Helical Hypersurfaces in Minkowski Geometry E 41

: We deﬁne helical (i.e., helicoidal) hypersurfaces depending on the axis of rotation in Minkowski four-space E 41 . There are three types of helicoidal hypersurfaces. We derive equations for the curvatures (i.e., Gaussian and mean) and give some examples of these hypersurfaces. Finally, we obtain a theorem classifying the helicoidal hypersurface with timelike axes satisfying ∆ I H = A H .


Preliminaries
In this section, we introduce the first and the second fundamental forms, matrix of the shape operator S, Gaussian curvature K, and the mean curvature H of hypersurface M = M(u, v, w) in Minkowski four-space E 4 1 . Throughout the paper, we shall identify a vector (a,b,c,d) with its transpose (a,b,c,d) t .
Let M = M(u, v, w) be an isometric immersion of a hypersurface from M 3 1 to E 4 1 = (R 4 , ds 2 ), where ds 2 = dx 2 1 + dx 2 2 + dx 2 3 − dx 2 4 is an element of length (Lorentz metric) and x i are the pseudo-Euclidean coordinates of type (3,1). The vector product of − → x = (x 1 , x 2 , x 3 , x 4 ), − → y = (y 1 , y 2 , y 3 , y 4 ), − → z = (z 1 , z 2 , z 3 , z 4 ) in E 4 1 is defined as follows · e, M = M uv · e, N = M vv · e, P = M uw · e, T = M vw · e, V = M ww · e, e is the Gauss map (i.e., the unit normal vector) I −1 I I gives the matrix of the shape operator S. Now, we have the formulas of the Gaussian curvature K = det(S) = det I I det I , and the mean curvature H = 1 3 tr (S), respectively, as follows A hypersurface M is minimal if H = 0 identically on M.
Let γ : I ⊂ R −→ Π be a curve in a plane Π and be a straight line in Π of E 4 1 . A rotational hypersurface in E 4 1 is defined as a hypersurface rotating a curve (profile) γ around a line (axis) . When the profile curve γ rotates around the axis , it simultaneously displaces parallel lines orthogonal to the axis , so that the speed of displacement is proportional to the speed of rotation. Resulting hypersurface is called the helicoidal hypersurface with axis and pitches a, b ∈ R\{0}.
Therefore, we introduce three type of the helicoidal hypersurfaces in E 4 1 throughout next three sections.

Helicoidal Hypersurfaces with Spacelike Axis
Supposing 1 is the line spanned by the spacelike vector (1, 0, 0, 0) t , the orthogonal matrix is given by where v, w ∈ R. The matrix A 1 can be found by solving the following equations, simultaneously, where ε = diag(1, 1, 1, −1). When the axis of rotation is 1 , there is an Minkowskian transformation by which the axis is 1 transformed to the x 1 -axis of E 4 1 . A parametrization of the profile curve is given by where ϕ (u) : I ⊂ R −→ R is a differentiable function for all u ∈ I. Thus, the helicoidal hypersurface which is spanned by the vector (1, 0, 0, 0) with pitches a, b ∈ R\{0}, is where u ∈ I, v, w ∈ R. If w = 0, we get helicoidal surface with spacelike axis as in the three dimensional Minkowski space E 3 1 . When a = b = 0, the surface is just a rotational hypersurface with timelike axis: Next, we obtain the curvatures of a helicoidal hypersurface with spacelike axis where u, a, b ∈ R \ {0} and 0 ≤ v, w ≤ 2π. See Figures 1 and 2 to projections of H with spacelike axis into three-space.
Computing the first differentials of (1), we get the first quantities as follows where ϕ = ϕ(u), ϕ = dϕ du . Thus, we have With the second differentials with respect to u, v, w, we obtain the second quantities as follows and det I I = u 4 cosh w (det I) 3/2 −u 4 ϕ 2 cosh 4 w + bu 3 ϕ cosh 3 w sinh w + a 2 u 2 sin 2 w ϕ −u cosh 2 w a 2 + b 2 cosh 2 w ϕ + b 3 cosh 3 w sinh w − 2a 2 b cosh w sinh .
Hence, the Gauss map of the helicoidal hypersurface is given by Finally, we calculate the Gaussian curvature and the mean curvature of the helicoidal hypersurface with spacelike axis and state the results in the following propostion: Proposition 1. For a helicodal hypersurface with spacelike axis in E 4 1 the Gaussian and mean curvatures, respectively, are as follows

