ON BOUNDED AND UNBOUNDED CURVES DETERMINED BY THEIR CURVATURE AND TORSION OLEG ZUBELEVICH DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA, 9899, MOSCOW, MGU Abstract. We consider a curve in R 3 and provide sufficient conditions for the curve to be unbounded in terms of its curvature and torsion. We also present sufficient conditions on the curvatures for the curve to be bounded in R 4.. Introduction In this short note we concern 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 κ(s) and its torsion τ(s), the argument s is the arc-length parameter. The pair (κ(s), τ(s)) 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) = κ(s)n(s), n (s) = κ(s)v(s) + τ(s)b(s), (.) b (s) = τ(s)n(s). Mathematics Subject Classification. 53A4. Key words and phrases. curves, intrinsic equation, curvature, torsion, Frenet- Serret formulas. Partially supported by grants RFBR 8--68, Science Sch.-8784...
OLEG ZUBELEVICH The vectors v(s), n(s), b(s) stand for the Frenet-Serret frame at the curve s point with parameter s. Then the radius-vector of the curve is computed as follows r(s) = v(ξ)dξ + r(). If the curve γ is flat (γ R τ(s) = ) then the system (.) is integrated explicitly. In three dimensional case nobody can integrate this system with arbitrary sooth functions τ, κ. So we obtain a very natural and pretty problem: to restore the properties of the curve γ having the curvature κ(s) and the torsion τ(s). For example, under which conditions on the functions κ, τ the curve γ is closed? This is a hard open problem. There may by another question: Which are sufficient conditions for the whole curve to be contained in a sphere? This question is much simpler. Such a type questions have been discussed in [4], [3], [5]. There is a sufficient condition for the curve to be unbounded []. In this article the condition is formulated in terms of curvature only and this condition is valid in a big class of spaces, including Hilbert spaces and Riemannian manifolds of non-positive curvature. In general case, (.) is a linear system of ninth order with matrix depending on s. To describe the properties of γ one must study this system. In this note we formulate and prove some sufficient conditions for unboundedness of the curve γ. We also present sufficient conditions for the curve to be bounded in the four dimensional Euclidean space. It is interesting that in R m of odd m the curves are in generic case unbounded but for the even m they are generically bounded. Some justification of this very informal observation is given below.. Main Theorem We shall say that γ is unbounded if sup s r(s) =. Theorem.. Suppose there exists a function λ(s) such that functions k(s) = λ(s)κ(s), are monotone and belong to C[, ). Introduce a function T (s) = t(ξ)dξ. t(s) = λ(s)τ(s) i.e. one of these functions, for example k(s) is monotonically increased: s < s k(s ) k(s ), s, s [, ) while another one t(s) is monotonically decreased: s < s t(s ) t(s ), s, s [, ). The inverse situation is also allowed, or the both functions can be increased or decreased simultaneously.
ON BOUNDED AND UNBOUNDED CURVES 3 Suppose also that the following equalities hold lim T (s) =, lim s Then the curve γ is unbounded. k(s) s T (s) = lim t(s) s T (s) =. (.) The proof of this theorem is contained in Section 4.. Putting in this Theorem λ = /τ, we deduce the following corollary. Corollary. Suppose that a function κ(s)/τ(s) is monotone and lim s Then the curve γ is unbounded. κ(s) s τ(s) =. (.) Note that the geodesic curvature of the tantrix κ T (s) is equal to τ(s)/κ(s) [3]. So that formula (.) can be rewritten as follows lim κ T (s)s =. s Theorem. is not reduced to Corollary. Consider an example. Let the curve γ be given by κ(s) =, τ(s) = + s. Since τ(s) as s it may seem that this curve is about a circle with κ(s) =. Nevertheless applying Theorem. with λ = we see that the curve γ is unbounded. Consider a system which consists of (.) together with the equation r (s) = v(s). From viewpoint of stability theory, Theorem. states that under certain conditions this system is unstable. Since r(s) = O(s) as s, this instability is too mild to study it by standard methods such as the Lyapunov exponents method. 3. Supplementary Remarks: Bounded Curves in R 4 Actually the above developed technique can be generalized to the curves in any multidimensional Euclidean space R m. For the case of the odd m we can prove a theorem similar to Theorem.. But for the case 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 of big formulas we consider only the case m = 4. The tangential spherical image of the curve γ is the curve on the unit sphere. This curve has the radius-vector r (s).
