The Rotation Index of a Plane Curve

THE ROTATION INDEX OF A PLANE CURVE

AARON W. BROWN AND LORING W. TU

The rotation index of a smooth closed plane curve C is the number of complete rotations that a tangent vector to the curve makes as it goes around the curve. According to a classical theorem of Heinz Hopf [1], the rotation index of a piecewise smooth closed curve with no self-intersections is ˙1, depending on whether the curve is oriented clockwise or counterclockwise. Hopf’s theorem is a key ingredient in a proof of the Gauss–Bonnet theorem for a surface ([2, Proof of Th. 9.3, p. 164], [3, Proof of Th. 4.4, p. 187]). In this article, we generalize Hopf’s theorem by allowing the curve C to have self-intersections (Figure 0.1).

C1 C1 b b p2 p1 p4 b C1

p3 b 1

b c.0/ FIGURE 0.1. Local indices at self-intersection points.

We show that it is possible to deﬁne a local index p at each self-intersection p such that the rotation index rot.C/ of the curve C can be computed from these local indices:

rot.C/ D ˙1 C p; (0.1) Xp where ˙1 depends on the orientation of C and the sum runs over all self-intersection points of C . What is most remarkable about this formula is that it expresses the rotation index, a priori a global invariant of the curve, as a sum of local contributions at the self-intersections. Thus, our formula fits into the general framework of localization theorems such as the Hopf index theorem for a vector field, the Bott residue theorem, or the Lefschetz fixed-point theorem that express a global invariant as a sum of local invariants. Whereas the known localization theorems localize to zeros of a vector field, fixed points of a map, or fixed points of a group action, Formula (0.1) localizes to the self-intersection points of a curve. It is the only localization formula of this type that we are aware of. Our rotation index theorem expresses a fundamental fact about closed plane curves: the number of complete turns and the type of crossings of a closed plane curve are not arbitrary, but must satisfy the constraint imposed by (0.1). We first prove the formula for a piecewise smooth curve

March 21, version 7. 1 2 AARON W. BROWN AND LORING W. TU in the plane and then generalize it to a piecewise smooth curve in a Riemannian manifold of dimension 2.

1. ORIENTATIONS ON A PLANE CURVE By a curve in the plane, we will mean either a continuous map cW Œ0;` ! R2 or its oriented image C WD c.Œ0;`/; the context will make clear whether a curve is a map or a point set. The curve cW Œ0;` ! R2 is closed if c.0/ D c.`/. By the Jordan curve theorem, a simple closed curve C in the plane separates the plane into two regions, one bounded and the other unbounded; the bounded region is called the interior of the curve. When the simple closed curve is piecewise smooth, the union M of the curve C and its interior is a manifold with boundary and possibly with corners, and the positive orientation on C is the orientation in Stokes’s theorem, namely, if Np is an outward normal vector at p 2 C and Tp is a tangent vector at p that gives the orientation 2 of C , then the ordered pair .Np; Tp/ gives the standard orientation of R (Figure 1.1).

FIGURE 1.1. Positive orientation on a curve.

When the curve C has a self-intersection, such as the ﬁgure-eight (Figure 1.2), the region D bounded by the curve may have more than one components. Because the various components can give rise to incompatible orientations, it may not be possible to deﬁne the positive orientation as the boundary orientation of D [C . For a piecewise smooth curve that may have self-intersections, we introduce the concept of a positive orientation with respect to a smooth extremal point.

b b b b p q p q

FIGURE 1.2. Positive orientation with respect to p and q

Deﬁnition 1.1. Let cW Œ0;` ! R2 be a piecewise smooth curve. A point p 2 C WD c.Œ0;`/ is called an extremal point of C if the curve C lies entirely in a closed half plane bounded by a line through p (Figure 1.3).

Lemma 1.2. If p is a smooth extremal point of a piecewise smooth plane curve C and L is the line through p bounding a closed half plane containing C , then the line L is tangent to C at p (Figure 1.3). THEROTATIONINDEXOFAPLANECURVE 3

b s

b b r b q p

FIGURE 1.3. p and q not extremal; r and s are extremal.

PROOF. Let Ltan be the tangent line to C at p. If Ltan ¤ L, then Ltan intersects L transversally at p, so C also intersects L transversally at p (Figure 1.4), contradicting the hypothesis that C lies on one side of L. Hence, Ltan D L. C Ltan L

b p

FIGURE 1.4. Transversal intersection of Ltan and L.

Now suppose cW Œ0;` ! R2 is a piecewise smooth, closed curve. At a smooth extremal point p D c.t0/, there is a well-deﬁned unit outward normal vector Np to the curve, since C lies 0 entirely on one side of its tangent line. Let Tp D c .t0/ be the velocity vector at p D c.t0/. We say that the curve c is positively oriented with respect to p if the ordered pair .Np; Tp/ gives the counterclockwise orientation of R2 (Figure 1.5).

Tp Np

FIGURE 1.5. Positive orientation with respect to a smooth extremal point.

