On curves with nonnegative torsion
OON CURVES WITH NONNEGATIVE TORSION
HUBERT L. BRAY AND JEFFREY L. JAUREGUIA
BSTRACT . We provide new results and new proofs of results about the torsion of curves in R . Let γ be a smooth curve in R that is the graph over a simple closed curve in R with positive curvature.We give a new proof that if γ has nonnegative (or nonpositive) torsion, then γ has zero torsionand hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve,without any assumption on its curvature, cannot be perturbed to a closed space curve of constantnonzero torsion. We also prove similar statements for curves in Lorentzian R , which are relatedto important open questions about time flat surfaces in spacetimes and mass in general relativity.
1. I
NTRODUCTION
The curvature and torsion of curves in R are defined by the Frenet formulas found in mostundergraduate differential geometry texts (see [4], for instance). A curve in R with positiveconstant curvature and nonzero constant torsion must be a helix, seen in figure 1. Positive torsionis what causes the helix to screw in some direction, which in the short run prevents a curve withpositive torsion from closing up on itself. However, as also shown in figure 1, in the long run acurve with positive torsion can circle around and then close up on itself. In fact, Weiner even foundexamples of constant positive torsion curves in R (with non-constant positive curvature) whichclose up on themselves [12] (cf. [1]).Nevertheless, there is a clear intuition about torsion to try to exploit here: positive torsion tendsto cause curves to screw in some direction, thereby preventing the curve from being closed. Hence,it is reasonable to conjecture that a closed curve in R cannot have positive torsion everywhere, aslong as an additional hypothesis is added which rules out counterexamples like the one in figure 1.The additional hypothesis we add, that γ is a graph over a simple closed curve in R with positivecurvature, is shown in figure 2. Our first theorem, which is a corollary of results in [10], [11], and[6], verifies this conjecture: Theorem 1.
Let γ be a smooth curve in R which is the graph over a simple closed curve in R with positive curvature. If γ has nonnegative (or nonpositive) torsion, then γ has zero torsion andhence lies in a plane. A similar result, which is also a natural adaptation of results in [10], [11], and [6], is true inLorentzian R , , which has the metric dx + dy − dt . In what follows, a tangent vector v isspacelike if the metric evaluates to be positive on v , and a submanifold is spacelike if its nonzerotangent vectors are spacelike. Theorem 2.
Let γ be a smooth simple closed curve in R , with spacelike curvature vector whichlies in a complete spacelike hypersurface. If γ has nonnegative (or nonpositive) torsion, then γ haszero torsion and hence lies in a plane. Note that a complete spacelike hypersurface in R , must be a graph over R (see [7], for in-stance), so that γ is again a graph over a simple closed curve in R , which turns out to have positive The first named author was supported in part by NSF grant a r X i v : . [ m a t h . DG ] J a n HUBERT L. BRAY AND JEFFREY L. JAUREGUI F IGURE Two curves in R and their projections to the xy plane. A curve in R withpositive constant curvature and nonzero constant torsion must be a helix, as shown on theleft. This example suggests the conjecture that positive torsion, which is what causes thehelix to screw in some direction, should prevent the curve from closing up on itself, at leastin the short run. However, the curve on the right shows how a closed curve in R can havepositive torsion everywhere. Hence, the conjecture will need to exclude this example. curvature, by the spacelike curvature vector condition (cf. Lemma 2). Whereas the hypotheses ofTheorem 1 are somewhat ad hoc , the hypotheses of Theorem 2 are quite natural. We provide newproofs of these theorems in sections 2 and 3. These new proofs are very efficient and provide anew way of understanding these results. We also show in section 4 that these results continue tohold if the projection of γ is allowed to wind around the plane curve multiple times.In section 5, we prove a new rigidity result for constant torsion curves (separate from Theorems1 and 2). The precise statement is Theorem 5, but we state an informal version here: Theorem 3.
