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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Osculating curves in 4-dimensional semi-Euclidean space with index 2

Kazim İlarslan
/ Nihal Kiliç
/ Hatice Altin Erdem
Published Online: 2017-05-06 | DOI: https://doi.org/10.1515/math-2017-0050

Abstract

In this paper, we give the necessary and sufficient conditions for non-null curves with non-null normals in 4-dimensional Semi-Euclidian space with indeks 2 to be osculating curves. Also we give some examples of non-null osculating curves in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$.

MSC 2010: 53C40; 53C50

1 Introduction

In the Euclidian space 𝔼3, it is well known that to each unit speed curve α : I⊂→ 𝔼3, whose successive derivatives α′(s), α″(s) and α‴(s) are linearly independent vectors, one can associate the moving orthonormal Frenet frame {T, N, B}, consisting of the tangent, the principal normal and the binormal vector field, respectively. The planes spanned by {T, N}, {T, B} and {N, B} are respectively known as the osculating, rectifying and the normal plane. The rectifying curve in 𝔼3 is defined in [2] as a curve whose position vector (with respect to some chosen origin) always lies in its rectifying plane. It is shown in [1] that there exists a simple relationship between the rectifying curves and centrodes, which play some important roles in mechanics and kinematics. Some characterizations of rectifying curves in Minkowski space-time are given in [6].

It is well-known that the position vector of a curve in 𝔼3 always lies in its osculating plane B = Sp{T, N} if and only if its second curvature k2(s) is equal to zero for each s ([7]). The same property holds for timelike and spacelike curves (with non-null principal normal) in Minkowski 3-space. Osculating curves of first kind and second kind in Euclidian 4-space and Minkowski space time were studied by İlarslan and Nesovic in [4, 5].

In the light of the papers in [4, 5], in this paper we define the first kind and the second kind osculating curves in 4-dimensional semi-Euclidian space with index 2, by means of the orthogonal complements $\begin{array}{c}{B}_{2}^{\perp }\text{\hspace{0.17em}and\hspace{0.17em}}{B}_{1}^{\perp }\end{array}$ of binormal vector fields B2 and B1, respectively. We restrict our investigation of the first kind and the second kind osculating curves to timelike curves as well as to spacelike curves whose Frenet frame {T, N, B1, B2} contains only non-null vector fields. We characterize such osculating curves in terms of their curvature functions and find the necessary and the sufficient conditions for such curves to be the osculating curve.

2 Preliminaries

$\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ is the Euclidean 4-space 𝔼4 equipped with indefinite flat metric given by $g=−dx12−dx22+dx32+dx42$ where (x1, x2, x3, x4) is a rectangular coordinate system of $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$. Recall that an arbitrary vector v$\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$\ {0} can be spacelike, timelike or null(lightlike), if respectively holds g(v, v) > 0 or g(v, v) < 0 or g(v, v) = 0. In particular the vector v = 0 is a spacelike. The norm of a vector v is given by $\begin{array}{c}||v||=\sqrt{|g\left(v,v\right)|}\end{array}$ and two vectors v and w are said to be orthogonal if g(v, w) = 0. An arbitrary curve α(s) in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ , can locally be spacelike, timelike or null (lightlike), if all its velocity vectors α′(s) are respectively spacelike, timelike or null. A spacelike or timelike curve α(s) has unit speed, if g(α′(s), α′(s)) = ± 1. Recall that the pseudosphere, the pseudohyperbolic space and lightcone are hyperquadrics in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ , respectively defined by $\begin{array}{}{S}_{2}^{3}\left(m,r\right)=\left\{x\in {\mathbb{E}}_{2}^{4}:g\left(x-m,x-m\right)={r}^{2}\right\},{H}_{1}^{3}=\left\{x\in {\mathbb{E}}_{2}^{4}:g\left(x-m,x-m\right)=-{r}^{2}\right\},{C}^{3}\left(m\right)=\left\{x\in {\mathbb{E}}_{2}^{4}:g\left(x-m,x-m\right)=0\right\},\end{array}$ where r > 0 is the radius and m$\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ is the centre (or vertex) of hyperquadric ([8]).

Let {T, N, B1, B2} be the non-null moving Frenet frame along a unit speed non-null curve α in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ , consisting of the tangent, principal normal, first binormal and second binormal vector field, respectively. If α is a non-null curve with non-null vector fields, then {T, N, B1, B2} is an orthonormal frame. Accordingly, let us put $g(T,T)=ϵ0,g(N,N)=ϵ1,g(B1,B1)=ϵ2,g(B2,B2)=ϵ3,$(1) whereby ϵ0, ϵ1, ϵ2, ϵ3 ∊ {−1, 1}. Then the Frenet equations read, ([3]) $T′N′B1′B2′=0k100−ϵ0ϵ1k10k200−ϵ1ϵ2k20κ300−ϵ2ϵ3k30TNB1B2,$(2) where the following conditions are satisfied: $g(T,N)=g(T,B1)=g(T,B2)=g(N,B2)=g(B1,B2)=0,$(3) The curve α lies fully in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ if k3(s) ≠ 0 for each s.