4 OLEG ZUBELEVICH So let a curve γ R 4 be given by its curvatures κ i (s) C[, ), i =,, 3. And let v j (s), j =,, 3, 4 be the Frenet-Serret frame. Then the Frenet-Serret equations are d ds v v v 3 v 4 (s) = A(s) v v v 3 v 4 (s), κ (s) A(s) = κ (s) κ (s) κ (s) κ 3 (s) κ 3 (s) Theorem 3.. Suppose that the function κ (s)κ 3 (s) does not take the zero value. The functions are monotone and Then the curve γ is bounded. f (s) = κ (s), f (s) = κ (s) κ (s)κ 3 (s) sup f i (s) <, i =,. s The proof of this theorem is contained in Section 4.. 4. Proofs 4.. Proof of Theorem.. Let us expand the radius-vector by the Frenet-Serret frame r(s) = r (s)v(s) + r (s)n(s) + r 3 (s)b(s). Differentiating this formula we obtain v(s) = r (s)v(s) + r (s)n(s) + r 3(s)b(s) + r (s)v (s) + r (s)n (s) + r 3 (s)b (s). Using the Frenet-Serret equations, one yields r (s) = κ(s) κ(s) τ(s) r(s) +, r = r r. (4.) τ(s) Let us multiply both sides of system (4.) by the row-vector λ(s) ( τ(s),, κ(s) ) r 3
from the left: ON BOUNDED AND UNBOUNDED CURVES 5 Then we integrate this equality: t(s)r (s) + k(s)r 3(s) = t(s). t(a)r (a)da + k(a)r 3(a)da = T (s). (4.) From the Second Mean Value Theorem [], we know that there is a parameter ξ [, s] such that t(a)r (a)da = t() ξ r (a)da + t(s) ξ r (a)da = t() ( r (ξ) r () ) + t(s) ( r (s) r (ξ) ) By the same argument for some η [, s] we have k(a)r 3(a)da = k() ( r 3 (η) r 3 () ) + k(s) ( r 3 (s) r 3 (η) ). Thus formula (4.) takes the form t() ( r (ξ) r () ) + t(s) ( r (s) r (ξ) ) + k() ( r 3 (η) r 3 () ) + k(s) ( r 3 (s) r 3 (η) ) = T (s). (4.3) Since the Frenet-Serret frame is orthonormal we have r(s) = r (s) + r (s) + r 3(s) = r(s). Assume the converse: the curve γ is bounded i.e. sup s r(s) <. Then due to conditions (.) the left side of formula (4.3) is o(t (s)) as s. This contradiction proves the theorem. The Theorem is proved. 4.. Proof of Theorem 3.. Let r(s) be a radius-vector of the curve γ. Then one can write r(s) = 4 r i v i (s), i= r (s) = v (s). Similarly as in the previous section, due to the Frenet-Serret equations this gives r r (s) = A(s)r(s) +, r = r r 3. r 4
6 OLEG ZUBELEVICH First we multiply this equation by r T (s)a (s), (det A = (κ κ 3 ) ): r T (s)a (s)r (s) = r T (s)r(s) + r T (s)a (s). (4.4) 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 (s)f (s) + r 4(s)f (s). Then formula (4.4) takes the form ( ) r(s) = r (s)f (s) + r 4(s)f (s). Integrating this formula we obtain ) ( r(s) r() = r (a)f (a) + r 4(a)f (a)da. By the same argument which was employed to obtain formula (4.3), it follows that ) ( r(s) r() = f () ( r (ξ) r () ) + f (s) ( r (s) r (ξ) ) + f () ( r 4 (η) r 4 () ) + f (s) ( r 4 (s) r 4 (η) ), (4.5) here ξ, η [, s]. To proceed with the proof assume the converse. Let the curve γ be unbounded: sup s r(s) =. Take a sequence s k such that It is easy to see that r(s k ) = max r(s), k N, s k [, k]. s [,k] s k, r(s) r(s k ), s [, s k ] and r(s k ) as k. Substitute this sequence to formula (4.5): ) ( r(s k ) r() = f () ( r (ξ k ) r () ) + f (s k ) ( r (s k ) r (ξ k ) ) + f () ( r 4 (η k ) r 4 () ) + f (s k ) ( r 4 (s k ) r 4 (η k ) ), (4.6)
ON BOUNDED AND UNBOUNDED CURVES 7 here ξ k, η k [, s k ] and thus r (ξ k ) r(s k ), r 4 (η k ) r(s k ). Due to conditions of the Theorem and the choice of the sequence s k the right-hand side of formula (4.6) grows not faster than O( r(s k ) ) as k. But the left-hand one is of order r(s k ) /. This gives a contradiction. The Theorem is proved. References [] S. Alexander, R. Bishop, and R. Ghrist, Total curvature and simple pursuit on domains of curvature bounded above, Geometriae Dedicata, 47(), 75-9. [] 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., 57(95), 44-54. [4] P.W. Gifford, Some refinements in theory of specialized space curves, Amer. Math. Monthly, 6(953), 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.