Helicoidal Hypersurfaces with Timelike Axis
Taking 2 is the line spanned by the timelike vector (0, 0, 0, 1) t , the orthogonal matrix is given by where v, w ∈ R. The matrix A 2 can be found by where ε = diag(1, 1, 1, −1). When the axis of rotation is 2 , there is an Minkowskian transformation by which the axis is 2 transformed to the x 4 -axis of E 4 1 . Parametrization of the profile curve is given by where ϕ (u) is a differentiable function for all u ∈ I. Thus, the helicoidal hypersurface which is spanned by the vector (0, 0, 0, 1) with pitches a, b ∈ R\{0}, is as follows If w = 0, we get helicoidal surface with timelike axis as in the three dimensional Minkowski space E 3 1 . When a = b = 0, the surface is just a rotational hypersurface with timelike axis as follows R(u, v, w) = (u cos v cos w, u sin v cos w, u sin w, ϕ(u)) . Now, we obtain the mean curvature and the Gaussian curvature of a helicoidal hypersurface with timelike axis where u, a, b ∈ R \ {0} and 0 ≤ v, w ≤ 2π. See Figures 3 and 4 to projections of H with timelike axis into three-space.
Computing the first differentials of (2), we find the first quantities With the second differentials with respect to u, v, w, we have the second quantities Then, the Gauss map of the helicoidal hypersurface is given by Finally, we calculate the Gaussian curvature and the mean curvature of the helicoidal hypersurface with timelike axis and state the results in the following propostion.

Proposition 2.
For a helicodal hypersurface with timelike axis in E 4 1 the Gaussian and mean curvatures, respectively, are as follows Corollary 3. When ϕ = c = const., then we have

Corollary 4.
When ϕ = c = const. and b = 0, we have the same situation of Corollary 2, i.e. K and H vanish.

Helicoidal Hypersurfaces with Lightlike Axis
Considering 3 is the line spanned by the lightlike vector (0, 0, 1, 1) t , the orthogonal matrix is given by where v, w ∈ R. The matrix A 3 can be found by where ε = diag(1, 1, 1, −1). When the axis of rotation is 3 , there is an Minkowskian transformation by which the axis is 3 transformed to the x 3 x 4 -axis of E 4 1 . Parametrization of the profile curve is given by where ϕ (u) : I ⊂ R −→ R is a differentiable function for all u ∈ I. So, the helicoidal hypersurface which is spanned by the lightlike vector (0, 0, 1, 1) with pitches a, b ∈ R\{0}, is as follows: When w = 0, we get helicoidal surface with lightlike axis as in the three dimensional Minkowski space E 3 1 . When a = b = 0, the surface is just a rotational hypersurface with lightlike axis as follows Next, we obtain the curvatures of a helicoidal hypersurface with lightlike axis where u, a, b ∈ R \ {0} and v, w ∈ R. See Figures 5 and 6 to projections of H with lightlike axis into three-space.
Calculating the first differentials of (3), we obtain the first quantities With the second differentials with respect to u, v, w, we have the second quantities Hence, the Gauss map of the hypersurface is given by Finally, we calculate the Gaussian curvature and the mean curvature of the helicoidal hypersurface with lightlike axis, respectively, as follows and We assume that det I > 0. Therefore, the problem now is reduced to finding the solution of this differential equation in ϕ = ϕ(u), where the function K = K(u) is the known smooth function given.
In order to get an idea for these hypersurfaces, we study K = 0, K < 0, K > 0, K = const. and H = 0 for some special functional forms of the curvatures. Case 1. K(u) = 0. Equation (6) takes the form Suppose that Then Equation (7) reduces to The solution of this equation is given by If c 1 = 0, then h(u) = c 2 2(u+c 2 ) , and find Moreover, we define following one-parameter family of curves Therefore, the equation of these helicoidal hypersurfaces H(u, v, w) is given by where c = √ a 2 + b 2 .
If c 1 = 0 then h(u) = u+c 2 ± √ (u+c 2 ) 2 −2c 1 c 2 2c 1 , and we obtain Then, we define following two-parameter family of curves Hence, the equation of these helicoidal hypersurfaces is given by Finally, we observe that given the function K(u) = 0, we can determine a one or two-parameter family of curves given by (9) or (11), respectively, and define the corresponding Equations (10) or (12) of the helicoidal hypersurfaces with lightlike axis immersed in E 4 1 . Case 2(a). When c 1 < 0 and det I > 0, Equation (6) takes the form which is satisfied by the function h(u) = c 1 u + c 2 and therefore ϕ = (c 1 . So, given the function K = K(u) by (13) following the same process there exists a family of helicoidal hypersurfaces H(u, v, w) immersed in E 4 1 , the equation of which is Similarly, when c 1 > 0 and det I > 0, Equation (6) reduces to K(u) > 0. Case 2(b). Equation (6) takes the form which is satisfied by the function h(u) = c 1 u 2 + c 2 u + c 3 and therefore ϕ = c 1 u 2 + (c 2 + 1)u + c 3 , where c i ∈ R. So, given the function K = K(u) by (14) following the same process there exists a family of helicoidal hypersurfaces H(u, v, w) immersed in E 4 1 , the equation of which is