A simple closed plane curve, not necessarily convex, cannot be perturbed in the C sense to a space curve of constant nonzero torsion. Our interest in this work arose from the study of time flat surfaces [3], which are spacelikecodimension-2 submanifolds of a spacetime satisfying a special geometric condition. For one-dimensional submanifolds, the condition is simply that of constant torsion.For the reader’s convenience, we recall the Frenet formulas for a curve γ ( s ) in R parametrizedby arc length, with curvature κ := | γ (cid:48)(cid:48) ( s ) | nonzero. The tangent T , normal N , and binormal B satisfy: T (cid:48) = κNN (cid:48) = − κT + τ BB (cid:48) = − τ N, which serves as a definition of the torsion, τ . Note that conventions for the sign of τ vary; ours isthe opposite of do Carmo [4].While we provide new proofs of Theorems 1 and 2 in this paper, these two theorems are alsoimplied by previously known four-vertex theorems. The curves γ above are convex in the sensethat each point has a tangent plane of which γ lies entirely on one side. The four-vertex theoremof Sedykh [10] for closed convex curves γ in R asserts that the torsion of γ has at least fourzeroes. A refinement of this result due to Thorbergsson and Umehara [11] shows that the torsionmust change sign at least four times (if γ is not a plane curve), which implies Theorem 1 (see also[6]). Theorem 2 follows as well because the sign of the torsion is determined by the sign of theexpression, ( γ (cid:48) × γ (cid:48)(cid:48) ) · γ (cid:48)(cid:48)(cid:48) , which is independent of whether × and · are taken with respect to the R or R , metric. N CURVES WITH NONNEGATIVE TORSION 3
Acknowledgements.
The authors would like to thank Daniel Stern and Mark Stern for helpfulconversations and are very grateful to Mohammad Ghomi for pointing out the above referencesand providing insightful feedback.2. C
URVES IN R WITH NONNEGATIVE TORSION
As depicted in figure 2, let γ be a smooth curve in R which is the graph, with height function h , over a simple closed curve γ in R with curvature κ > . We begin our discussion bycharacterizing the geometry of γ .F IGURE We restrict to curves γ in R that are graphs over simple closed curves γ in R with positive curvature. Since γ has curvature κ > , we may parametrize it by thedirection θ of its unit tangent vector T = (cos θ, sin θ ) in R . Since γ is a graph over γ with height function h , we parametrize γ and h by θ as well. Let the unit tangent vector of the base curve γ in R be T = γ (cid:48) ( s ) = (cos θ ( s ) , sin θ ( s )) , where s is the arc length of γ . A standard exercise is that the (signed) curvature κ of γ is givenby κ = dθds (upon reversing the orientation of γ if necessary). Since κ > and γ is simple byhypothesis, θ is strictly increasing on γ and may be taken to go from to π when going around γ once. Whereas it is more common to parametrize curves by their arc length parameter s , it isessential to our argument to parametrize γ (and hence γ and h later on) as periodic functions of θ with period π .Note dγ dθ = 1 κ dγ ds = 1 κ (cos θ, sin θ ) , and, since γ is closed, (cid:82) π dγ dθ dθ . Hence, (cid:90) π κ ( θ ) cos θ dθ (1) (cid:90) π κ ( θ ) sin θ dθ. (2)Thus, the geometry of γ is captured by κ ( θ ) > , periodic in θ with period π , subject toequations (1) and (2) above. HUBERT L. BRAY AND JEFFREY L. JAUREGUI
Proof of Theorem 1.
The next step to proving Theorem 1 is to compute the formula for thetorsion of the curve γ in R in terms of the height function h ( θ ) and the curvature κ ( θ ) of the basecurve γ . For the remainder of the proof, all derivatives are with respect to θ : γ = ( γ ; h ) γ (cid:48) = ( 1 κ cos θ, κ sin θ, h (cid:48) ) κ γ (cid:48) = (cos θ, sin θ, κ h (cid:48) )( κ γ (cid:48) ) (cid:48) = ( − sin θ, cos θ, ( κ h (cid:48) ) (cid:48) )( κ γ (cid:48) ) (cid:48)(cid:48) = ( − cos θ, − sin θ, ( κ h (cid:48) ) (cid:48)(cid:48) ) so that(3) κ γ (cid:48) × ( κ γ (cid:48) ) (cid:48) = (( κ h (cid:48) ) (cid:48) sin θ − κ h (cid:48) cos θ, − ( κ h (cid:48) ) (cid:48) cos θ − κ h (cid:48) sin θ, Plugging this into the well known [4] formula (4) for the torsion τ of a curve γ in R and usingstandard properties of the cross product gives τ = ( γ (cid:48) × γ (cid:48)(cid:48) ) · γ (cid:48)(cid:48)(cid:48) | γ (cid:48) × γ (cid:48)(cid:48) | (4) = κ ( κ γ (cid:48) × ( κ γ (cid:48) ) (cid:48) ) · ( κ γ (cid:48) ) (cid:48)(cid:48) | κ γ (cid:48) × ( κ γ (cid:48) ) (cid:48) | = κ ( κ h (cid:48) ) (cid:48)(cid:48) + κ h (cid:48) (( κ h (cid:48) ) (cid:48) ) + ( κ h (cid:48) ) + 1 . (5)For the purposes of intuition, consider the simplest case when the base curve γ is the unit circlein R with constant curvature κ ≡ . Then τ = h (cid:48)(cid:48)(cid:48) + h (cid:48) ( h (cid:48)(cid:48) ) + ( h (cid:48) ) + 1 so that (cid:90) π ( h (cid:48)(cid:48)(cid:48) ( θ ) + h (cid:48) ( θ )) dθ = (cid:90) π τ (cid:2) ( h (cid:48)(cid:48) ) + ( h (cid:48) ) + 1 (cid:3) dθ from which it is clear that if τ ≥ (or τ ≤ ), then τ ≡ . Interestingly, this argument generalizesfor all simple, closed base curves γ that have κ > . Lemma 1.
