On bounded and unbounded curves in Euclidean space Oleg Zubelevich Abstract. We provide sufficient conditions for curves in R 3 to be unbounded in terms of its curvature and torsion. We present as well sufficient conditions on the curvatures for boundedness, for curves in R 4. M.S.C. 2: 53A4. Key words: curves; intrinsic equation; curvature; torsion; Frenet-Serret formulas. Introduction This short note concerns a smooth curve γ in the standard three-dimensional Euclidean space R 3. It is well known that the curve is uniquely defined up to translations and rotations of R 3 ) by its curvature κs) and its torsion τs), the argument s is the arc-length parameter. The pair κs), τs)) is called the intrinsic equation of the curve. In the sequel we assume that κ, τ C[, + ). To obtain the radius-vector of the curve γ one must solve the system of Frenet- Serret equations: v s) = κs)ns),.) n s) = κs)vs) + τs)bs), b s) = τs)ns). The vectors vs), ns), bs) stand for the Frenet-Serret frame at the curve s point with parameter s. Then the radius-vector of the curve is given by rs) = vξ)dξ + r). If the curve γ is flat this happens iff τs) = ) then the system.) is explicitly integrated. In the three dimensional case it is difficult to integrate this system with arbitrary smooth functions τ, κ. Hence there follows the natural problem of restoring the properties of the curve γ having the curvature κs) and the torsion τs). A natural question might be under which conditions for κ and τ, is the curve γ closed - which is a difficult open problem. Another question might be: which are sufficient conditions for the whole curve to be contained in a sphere - which is much simpler. Such questions have been discussed in [4], [3], [5]. Differential Geometry - Dynamical Systems, Vol.9, 27, pp. 36-4. c Balkan Society of Geometers, Geometry Balkan Press 27.
On bounded and unbounded curves in Euclidean space 37 A sufficient condition for the curve to be unbounded has been obtained in []. In this article we determine such a condition in terms of curvature only, and the result is shown to be valid in a large class of spaces, including Hilbert spaces and Riemannian manifolds of non-positive curvature. In general,.) is a linear system of ninth order with the matrix depending on s. To describe the properties of γ, one must study this system. Also, in this note we formulate and prove several sufficient conditions for the unboundedness of the curve γ, and also present sufficient conditions for the curve to be bounded in the four dimensional Euclidean space. It must be observed that in R m with odd m, the curves are generally unbounded, but for even m, they are bounded. We justify below this very informal remark. 2 The main Theorem We shall say that γ is unbounded iff sup s rs) =. Theorem 2.. Suppose there exists a function λs) such that the functions ks) = λs)κs), and ts) = λs)τs) are monotone and belong to C[, ). Considering the function T s) = tξ)dξ, assume also that the following equalities hold 2.) lim T s) =, lim s ks) s T s) = lim ts) s T s) =. Then the curve γ is unbounded. The proof of this theorem is contained in Section 4.. By putting λ = /τ in this Theorem, we deduce the following Corollary 2.2. Assume that the function κs)/τs) is monotone and κs) 2.2) lim s s τs) =. Then the curve γ is unbounded. Note that the geodesic curvature of the tantrix 2 κ T s) is equal to τs)/κs) [3]. So that formula 2.2) can be rewritten as follows lim κ T s)s =. s E.g., one of these functions, ks) is monotonically increasing: s < s ks ) ks ), s, s [, ) while the other one ts) is monotonically decreasing: s < s ts ) ts ), s, s [, ). The inverse situation is allowed as well, or the both functions can be increasing or decreasing simultaneously. 2 The tangential spherical image of the curve γ is the curve on the unit sphere. This curve has the radius-vector r s).
38 Oleg Zubelevich The Theorem 2. does not reduce to Corollary 2.2. To justify this, we consider the following example. Let the curve γ be given by κs) =, τs) = + s. Since τs) as s, it may seem that this curve is a circle with κs) =. Still, by applying Theorem 2. with λ = we see that the curve γ is unbounded. We consider a system which consists of.) together with the equation r s) = vs). From the viewpoint of stability theory, Theorem 2. states that under certain conditions this system is unstable. Since rs) = Os) as s, this instability is too weak to be studied by standard methods, like the Lyapunov exponents one. 3 Additional remarks. Bounded curves in R 4 The technique developed above can be generalized to the curves in any multidimensional Euclidean space R m. For the case when m is odd, we can prove a result similar to Theorem 2.. However, when m is even, our method allows to obtain sufficient conditions for the curve to be bounded. In this section we illustrate such an effect. To avoid big formulas we consider only the case m = 4. So let a curve γ R 4 be given by its curvatures κ i s) C[, ), i {, 2, 3}, and let v j s), j =, 2, 3, 4 be the Frenet-Serret frame. Then the Frenet-Serret equations are v v κ s) d v 2 ds v 3 s) = As) v 2 v 3 s), As) = κ s) κ 2 s) κ 2 s) κ 3 s). v 4 v 4 κ 3 s) Theorem 3.. Suppose that the function κ s)κ 3 s) is nowhere vanishing, and that the functions f s) = κ s), f 2s) = κ 2s) κ s)κ 3 s) are monotone and Then the curve γ is bounded. sup f i s) <, i =, 2. s The proof of this theorem is contained in Section 4.2. 4 Proofs 4. Proof of Theorem 2. Let us expand the radius-vector by the Frenet-Serret frame, rs) = r s)vs) + r 2 s)ns) + r 3 s)bs).