Let α be a non-null curve with non-null normals in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$. We define that α is the first or the second kind osculating curve in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$, if its position vector with respect to some chosen origin always lies in the orthogonal complement $\begin{array}{}{B}_{2}^{\perp }\text{\hspace{0.17em}or\hspace{0.17em}}{B}_{1}^{\perp }\end{array}$ , respectively. The orthogonal complements $\begin{array}{}{B}_{1}^{\perp }\text{\hspace{0.17em}and\hspace{0.17em}}{B}_{2}^{\perp }\end{array}$ are non-degenerate hyperplanes of $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ , spanned by {T, N, B2} and {T, N, B1}, respectively.

Consequently, the position vector of the timelike and the spacelike first kind osculating curve (with non-null vector fields N and B1), satisfies the equation $α(s)=a(s)T(s)+b(s)N(s)+c(s)B1(s),$(4) and the position vector of the timelike and the spacelike second kind osculating curve (with non-null vector fields N and B1), satisfies the equation $α(s)=a(s)N(s)+b(s)B1(s)+c(s)B2(s),$(5) for some differentiable functions a(s), b(s) and c(s) in arclength function s.

3 Timelike and spacelike first kind osculating curves in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$

In this section we show that a non-null curve with non-null normals is the first kind osculating curve if and only if it lies fully in non-degenerate hyperplane of $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ . In relation to that we give the following theorem.

Theorem 3.1

Let α be a non-null curve in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ with non-null vector fields N and B1. Then α is congruent to the first kind osculating curve if and only if k3(s) = 0 for each s.

Proof

First assume that α is the first kind osculating curve in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ . Then its position vector satisfies relation (4). Differentiating relation (4) with respect to s and using Frenet equations (2), we easily find k3(s) = 0.

Conversely, assume that the third curvature k3(s) = 0 for each s. Let us decompose the position vector of α with respect to the orthonormal frame {T, N, B1, B2} by $α=ϵ0g(α,T)T+ϵ1g(α,N)N+ϵ2g(α,T)B1+ϵ3g(α,B2)B2,$(6) Since k3(s) = 0, relation (2) implies that B2 is a constant vector and g(α, B2) = constant. Substituting this in (6), we conclude that α is congruent to the first kind osculating curve. This completes the proof of the theorem. □

Corollary 3.2

Every non-null curve with non-null vector fields N and B1 lying fully in non-degenerate hyperplane in $\begin{array}{c}{\mathbb{E}}_{2}^{4}\end{array}$ is the first kind osculating curve.

4 Timelike and spacelike second kind osculating curves in ${\mathrm{E}}_{2}^{4}$

In this section, we characterize non-null second kind osculating curves in ${\mathrm{E}}_{2}^{4}$ with non-null vector fields N and B1 in terms of their curvatures. Let α = α(s) be the unit speed non-null second kind osculating curve in ${\mathrm{E}}_{2}^{4}$ with non-null vector fields N and B1 and non-zero curvatures k1(s), k2(s) and k3(s). By definition, the position vector of the curve α satisfies the equation (5), for some differentiable functions a(s), b(s) and c(s). Differentiating equation (5) with respect to s and using the Frenet equations (2), we obtain $T=(a′−ϵ0ϵ1bk1)T+(ak1+b′)N+(bk2−ϵ2ϵ3ck3)B1+c′B2.$

It follows that $a′−ϵ0ϵ1bk1=1,ak1+b′=0,bk2−ϵ2ϵ3ck3=0,c′=0,$(7)

and therefore $a(s)=−ϵ2ϵ3c0k1(k3k2),b(s)=ϵ2ϵ3c0(k3k2),c(s)=c0,$(8)

where c0 ∈ ℝ0. In this way functions a(s), b(s) and c(s) are expressed in terms of curvature functions k1(s), k2(s) and k3(s) of the curve α. Moreover, by using the first equation in (7) and relation (8), we easily find that the curvatures k1(s), k2(s) and k3(s) satisfy the equation $ϵ2ϵ31k1(k3k2)′′+(k3k2)k1=−1c0,c0∈R0.$

In this way, we obtain the following theorem.