Given a smooth function κ ( θ ) > with period π satisfying equations (1) and (2),there exists f ( θ ) > with period π such that (6) f (cid:48)(cid:48) ( θ ) + f ( θ ) = 1 κ ( θ ) . We will prove this lemma momentarily. Note that in the example above when the base curve γ is the unit circle in R , f ≡ satisfies (6). This lemma is the key step in the proof of Theorem 1 N CURVES WITH NONNEGATIVE TORSION 5 since we can then write (cid:90) π (cid:20) f (cid:48)(cid:48) ( θ ) + f ( θ ) − κ (cid:21) κ h (cid:48) ( θ ) dθ = (cid:90) π [( κ h (cid:48) ) (cid:48)(cid:48) + κ h (cid:48) ] f dθ = (cid:90) π τ (cid:2) (( κ h (cid:48) ) (cid:48) ) + ( κ h (cid:48) ) + 1 (cid:3) fκ dθ, (7)having integrated by parts and using (5). From this identity, it is again clear that if τ ≥ (or τ ≤ ), then τ ≡ . As it is a standard result that a curve with nonzero curvature and zero torsionlies in a plane [4], this proves Theorem 1.2.2. Proof of Lemma 1.
First note that given κ , a solution f to equation (6) does not exist unless κ satisfies the constraints in equations (1) and (2), as can be seen by integrating by parts twice.However, given equations (1) and (2), a positive solution f to equation (6) always exists.To prove existence, we write down the formula for f which comes from identifying the relevantkernel and verify that this f satisfies equation (6). For convenience, let p = 1 /κ , and recall that κ , p , and f are periodic functions with period π . Let(8) f ( θ ) = (cid:90) π − π k ( β ) p ( β + θ + π ) dβ where k ( β ) = β sin β π . Then taking two derivatives in θ and integrating by parts twice gives us f (cid:48)(cid:48) ( θ ) = p ( θ ) + (cid:90) π − π (cid:18) cos βπ − β sin β π (cid:19) p ( β + θ + π ) dβ so that f (cid:48)(cid:48) ( θ ) + f ( θ ) = p ( θ ) + 1 π (cid:90) π − π p ( β + θ + π ) cos β dβ = p ( θ ) + 1 π (cid:90) θ +2 πθ p ( β ) cos( β − θ − π ) dβ = p ( θ ) − π (cid:90) θ +2 πθ p ( β ) (cos β cos θ + sin β sin θ ) dβ = p ( θ ) by equations (1) and (2). Since p = 1 /κ > and k ( β ) > for β ∈ ( − π, π ) \{ } , it follows fromequation (8) that f > , proving the lemma.2.3. Counterexamples to stronger statements.
