In this section, we characterize non-null second kind osculating curves in
${\mathrm{E}}_{2}^{4}$ with non-null vector fields *N* and *B*_{1} 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 *B*_{1} and non-zero curvatures *k*_{1}(*s*), *k*_{2}(*s*) and *k*_{3}(*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
$$T=({a}^{\prime}-{\u03f5}_{0}{\u03f5}_{1}b{k}_{1})T+(a{k}_{1}+{b}^{\prime})N+(b{k}_{2}-{\u03f5}_{2}{\u03f5}_{3}c{k}_{3}){B}_{1}+{c}^{\prime}{B}_{2}.$$

It follows that
$${a}^{\prime}-{\u03f5}_{0}{\u03f5}_{1}b{k}_{1}=1,\phantom{\rule{1em}{0ex}}a{k}_{1}+{b}^{\prime}=0,\phantom{\rule{1em}{0ex}}b{k}_{2}-{\u03f5}_{2}{\u03f5}_{3}c{k}_{3}=0,\phantom{\rule{1em}{0ex}}{c}^{\prime}=0,$$(7)

and therefore
$$a(s)=-{\displaystyle \frac{{\u03f5}_{2}{\u03f5}_{3}{c}_{0}}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}),\phantom{\rule{1em}{0ex}}b(s)={\u03f5}_{2}{\u03f5}_{3}{c}_{0}(\frac{{k}_{3}}{{k}_{2}}),\phantom{\rule{1em}{0ex}}c(s)={c}_{0},}$$(8)

where *c*_{0} ∈ ℝ_{0}. In this way functions *a*(*s*), *b*(*s*) and *c*(*s*) are expressed in terms of curvature functions *k*_{1}(*s*), *k*_{2}(*s*) and *k*_{3}(*s*) of the curve *α*. Moreover, by using the first equation in (7) and relation (8), we easily find that the curvatures *k*_{1}(*s*), *k*_{2}(*s*) and *k*_{3}(*s*) satisfy the equation
$${\u03f5}_{2}{\u03f5}_{3}{\left(\frac{1}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}{)}^{\prime}\right)}^{\prime}+(\frac{{k}_{3}}{{k}_{2}}){k}_{1}=-\frac{1}{{c}_{0}},{c}_{0}\in {\mathbb{R}}_{0}.$$

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*, *B*_{1} *and* *B*_{2} *with curvatures* *k*_{1}(*s*), *k*_{2}(*s*) *and* *k*_{3}(*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*
$$\u03f5}_{2}{\u03f5}_{3}{\left(\frac{1}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}{)}^{\prime}\right)}^{\prime}+(\frac{{k}_{3}}{{k}_{2}}){k}_{1}=-\frac{1}{{c}_{0}$$(9)

*where* ϵ_{2}ϵ_{3} = ± 1, *c*_{0} ∈ 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
$$X(s)=\alpha (s)+\frac{{\u03f5}_{2}{\u03f5}_{3}{c}_{0}}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}{)}^{\prime}T(s)-{\u03f5}_{2}{\u03f5}_{3}{c}_{0}(\frac{{k}_{3}}{{k}_{2}})N(s)-{c}_{0}{B}_{2}(s)$$

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*, *B*_{1} *and* *B*_{2} *with curvatures* *k*_{1}(*s*), *k*_{2}(*s*) *and* *k*_{3}(*s*) ≠ 0 *lying fully in*
${\mathrm{E}}_{2}^{4}$ *is congruent to the second kind osculating curve*.

#### Example 4.3

*Let* *α*(*s*) *be a unit speed spacelike curve in* ${\mathrm{E}}_{2}^{4}$ *given by*
$$\alpha (s)=\frac{1}{15\sqrt{2}}(\mathrm{sinh}(3\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}9\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}9\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(3\sqrt{5}s))$$

*We easily obtain the Frenet vectors and curvatures as follows:*
$$\begin{array}{}T(s)=\frac{1}{\sqrt{10}}(\mathrm{cosh}(3\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}3\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}3\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{sinh}(3\sqrt{5}s)),\\ N(s)=\frac{\sqrt{2}}{2}(\mathrm{sinh}(3\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(3\sqrt{5}s)),\\ {B}_{1}(s)=\frac{\sqrt{10}}{10}(3\mathrm{cosh}(3\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\frac{1}{4}\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\frac{1}{4}\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}3\mathrm{sinh}(3\sqrt{5}s)),\\ {B}_{2}(s)=\frac{\sqrt{2}}{2}(\mathrm{sinh}(3\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}-\frac{3}{4}\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}-\frac{3}{4}\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(3\sqrt{5}s)),\end{array}$$

