual subset of If Q is closed subset, then is open.
Let be a function germ which has an -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 f has an -singularity, then the codimension of the orbit is . There is another characterization of versal unfoldings as follows.
Proposition 15. Let be an r-parameter unfolding of f: which has an -singularity at 0. Then F is a versal unfolding if and only if is transversal to the orbit for Here, is the -jet extension of given by
We can prove the following generic classification theorem.
Theorem 16. There exists an open and dense subset such that, for any , the indefinite hypersurface is locally diffeomorphic to , , or , the indefinite surface is locally diffeomorphic to or at its singularities, and the locus of the center point of the osculating nullcone at is locally diffeomorphic to -cusp if the point is singular.
Proof. For , we consider the decomposition of the jet space into orbits. We now define a semialgebraic set byThen the codimension of is 5. Therefore, the codimension of is 6. We have the orbit decomposition of intowhere is the orbit through an -singularity. Thus, the codimension of is . We consider the -jet extension of the function . By Proposition 14, there exists an open and dense subset such that is transversal to () and the orbit decomposition of . This means that and is an -versal unfolding of at any point . By Theorem 9, the indefinite hypersurface (the discriminant set of order 1 of ) is locally diffeomorphic to , , or at its singularities, the indefinite surface (the discriminant set of order 2 of ) is locally diffeomorphic to or at its singularities, and the locus (the discriminant set of order 3 of ) of the center point of the osculating nullcone is locally diffeomorphic to -cusp at its singularities.
6. Example
In this section, to better illustrate our results, we give an example of a regular timelike curve on nullcone, and then we want to show further the indefinite hypersurfaces , the indefinite surface , and the curve involved in the timelike curve. We consider a unit speed timelike curve defined bywhere is the open interval . We calculateLet and , and we get andUsing the Frenet equations (7), we obtainTherefore, we havewhere Moreover, we obtain , where Thus, the surface is given bywhereThus, the hypersurface is given bywhereThe curve is given bywhereWe calculate the geometric invariantIt can be verified easily that there are no real roots when for ; that is, for any . Hence, according to Theorem 4, we have the following: the indefinite hypersurface at is locally diffeomorphic to for any , the indefinite surface at is locally diffeomorphic to for any , and at is locally diffeomorphic to a line for any .
We draw the projections of , , and to 3-space in Figures 7–14. It is shown clearly that is the intersection of and as well as the critical value set of . There is a good agreement between the results of Theorem 4 and these intuitive figures in terms of this example.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the China Postdoctoral Science Foundation (Grants nos. 2014M551168 and 2016T90244) and the Natural Science Foundation of Heilongjiang Province of China (Grant no. A201410).