Theorem 1 is not true without the assumption of κ > for the base curve γ . For instance, the closed space curve on the right in figure 1 haspositive torsion and is a graph over a simple closed plane curve whose curvature changes sign. Todemonstrate the sharpness of Theorem 1, figure 3 shows that a simple closed curve with even the HUBERT L. BRAY AND JEFFREY L. JAUREGUI F IGURE The rounded triangle on the left, a hypotrochoid, has only a very slight amountof negative curvature, yet the graph over it on the right has positive torsion. This exampledemonstrates the sharpness of Theorem 1. slightest amount of negative curvature can admit a graph with positive torsion. For t ∈ [0 , π ) , consider the base curve(9) γ ( t ) = (cid:18) ( a + b ) cos( t ) − c cos (cid:18) ( a + b ) tb (cid:19) , ( a + b ) sin( t ) − c sin (cid:18) ( a + b ) tb (cid:19) , (cid:19) , with a = 1 , b = − / , and c = 1 / − (cid:15) , which describes a hypotrochoid . For (cid:15) > small, thecurve has κ > , while for (cid:15) < small, κ is slightly negative near the midpoints of the threesides. The plane curve in figure 3 shows γ with the choice (cid:15) = − . ; the graph on the rightuses the height function h ( t ) = sin(3 t ) . By direct computation in Mathematica, γ has nonzerocurvature and positive torsion (and this behavior persists for all (cid:15) < small).Theorem 1 is also not true without the assumption that the base curve is simple, that is, does notintersect itself, even if we keep the requirement that the curvature of the base curve κ > .A counterexample can be found using the base curve in (9), for t ∈ [0 , π ) , with the values a = 1 , b = 1 / , and c = 1 . b . This curve, an epitrochoid, has positive curvature and is shown onthe left in figure 4 . The height function h = sin(5 t ) yields the curve shown on the right in figure4. By direct computation in Mathematica, γ has nonzero curvature and positive torsion.3. C URVES IN R , WITH NONNEGATIVE TORSION
Since the proof of Theorem 2 is very similar to the proof of Theorem 1, we will only point outthe differences. A more detailed discussion of the Lorentzian case can be found in [2].Complete spacelike hypersurfaces in Lorentzian space R , with metric dx + dy − dt are wellknown to be graphs over the xy plane (see [7] for instance). Hence, it follows that the smoothsimple closed curve γ is a graph over a smooth simple closed curve γ in the xy plane, as in figure2. Now consider the osculating plane [4] defined at each point of γ ( s ) by the span of γ (cid:48) ( s ) and γ (cid:48)(cid:48) ( s ) , where s is the arc length parameter. The tangent vector γ (cid:48) ( s ) is spacelike since γ lies in aspacelike hypersurface, and γ (cid:48)(cid:48) ( s ) , the curvature vector, is spacelike by assumption (and hence hasnonzero length). Parametrizing by arc length implies that γ (cid:48)(cid:48) ( s ) is perpendicular to γ (cid:48) ( s ) , so everyosculating plane is well-defined and spacelike, and hence not vertical. We acknowledge the following website of Mohammad Ghomi, which contains a vast library of plane and spacecurves: http://people.math.gatech.edu/~ghomi/MathematicaNBs/
N CURVES WITH NONNEGATIVE TORSION 7 F IGURE The closed plane curve γ on the left, an epitrochoid, is parametrized by (9)and has positive curvature but is not simple. The space curve γ on the right is a graph over γ with positive torsion. This example shows that Theorem 1 is not true without assumingthe base curve is simple. Lemma 2.