Theorem 4.1

Let α(s) be a unit speed non-null curve with non-null vector fields N, B1 and B2 with curvatures k1(s), k2(s) and k3(s) ≠ 0 lying fully in ${\mathrm{E}}_{2}^{4}$. Then α is congruent to the second kind osculating curve if and only if there holds $ϵ2ϵ31k1(k3k2)′′+(k3k2)k1=−1c0$(9)

where ϵ2ϵ3 = ± 1, c0 ∈ 0.

Proof

First assume that α(s) is congruent to the second kind osculating curve in ${\mathrm{E}}_{2}^{4}$. By using (8) and the first equation in relation (7), we easily find that relation (9) holds.

Conversely, assume that equation (9) is satisfied. Let us consider the vector X${\mathrm{E}}_{2}^{4}$ given by $X(s)=α(s)+ϵ2ϵ3c0k1(k3k2)′T(s)−ϵ2ϵ3c0(k3k2)N(s)−c0B2(s)$

By using relations (2) and (9) we easily find X′(s) = 0, which means that X is a constant vector. Consequently, α is congruent to the second kind osculating curve. □

Recall that a unit speed non-null curve in ${\mathrm{E}}_{2}^{4}$ is called a W-curve, if it has constant curvature functions (see [9]). The following theorem gives the characterization of a non-null W-curves in ${\mathrm{E}}_{2}^{4}$ in terms of osculating curves.

Theorem 4.2

Every non-null W-curve, with non-null vector fields N, B1 and B2 with curvatures k1(s), k2(s) and k3(s) ≠ 0 lying fully in ${\mathrm{E}}_{2}^{4}$ is congruent to the second kind osculating curve.

Proof

It is clear from Theorem 4.1. □

Example 4.3

Let α(s) be a unit speed spacelike curve in ${\mathrm{E}}_{2}^{4}$ given by $α(s)=1152(sinh⁡(35s),9cosh⁡(5s),9sinh⁡(5s),cosh⁡(35s))$

We easily obtain the Frenet vectors and curvatures as follows: $T(s)=110(cosh⁡(35s),3sinh⁡(5s),3cosh⁡(5s),sinh⁡(35s)),N(s)=22(sinh⁡(35s),cosh⁡(5s),sinh⁡(5s),cosh⁡(35s)),B1(s)=1010(3cosh⁡(35s),14sinh⁡(5s),14cosh⁡(5s),3sinh⁡(35s)),B2(s)=22(sinh⁡(35s),−34cosh⁡(5s),−34sinh⁡(5s),cosh⁡(35s)),$

where T and B1 are spacelike vectors, N and B2 are timelike vectors, k1(s) = 3, k2(s) = 4 and k3(s) = 5. Since g(α, B2) = 0, α is congruent to second kind osculating curve. Also from Theorem 4.1. we find ${c}_{0}=-\frac{4}{15}.$ Thus we can write $α(s)=13N(s)−415B2(s)$

Example 4.4

Let α(s) be a unit speed timelike curve in ${\mathrm{E}}_{2}^{4}$ with the equation $α(s)=115(sinh⁡(25s),cosh⁡(5s),sinh⁡(5s),cosh⁡(25s))$

We easily obtain the Frenet vectors and curvatures as follows: $T(s)=13(2cosh⁡(25s),sinh⁡(5s),cosh⁡(5s),2sinh⁡(25s)),N(s)=115(4sinh⁡(25s),cosh⁡(5s),sinh⁡(5s),4cosh⁡(25s)),B1(s)=−13(cosh⁡(25s),2sinh⁡(5s),2cosh⁡(5s),sinh⁡(25s)),B2(s)=−115(sinh⁡(25s),4cosh⁡(5s),4sinh⁡(5s),cosh⁡(25s)),$

where N and B1 are spacelike vectors T and B2 are timelike vectors, k1(s) = 5, k2(s) = 2 and k3(s) = 2. It can be easily verified that g(α, B1) = 0, which means that α is congruent to the second kind osculating curve. Also from Theorem 4.1. we find ${c}_{0}=-\frac{1}{15}.$ Thus we can write $α(s)=−15N(s)+15B2(s)$

Remark

The curve given in Example 4.3 liesfully in the pseudohyperbolic space ${H}_{1}^{3}$ with the equation $-{x}_{1}^{2}-{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}=\frac{-8}{45}.$ The curve given in Example 4.4 liesfully in the light cone C3 with the equation $-{x}_{1}^{2}-{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}=0.$

Theorem 4.5

Let α(s) be a unit speed non-null curve with non-null vector fields N, B1 and B2 with curvatures k1(s), k2(s) and k3(s) ≠ 0 lying fully in ${\mathrm{E}}_{2}^{4}$. If α is the second kind osculating curve, then the following statements hold:

1. The tangential and the principal normal component of the position vector α are respectively given by $〈α(s),T(s)〉=ϵ0ϵ2ϵ3c0k1(k3k2)′,〈α(s),N(s)〉=ϵ1ϵ2ϵ3c0(k3k2),c0∈R0.$(10)

2. The second binornmal component of the position vector α is a non-zero constant, i.e. $〈α(s),B2(s)〉=c0ϵ3,c0∈R0$(11)

Conversely, if α(s) is a unit speed non-null curve with non-null vector fields N, B1 and B2, lying fully in ${\mathrm{E}}_{2}^{4}$ and one of the statements (i) or (ii) hold, then α is congruent to the second kind osculating curve.