*where T and* *B*_{1} *are spacelike vectors*, *N* *and* *B*_{2} *are timelike vectors*, *k*_{1}(*s*) = 3, *k*_{2}(*s*) = 4 *and* *k*_{3}(*s*) = 5. *Since g*(*α*, *B*_{2}) = 0, *α* *is congruent to second kind osculating curve. Also from Theorem 4.1. we find*
${c}_{0}=-{\displaystyle \frac{4}{15}.}$ *Thus we can write*
$$\alpha (s)=\frac{1}{3}N(s)-\frac{4}{15}{B}_{2}(s)$$

#### Example 4.4

*Let* *α*(*s*) *be a unit speed timelike curve in*
${\mathrm{E}}_{2}^{4}$ *with the equation*
$$\alpha (s)=\frac{1}{\sqrt{15}}(\mathrm{sinh}(2\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(2\sqrt{5}s))$$

*We easily obtain the Frenet vectors and curvatures as follows:*
$$\begin{array}{}T(s)=\frac{1}{\sqrt{3}}(2\mathrm{cosh}(2\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}2\mathrm{sinh}(2\sqrt{5}s)),\\ N(s)=\frac{1}{\sqrt{15}}(4\mathrm{sinh}(2\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}4\mathrm{cosh}(2\sqrt{5}s)),\\ {B}_{1}(s)=\frac{-1}{\sqrt{3}}(\mathrm{cosh}(2\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}2\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}2\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{sinh}(2\sqrt{5}s)),\\ {B}_{2}(s)=\frac{-1}{\sqrt{15}}(\mathrm{sinh}(2\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}4\mathrm{cosh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}4\mathrm{sinh}(\sqrt{5}s),\phantom{\rule{thinmathspace}{0ex}}\mathrm{cosh}(2\sqrt{5}s)),\end{array}$$

*where* *N* *and* *B*_{1} *are spacelike vectors* *T* *and* *B*_{2} *are timelike vectors*, *k*_{1}(*s*) = 5, *k*_{2}(*s*) = 2 *and* *k*_{3}(*s*) = 2. *It can be easily verified that g*(*α*, *B*_{1}) = 0, *which means that* *α* *is congruent to the second kind osculating curve. Also from Theorem 4.1. we find*
${c}_{0}=-{\displaystyle \frac{1}{15}.}$ *Thus we can write*
$$\alpha (s)=-\frac{1}{5}N(s)+\frac{1}{5}{B}_{2}(s)$$

#### Theorem 4.5

*Let* *α*(*s*) *be a unit speed non-null curve with non-null vector fields N*, *B*_{1} *and* *B*_{2} *with curvatures* *k*_{1}(*s*), *k*_{2}(*s*) *and* *k*_{3}(*s*) ≠ 0 *lying fully in* ${\mathrm{E}}_{2}^{4}$. *If* *α* *is the second kind osculating curve, then the following statements hold:*

*The tangential and the principal normal component of the position vector* *α* *are respectively given by*
$$\u3008\alpha (s),T(s)\u3009=\frac{{\u03f5}_{0}{\u03f5}_{2}{\u03f5}_{3}{c}_{0}}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}{)}^{\prime},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\u3008\alpha (s),N(s)\u3009={\u03f5}_{1}{\u03f5}_{2}{\u03f5}_{3}{c}_{0}(\frac{{k}_{3}}{{k}_{2}}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{0}\in {\mathbb{R}}_{0}.$$(10)

*The second binornmal component of the position vector* *α* *is a non-zero constant, i.e*.
$$\u3008\alpha (s),\phantom{\rule{thinmathspace}{0ex}}{B}_{2}(s)\u3009\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{c}_{0}{\u03f5}_{3},{c}_{0}\in {\mathbb{R}}_{0}$$(11)