Consider a curve γ in R or R , , and suppose the osculating plane is well-definedand is not vertical at some point γ ( s ) . Then the projection γ of γ to the xy plane has nonzerocurvature at γ ( s ) .Proof. Inside the osculating plane at γ ( s ) , there exists a unique osculating circle, which agreeswith γ to second order at γ ( s ) . Hence, when we project γ and its osculating circle to the xy plane,the resulting ellipse will agree with the base curve γ to second order at every point. In particular, γ has nonzero curvature at s . (cid:3) Applying the lemma at every point, and appealing to continuity, the projected curve γ hascurvature κ > . Hence, we may parametrize everything by θ , just as before.The cross product in R , is defined by simply changing the sign of the t component of the usualcross product in R . Hence, equation (3) becomes κ γ (cid:48) × ( κ γ (cid:48) ) (cid:48) = (( κ h (cid:48) ) (cid:48) sin θ − κ h (cid:48) cos θ, − ( κ h (cid:48) ) (cid:48) cos θ − κ h (cid:48) sin θ, − . (Note that equation (4) also gives the torsion for curves in R , , up to sign.) Also, the other maindifference is that now κ | γ (cid:48) × γ (cid:48)(cid:48) | = | κ γ (cid:48) × ( κ γ (cid:48) ) (cid:48) | = (( κ h (cid:48) ) (cid:48) ) + ( κ h (cid:48) ) − , where | v | = v · v is negative for timelike vectors and positive for spacelike vectors.Since γ (cid:48) and γ (cid:48)(cid:48) are linearly independent and spacelike, their cross product γ (cid:48) × γ (cid:48)(cid:48) must betimelike. Hence, the above equation implies that we still have a sign on the new term (( κ h (cid:48) ) (cid:48) ) + ( κ h (cid:48) ) − < . Thus, the same argument as before using Lemma 1 implies that there exists an f > such that(10) (cid:90) π τ (cid:2) (( κ h (cid:48) ) (cid:48) ) + ( κ h (cid:48) ) − (cid:3) fκ dθ from which it is again clear that if τ ≥ (or τ ≤ ), then τ ≡ . Since the tangent vectorand the curvature vector to γ are both spacelike, the Frenet frame is well-defined, and so a trivialmodification of the usual proof that a curve with zero torsion in R lies in a plane [4] works in R , as well. This proves Theorem 2. HUBERT L. BRAY AND JEFFREY L. JAUREGUI F IGURE On the left is a depiction of a plane curve γ with κ > . The space curvesin the center and on the right are local graphs over γ , winding around k = 3 times. In thecenter, the torsion of the curve changes signs. On the right, the torsion is positive, whichprevents the curve from being closed (that is, the height function h is not πk -periodic).
4. C
URVES OF HIGHER WINDING NUMBER
Theorems 1 and 2 also have nice generalizations when we only assume that the curve γ is a localgraph over the base curve γ . In other words, while γ must still project down to γ , it is allowed towrap around the cylinder over γ more than once, as in figure 5. The precise statement, which is acorollary to results in [5], goes as follows: Theorem 4.
Let γ be a simple closed curve in R with positive curvature. Let γ be a smoothclosed curve in R that is locally a graph over γ . Then if γ has nonnegative (or nonpositive)torsion, then γ has zero torsion and hence lies in a plane. The above theorem is also true if R is replaced by R , , if we assume γ and its curvature vectorare spacelike. Theorem 4 is depicted in figure 5. The idea is that positive torsion causes the curve γ to “screw upwards” in the long run like a helix, thereby making it impossible for it to close upon itself.Our new proof of Theorem 4 is nearly exactly the same as our proofs of Theorems 1 and 2. Ourproof is made clear by a modification of equation (7), which now becomes: (cid:90) πk τ (cid:2) (( κ h (cid:48) ) (cid:48) ) + ( κ h (cid:48) ) + 1 (cid:3) fκ dθ. The integer k is the number of times γ wraps around the cylinder, where now h and τ are periodicwith period πk . Note that κ and f still have period π and are defined precisely as before.An alternate approach is the generalized four-vertex theorem of Romero Fuster and Sedykh [5].Their result shows that on a smooth arc of γ joining consecutive self-intersections, at least one zeroof the torsion must occur.5. R IGIDITY FOR CONSTANT TORSION CURVES