Case 2(c).
We consider K = d = const., d ∈ R\{0}. Then we get Using the substitution h = t, the equation reduces to We could not compute this equation using analytical methods. It is the future problem for us. Case 3. Now, we think ϕ = ϕ(u) such that h (u) = ϕ (u) − 1 = 0 for every u ∈ R\{0}. So, we can consider the inverse function u = u(h). Then, Equation (6) can be written as Taking h = t, it takes the form If we do not know some particular solution, we can not get its general solution. Case 4. The mean curvature of the helicoidal hypersurface given by (3) in the Minkowski space E 4 1 is given by (5) . The problem now is to find the solution of this equation in ϕ = ϕ(u), where the function H = H(u) is the known smooth function given. Since we may give the solution of the equation we can find the helicoidal minimal hypersurfaces. Taking h(u) = ϕ(u) − u, δ = h (u) = ϕ (u) − 1, h (u) = ϕ (u) then this equation takes the form where c 2 = a 2 + b 2 . So, using h (u) = t(u) it reduces to

Solution of above equation is
Therefore, we see that h = h(u) (resp. ϕ = ϕ(u)) satisfy the following equations: Hence, for every function ϕ = ϕ(u) which satisfies the last equation, there exists a helicoidal minimal hypersurface with lightlike axis in E 4 1 whose parametric representation is given by (3). We were not able to find the solution of Equation (5) by using analytical methods, so, it is for us, an open problem. Nevertheless, one could consider special values for the function H = H(u) as we did earlier for the function K = K(u), and then give solutions of the corresponding equations. For example, if This equation is satisfied by the function h(u) = e u and then ϕ(u) = e u + u. Here, when H = 0 then Given the function H = H(u) by (15), there exists a helicoidal hypersurface with lightlike axis immersed in E 4 1 the equation of which is given by Finally, we give the following theorem: Theorem 1. Let γ(u) = (0, 0, ϕ (u) , u), u ∈ I ⊂ R be a profile curve of the helicoidal hypersurface M immersed in E 4 1 given by (3). Then the Gaussian and the mean curvature at the point (0, 0, ϕ (u) , u) are functions of the same variable u, i.e., K = K(u), H = H(u). Moreover, given constants a, b ∈ I ⊂ R + , c 1 , c 2 ∈ R and a smooth function K = K(u) (resp. H = H(u)), u ∈ I we define the family of curves γ(u) ≡ γ(K(u), c; c 1 , c 2 ) (resp. γ(u) ≡ γ(H(u), c; c 1 , c 2 )).
From (16), we get −a 11 u cos v cos w − a 12 u cos w sin v − a 13 u sin w = 0, −a 21 u sin v cos w − a 22 u cos w cos v − a 23 u sin w = 0, −a 31 u cos v cos w − a 32 u sin v − a 33 u sin w = 0, a 41 u cos v cos w + a 42 u sin v cos w + a 43 u sin w = 0.
cosine and sine are linearly independent functions of v, then we see a ij = 0. Since Ω (u, w) = 3H W , we have H = 0. Consequently, H is a minimal hypersurface with timelike axis. Therefore, we have following theorem: Theorem 2. Let timelike H : M 3 1 −→ E 4 1 be an isometric immersion given by (2). Then ∆ I H = AH, where A is a 4 × 4 matrix iff the mean curvature of H vanishes.

Open Problems
An umbilical point is an significant geometric qualification, related to lines of curvature. Since a line of curvature will end at such points, it is a singularity of a line of curvature. It can partially be because there is an powerful criterion for a smooth (hyper)surface defined by a formula, for both parametric or implicit (hyper)surfaces: Lemma 1. A point is an umbilical point iff H 2 − K = 0 at this point.