*Conversely, if* *α*(*s*) *is a unit speed non-null curve with non-null vector fields N*, *B*_{1} *and* *B*_{2}, *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
$$\begin{array}{}{\displaystyle \alpha (s)=-\frac{{\u03f5}_{2}{\u03f5}_{3}{c}_{0}}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}{)}^{\prime}T(s)+{\u03f5}_{2}{\u03f5}_{3}{c}_{0}(\frac{{k}_{3}}{{k}_{2}})N(s)+{c}_{0}{B}_{2}(s)}\end{array}$$(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*)〉 =
${\u03f5}_{1}{\u03f5}_{2}{\u03f5}_{3}{c}_{0}(\frac{{k}_{3}}{{k}_{2}}),$ with respect to *s* and using (2) we get 〈*α*, *B*_{1}〉 = 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* *k*_{1}(*s*), *k*_{2}(*s*) *and* *k*_{3}(*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* *B*_{1} *have the opposite casual characters then*
$$\begin{array}{}{\displaystyle \frac{{k}_{3}(s)}{{k}_{2}(s)}=-\frac{1}{2{c}_{0}}(\int {e}^{-\int {k}_{1}(s)ds}ds+{c}_{1}){e}^{\int {k}_{1}(s)ds}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle +\frac{1}{2{c}_{0}}(\int {e}^{\int {k}_{1}(s)ds}ds+{c}_{2}){e}^{-\int {k}_{1}(s)ds}}\end{array}$$(13)

*where* *c*_{0} ∈ ℝ_{0} *and* *c*_{1}, *c*_{2} ∈ ℝ *ii) If* *N* *and* *B*_{1} *have the same casual characters then*
$$\begin{array}{}{\displaystyle \frac{{k}_{3}(s)}{{k}_{2}(s)}=\frac{1}{{c}_{0}}(\int \mathrm{sin}\varphi (s)ds+{c}_{1})\mathrm{cos}ffi(s)-\frac{1}{{c}_{0}}(\int \mathrm{cos}\varphi (s)ds+{c}_{2})\mathrm{sin}ffi(s)}\end{array}$$(14)

*where*
$\varphi (s)=\int {k}_{1}(s)ds,\phantom{\rule{thinmathspace}{0ex}}{c}_{0}\in {\mathbb{R}}_{0}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{1},{c}_{2}\in \mathbb{R}.$ *Conversely, if* *α*(*s*) *is a unit speed non-null curve with non-null vector fields N*, *B*_{1}, *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
$$\u03f5}_{2}{\u03f5}_{3}{\left(\frac{1}{{k}_{1}}(\frac{{k}_{3}}{{k}_{2}}{)}^{\prime}\right)}^{\prime}+(\frac{{k}_{3}}{{k}_{2}}){k}_{1}=-\frac{1}{{c}_{0}$$

Up to casual character of *N* and *B*_{1}we have the following cases:

*N* and *B*_{1} have the opposite casual character. Putting
${\u03f5}_{2}{\u03f5}_{3}=-1,\phantom{\rule{thinmathspace}{0ex}}y(s)=\frac{{k}_{3}(s)}{{k}_{2}(s)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}p(s)=\frac{1}{{k}_{1}(s)}$ in equation (9), we have,
$$-\frac{d}{ds}(p(s)\frac{dy}{ds})+\frac{y(s)}{p(s)}=-\frac{1}{{c}_{0}},\phantom{\rule{thinmathspace}{0ex}}{c}_{0}\phantom{\rule{thinmathspace}{0ex}}\in \phantom{\rule{thinmathspace}{0ex}}{\mathbb{R}}_{0}$$

Next, by changing the variables in the above equation by
$t(s)=\int \frac{1}{p(s)}ds,$ we find
$$-\frac{{d}^{2}y}{d{t}^{2}}+y\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}-\frac{p(t)}{{c}_{0}},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{0}\phantom{\rule{thinmathspace}{0ex}}\in \phantom{\rule{thinmathspace}{0ex}}{\mathbb{R}}_{0}$$

The solution of the previous differential equation is given by
$$y=-\frac{1}{2{c}_{0}}(\int p(t){e}^{-t}\phantom{\rule{thinmathspace}{0ex}}dt\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{c}_{1})\phantom{\rule{thinmathspace}{0ex}}{e}^{t}+\frac{1}{2{c}_{0}}(\int p(t){e}^{t}\phantom{\rule{thinmathspace}{0ex}}dt\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{c}_{2}){e}^{-t}$$

where *c*_{0} ∈ ℝ_{0}, *c*_{1}, *c*_{2} ∈ ℝ: Finally, since
$$y(s)=\frac{{k}_{3}(s)}{{k}_{2}(s)}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}t(s)=\int \frac{1}{p(s)}ds,$$

we obtain
$$\begin{array}{}{\displaystyle \frac{{k}_{3}(s)}{{k}_{2}(s)}=-\frac{1}{2{c}_{0}}(\int {e}^{-\int {k}_{1}(s)ds}ds+{c}_{1}){e}^{\int {k}_{1}(s)ds}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle +\frac{1}{2{c}_{0}}(\int {e}^{\int {k}_{1}(s)ds}ds+{c}_{2}){e}^{-\int {k}_{1}(s)ds}.}\end{array}$$

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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