Proof

First assume that α is congruent to the second kind osculating curve in ${\mathrm{E}}_{2}^{4}$. By using relation (4) and (8), the position vector of α can be written as $α(s)=−ϵ2ϵ3c0k1(k3k2)′T(s)+ϵ2ϵ3c0(k3k2)N(s)+c0B2(s)$(12)

Relation (12) easily implies that relations (10) and (11) are satisfied, which proves statements (i) and (ii).

Conversely, assume that the statement (i) holds. By taking the derivative of the equations 〈α(s), N(s)〉 = ${ϵ}_{1}{ϵ}_{2}{ϵ}_{3}{c}_{0}\left(\frac{{k}_{3}}{{k}_{2}}\right),$ with respect to s and using (2) we get 〈α, B1〉 = 0, which means that α is congruent to the second kind osculating curve.

If the statement (ii) holds, in a similar way we conclude that α is congruent to the second kind osculating curve. □

Theorem 4.6

Let α(s) be a non-null unit speed curve with non-null Frenet vectors and with curvatures k1(s), k2(s) and k3(s) ≠ 0 lying fully in ${\mathrm{E}}_{2}^{4}$. If α is congruent to the second kind osculating curve then the following statements hold:

i) If N and B1 have the opposite casual characters then $k3(s)k2(s)=−12c0(∫e−∫k1(s)dsds+c1)e∫k1(s)ds+12c0(∫e∫k1(s)dsds+c2)e−∫k1(s)ds$(13)

where c0 ∈ ℝ0 and c1, c2 ∈ ℝ ii) If N and B1 have the same casual characters then $k3(s)k2(s)=1c0(∫sin⁡ϕ(s)ds+c1)cos⁡ffi(s)−1c0(∫cos⁡ϕ(s)ds+c2)sin⁡ffi(s)$(14)

where $\varphi \left(s\right)=\int {k}_{1}\left(s\right)ds,\phantom{\rule{thinmathspace}{0ex}}{c}_{0}\in {\mathbb{R}}_{0}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1},{c}_{2}\in \mathbb{R}.$ Conversely, if α(s) is a unit speed non-null curve with non-null vector fields N, B1, lying fully in ${\mathrm{E}}_{2}^{4}$ and one of the statements (i) or (ii) holds, then α is congruent to the second kind osculating curve.

Proof

Let us first assume that α(s) is the unit speed non-null second kind osculating curve with non-null Frenet vector fields. From Theorem 4.1, the curvature functions of α satisfy the equation $ϵ2ϵ31k1(k3k2)′′+(k3k2)k1=−1c0$

Up to casual character of N and B1we have the following cases:

1. N and B1 have the opposite casual character. Putting ${ϵ}_{2}{ϵ}_{3}=-1,\phantom{\rule{thinmathspace}{0ex}}y\left(s\right)=\frac{{k}_{3}\left(s\right)}{{k}_{2}\left(s\right)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}p\left(s\right)=\frac{1}{{k}_{1}\left(s\right)}$ in equation (9), we have, $−dds(p(s)dyds)+y(s)p(s)=−1c0,c0∈R0$

Next, by changing the variables in the above equation by $t\left(s\right)=\int \frac{1}{p\left(s\right)}ds,$ we find $−d2ydt2+y=−p(t)c0,c0∈R0$

The solution of the previous differential equation is given by $y=−12c0(∫p(t)e−tdt+c1)et+12c0(∫p(t)etdt+c2)e−t$

where c0 ∈ ℝ0, c1, c2 ∈ ℝ: Finally, since $y(s)=k3(s)k2(s)andt(s)=∫1p(s)ds,$

we obtain $k3(s)k2(s)=−12c0(∫e−∫k1(s)dsds+c1)e∫k1(s)ds+12c0(∫e∫k1(s)dsds+c2)e−∫k1(s)ds.$

Conversely, if relation (13) holds, by taking the derivative of relation (13) two times with respect to s, we obtain that relation (9) is satisfied. Hence Theorem 4.1 implies that α is congruent to the first kind osculating curve.

In a similar way the statement (ii) can be proved.

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Accepted: 2016-01-29

Published Online: 2017-05-06

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 562–567, ISSN (Online) 2391-5455,

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