In this section we pose a rigidity question regarding closed curves of constant torsion : is itpossible to perturb a plane curve γ to a space curve with constant nonzero torsion in R (or R , ) ?We consider perturbations in the C sense, which is natural because torsion depends on threederivatives of a parametrization. If γ has positive curvature, the answer is no, by Theorems 1and 2, as any small perturbation of such a curve remains a graph over a simple closed plane curveof positive curvature. In Theorem 5 below, we show that the answer remains “no” without anyhypothesis on the curvature of γ . N CURVES WITH NONNEGATIVE TORSION 9 F IGURE The plane curve β used in the construction of a constant torsion curve. Thekey features are that β has a ◦ rotation symmetry and encloses zero area, counted withmultiplicity. Together, these imply that γ r is closed (see [1, 12]). Remark . It is possible to perturb a simple, closed (non-convex) plane curve to have positive (but non-constant) torsion. Consider the coiled helix on the right in figure 1. Its height functionmay be scaled by a constant (cid:15) > , and the torsion of the resulting curve is always positive (andthe curvature is positive as well, so the torsion is indeed well-defined). Yet as (cid:15) → , the curveconverges to its projection to the xy plane, which is simple. Remark . If we drop the hypothesis that γ is simple (but still require κ > ), such constanttorsion perturbations exist. Indeed, Weiner’s example of a closed curve of constant nonzero torsionbelongs to a family of such curves that converges to a self-intersecting plane curve.We briefly review this construction (cf. [1]) here, as it informs the proof of Theorem 5 below.For r ∈ (0 , , define the plane curve β r ( t ) = r (cid:32)
12 cos( t ) + √
24 cos(2 t ) ,
12 sin( t ) − √
24 sin(2 t ) , (cid:33) , depicted in figure 6. Let B r ( t ) be the vertical lift of β r ( t ) to the upper hemisphere of the unitsphere. By an observation of Koenigs in 1887 [8], the curve γ r ( t ) = 1 r (cid:90) t B r ( t ) × B (cid:48) r ( t ) dt has constant torsion equal to r and binormal equal to B r ( t ) . It is possible to verify that γ r is closedwith nonvanishing curvature. Since the binormal B r ( t ) concentrates at the north pole as r (cid:38) , itis not hard to see that γ r converges to a plane curve γ as r (cid:38) . It turns out that γ agrees with β modulo an isometry of R . See figure 7 for illustrations.Informally, the following theorem states that a simple, closed plane curve cannot be perturbedin the C sense to a closed space curve of constant nonzero torsion. Theorem 5.
Let γ n : S → R be a sequence of C maps converging in the C sense to γ : S → R . Assume that: (i) the image of γ lies in a plane and is not a point, (ii) γ n has constant speed c n > , non-vanishing curvature κ n , and constant nonzero torsion τ n (cid:54) = 0 .Then γ is not an embedding. In particular, its image has a self-intersection. Theorem 3 follows, because a simple closed curve is not embedded. F IGURE Depictions of curves of constant torsion constructed by Weiner [12] (cf. [1]).The top row is γ , shown both from above and from the side, with its projection to the xy plane on the far right. The bottom row gives the same views for γ / . As r → , γ r converges to the plane curve β , up to a rotation. Proof of Theorem 5.
Let t denote the S parameter, and let a prime denote ddt . By the Frenetformulas, the normal N n and binormal B n of γ n satisfy(11) c n B (cid:48) n = − τ n N n . Then B n × B (cid:48) n = τ n γ (cid:48) n , since the unit tangent T n equals c n γ (cid:48) n . Assuming without loss of generalitythat γ n (0) = (cid:126) for all n , we arrive at Koenigs’ formula: γ n ( t ) = 1 τ n (cid:90) t B n ( t ) × B (cid:48) n ( t ) dt. Without loss of generality, assume that the image of γ lies in the z = 0 plane in R . Lemma 3.
The binormal indicatrix B n : S → S converges uniformly, as n → ∞ , to the constantmap with value (0 , , (possibly reversing orientation, if necessary).Proof. By C convergence, γ has constant speed c = lim n →∞ c n ≥ ; by hypothesis (i), c > .Then there exists a point p ∈ S for which the curvature κ of γ is not zero, so γ (cid:48) ( p ) × γ (cid:48)(cid:48) ( p ) (cid:54) = (cid:126) .Then lim n →∞ B n ( p ) = lim n →∞ γ (cid:48) n × γ (cid:48)(cid:48) n | γ (cid:48) n × γ (cid:48)(cid:48) n | ( p ) = γ (cid:48) × γ (cid:48)(cid:48) | γ (cid:48) × γ (cid:48)(cid:48) | ( p ) = (0 , , ± n →∞ τ n ( p ) = lim n →∞ γ (cid:48)(cid:48)(cid:48) n · ( γ (cid:48) n × γ (cid:48)(cid:48) n ) | γ (cid:48) n × γ (cid:48)(cid:48) n | ( p ) = γ (cid:48)(cid:48)(cid:48) · ( γ (cid:48) × γ (cid:48)(cid:48) ) | γ (cid:48) × γ (cid:48)(cid:48) | ( p ) = 0 , by the C convergence of γ n to γ and the fact that γ lies in the z = 0 plane. By reversing