On bounded and unbounded curves in Euclidean space 39 By differentiating this formula, we obtain vs) = r s)vs) + r 2s)ns) + r 3s)bs) + r s)v s) + r 2 s)n s) + r 3 s)b s). Using the Frenet-Serret equations, one yields 4.) r s) = κs) κs) τs) rs) +, r = r r 2. τs) r 3 Let us multiply from left the both sides of the system 4.) by the row-vector λs)τs),, κs)); we infer: ts)r s) + ks)r 3s) = ts). Further, we integrate the equation 4.2) ta)r a)da + ka)r 3a)da = T s). From the Second Mean Value Theorem [2], we know that there is a parameter ξ [, s] such that ta)r a)da = t) ξ r a)da + ts) By the same argument for some η [, s] we have ξ r a)da = t) r ξ) r ) ) + ts) r s) r ξ) ) ka)r 3a)da = k) r 3 η) r 3 ) ) + ks) r 3 s) r 3 η) ). Thus formula 4.2) takes the form 4.3) t) r ξ) r ) ) + ts) r s) r ξ) ) + k) r 3 η) r 3 ) ) + ks) r 3 s) r 3 η) ) = T s). Since the Frenet-Serret frame is orthonormal we have rs) 2 = r 2 s) + r 2 2s) + r 2 3s) = rs) 2. If we assume that the Theorem does not hold, and that the curve γ is bounded, i.e. sup s rs) <. Then due to conditions 2.) the left side of formula 4.3) is ot s)) as s. This is a contradiction and the Theorem is proved. 4.2 Proof of Theorem 3. Let rs) be a radius-vector of the curve γ. Then one can write 4 rs) = r i v i s), i= r s) = v s).
4 Oleg Zubelevich Similarly as in the previous section, due to the Frenet-Serret equations this gives r r s) = As)rs) +, r = r 2 r 3. r 4 First we multiply this equation by r T s)a s), det A = κ κ 3 ) 2 ): 4.4) r T s)a s)r s) = r T s)rs) + r T s)a s). Since A is a skew-symmetric matrix we have r T s)a s)r s) =, and some calculation yields r T s)a s) = r 2s)f s) + r 4s)f 2 s). Then 4.4) takes the form 2 By integrating this formula, we obtain 2 rs) 2) = r 2 s)f s) + r 4s)f 2 s). rs) 2 r) 2) = r 2a)f a) + r 4a)f 2 a)da. By the same argument which was employed to obtain formula 4.3), it follows that 2 rs) 2 r) 2) = 4.5) f ) r 2 ξ) r 2 ) ) + f s) r 2 s) r 2 ξ) ) + f 2 ) r 4 η) r 4 ) ) + f 2 s) r 4 s) r 4 η) ), where ξ, η [, s]. To proceed with the proof assume that the curve γ be unbounded: sup s rs) =. We consider a sequence s k such that It is easy to see that and rs k ) as k. rs k ) = max s [,k] rs), k N, s k [, k]. s k, rs) rs k ), s [, s k ]
On bounded and unbounded curves in Euclidean space 4 We substitute this sequence into 4.5), and obtain: 2 rs k ) 2 r) 2) = 4.6) f ) r 2 ξ k ) r 2 ) ) + f s k ) r 2 s k ) r 2 ξ k ) ) + f 2 ) r 4 η k ) r 4 ) ) + f 2 s k ) r 4 s k ) r 4 η k ) ), here ξ k, η k [, s k ] and thus r 2 ξ k ) rs k ), r 4 η k ) rs k ). Due to the conditions of the Theorem and the choice of the sequence s k, the righthand side of formula 4.6) is O rs k ) ) as k. But the left-hand one is of order rs k ) 2 /2. This gives a contradiction, and the Theorem is proved. Acknowledgements. The author thanks to Professor Ya. V. Tatarinov, for the useful discussions on the topic. References [] S. Alexander, R. Bishop, and R. Ghrist, Total curvature and simple pursuit on domains of curvature bounded above, Geometriae Dedicata, 472), 275-29. [2] R. Courant, Differential and Integral Calculus. vol., John Wiley and Sons, 988. [3] W. Frenchel, On the differential geometry of closed space curves, Bull. Amer. Math. Soc., 5795), 44-54. [4] P.W. Gifford, Some refinements in theory of specialized space curves, Amer. Math. Monthly, 6953), 384-393. [5] Yung-Chow Wong and Hon-Fei Lai, A Critical Examination of the Theory of Curves in Three Dimensional Differential Geometry. Tohoku Math. Journ. Vol. 9, No., 967. Author s address: Oleg Zubelevich Dept. of Theoretical Mechanics, Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University MGU), Russia, 9899, Moscow, Russian Federation. E-mail: ozubel@yandex.ru