orientationif necessary, we may assume B n ( p ) limits to (0 , , . Since τ n is constant by hypothesis, we have N CURVES WITH NONNEGATIVE TORSION 11 lim n →∞ τ n = 0 . Now, by (11) the speed of the curve B n is | B (cid:48) n | = c n | τ n | , which converges to zero.In particular, B n converges uniformly to (0 , , . (cid:3) Write ( x n ( t ) , y n ( t ) , z n ( t )) for the components of B n ( t ) , and let β n ( t ) = ( x n ( t ) , y n ( t ) , be itsprojection to the z = 0 plane. Note that β n is a closed plane curve that concentrates at the originas n → ∞ . In what follows, the limits are in the sense of uniform convergence, and Lemma 3 isused on the third line: γ (cid:48) = lim n →∞ γ (cid:48) n = lim n →∞ τ n B n × B (cid:48) n = lim n →∞ τ n (0 , , × ( x (cid:48) n , y (cid:48) n , z (cid:48) n )= lim n →∞ τ n ( − y (cid:48) n , x (cid:48) n , A lim n →∞ τ n ( x (cid:48) n , y (cid:48) n , A lim n →∞ τ n β (cid:48) n , where A is the rotation matrix A = − . The above demonstrates that the rescaled curves τ n β n , converge uniformly to γ as n → ∞ , moduloan isometry of R . Using Lemma 4 below, we see that each β n encloses zero area, counted withmultiplicity, and thus the same goes for τ n β n and thus for γ itself. This implies that γ has self-intersections, which completes the proof of Theorem 5. (cid:3) Lemma 4.
For each n , (cid:90) π β n × β (cid:48) n dt = 0 . Recall that for a C closed curve α in the z = 0 plane, (0 , , · (cid:82) S α × α (cid:48) dt measures the areabounded by α , counted with (possibly positive and negative) multiplicity. Proof of Lemma 4.
Since γ n is a closed curve, we have for each n : , , · ( γ n (2 π ) − γ n (0))= (0 , , · (cid:90) π B n × B (cid:48) n dt = (cid:90) π x n ( t ) y (cid:48) n ( t ) − y n ( t ) x (cid:48) n ( t ) dt = (0 , , · (cid:90) π β n × β (cid:48) n dt. This completes the proof, since β n × β (cid:48) n is a scalar multiple of (0 , , for each t . (cid:3) Remark . Theorem 5 generalizes readily for curves in R , , assuming γ lies in a spacelike plane.The same proof works with trivial modifications; we remark that the binormal indicatrices B n takevalues in the “unit sphere” { v ∈ R , : | v | = − } .6. D ISCUSSION
Below are some obvious corollaries to Theorems 1 and 2 with interesting statements:
Corollary 1.
Let γ be a smooth curve in R which is the graph over a simple closed curve in R with positive curvature. If γ has either positive or negative torsion at a point, then the torsion musthave the other sign at some other point. A similar result is also true in Lorentzian R , . Corollary 2.
Let γ be a smooth simple closed curve in R , with spacelike curvature vector whichlies in a complete spacelike hypersurface. If γ has either positive or negative torsion at a point,then the torsion must have the other sign at some other point. Equations (7) and (10) may be interpreted as saying that the average value of the torsion on thecurve γ is zero, with respect to a particular choice of positive weighting. In this manner, one canview these results as a “weighted total torsion theorem” (cf. the classical total torsion theorem forcurves lying in a sphere [9]).Another pair of corollaries comes from considering curves with constant torsion; for the case ofLorentzian R , , we describe below connections to general relativity. Corollary 3.
Let γ be a smooth curve in R which is the graph over a simple closed curve in R with positive curvature. If γ has constant torsion, then γ has zero torsion and hence is containedin a plane. Corollary 4.
Let γ be a smooth simple closed curve in R , with spacelike curvature vector whichlies in a complete spacelike hypersurface. If γ has constant torsion, then γ has zero torsion andhence is contained in a plane. In [3] the authors define what it means for a codimension-2 spacelike submanifold of a Lorentzianspacetime to be “time flat”. For the case of curves in a (2+1)-dimensional spacetime, time flat isequivalent to constant torsion. Hence, the previous corollary says that given certain assumptions,time flat curves are contained in planes. This result supports the choice of terminology: time-flatcurves do not bend in timelike directions. In [3], we explain how the time flat condition is geomet-rically natural, along with its importance to understanding the evolution of the Hawking mass ingeneral relativity, and describe some interesting conjectures of a purely geometric nature.R
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