2. THE LOCAL INDEX AT A SELF-INTERSECTION

We say that a curve C1 crosses another curve C2 at an isolated intersection point p 2 C1 \ C2 if C1 passes from one side of C2 to the other side of C2 at p. Figure 2.1 shows two examples of curves that cross each other at p. Note that two curves can cross each other at p and still be tangent at p. Figure 2.2 shows four examples of curves that do not cross each other at p. Smooth curves meeting at p without crossing each other at p are necessarily tangent at p. 4 AARON W. BROWN AND LORING W. TU

FIGURE 2.1. Branches that cross each other at p. Double arrows and gray indi- cate the second branch.

p p p p b b b b

(i) (ii) (iii) (iv)

FIGURE 2.2. Branches that do not cross each other at p; local index p D 0.

Let cW Œ0;` ! R2 be a smooth plane curve with finitely many self-intersection points. In particular, all the self-intersection points are isolated. A point p D c.t/ is an m-fold point of c if p D c.t/ for exactly m values of t. We will define the local index at a self-intersection point in stages. First consider an isolated double (2-fold) point p where the two branches of C cross each other. Near p, the curve C intersects a sufficiently small circle centered at p in exactly four points. Label the initial point and the endpoint on the circle of the first branch (earlier branch) a and 1 respectively, and of the second branch (latter branch) b and 2 respectively (Figure 2.1). Because the two branches cross each other at p, points 1 and 2 are adjacent on the circle, i.e., not separated by a or b. As one walks on the circle from 1 to 2 without hitting a and b, the direction is either clockwise or counterclockwise. In this way, one can say whether the second branch is clockwise or counterclockwise from the first branch. The local index at a double point p is defined to be

C1 if the two branches cross each other at p and the second branch is 8 clockwise from the first branch, p D ˆ1 if the two branches cross each other at p and the second branch is (2.1) ˆ < counterclockwise from the first branch, ˆ 0 if the two branches do not cross each other at p: ˆ :ˆ At an m-fold point p, the curve has m branches, labelled say 1;2;:::;m in chronological order. For any pair 1 i < j m, branch i and branch j intersect at a double point with local index ij D ˙1 or 0 as above. The local index p at the m-fold point p is defined to be the sum of the THEROTATIONINDEXOFAPLANECURVE 5

m 2 pairs of local indices ij : p D ij : 1i

p DC3 3 p 2 b

1 r D1 b r b q q D1

b c.0/

FIGURE 2.3. A curve with a triple point.

Example 2.1. The curve C in Figure 2.3 has three branches at the triple point p, labelled 1; 2; 3 chronologically. Branches 1 and 2 deﬁne a double point with local index 12 D 1, since they cross each other at p and branch 2 is clockwise from branch 1. Similarly, 13 D 23 D 1. Hence, the local index at p is

p D 12 C 13 C 23 D 1 C 1 C 1 D 3: At the double point q, since branch 2 is counterclockwise from branch 1, the local index is q D1.

3. THE ROTATION INDEX THEOREM Let cW Œ0;` ! R2 be a smooth unit-speed curve such that c.0/ D c.`/. Since cW Œ0;` ! R2 is a closed curve, we can extend its domain to R without changing the image C by making c periodic of period `. In particular, the domain of c can be extended to the open interval ;`CŒ containing Œ0;`. We will assume c0.0/ D c0.`/. Let .t/N be the angle that the tangent vector c0.t/ makes with respect to the horizontal. Because the angle is defined only up to an integer multiple of 2, we have a well-defined smooth function NW Œ0;` ! R=2Z, called the tangent angle function. Since Œ0;` is simply connected, by the lifting criterion of covering space theory, N can be lifted to a smooth function W Œ0;` ! R. Provided .0/ is specified, the lifted function is unique. Definition 3.1. Since c0.`/ and c0.0/ are the same vector, .`/ and .0/ differ by an integer multiple of 2. The rotation angle of c is defined to be .`/ .0/, and the rotation index rot.c/ of c is defined to be 1 rot.c/ WD .`/ .0/ : (3.1) 2 Note that the right-hand side of (3.1) is independent of the choice of .0/. 6 AARON W. BROWN AND LORING W. TU

0 C c .t0 /

q c0.t / b 0 q D c.t0/

FIGURE 3.1. The jump angle q at q.

When the curve c is piecewise smooth, at each singular point q D c.t0/, we let

c0.t C/ D lim c0.t/ 0 C t!t0 be the outgoing tangent vector and

0 0 c .t0 / D lim c .t/ t!t0 the incoming tangent vector (Figure 3.1). The jump angle q at q is defined to be the angle 0 0 C from the incoming vector c .t0 / to the outgoing vector c .t0 /. For a piecewise smooth curve c, the rotation angle rot.c/ is defined to be the sum of the changes in the angles along each smooth segment plus the sum of the jump angles, and the rotation index is again defined to be .1=2/ rot.c/.

Theorem 3.2 (Rotation index theorem). Let cW Œ0;` ! R2 be a piecewise smooth immersion with ﬁnitely many self-intersections none of which are singularities of the map c. Assume c.0/ D c.`/ and let S be the set of self-intersections. Suppose that the initial point c.0/ is a smooth extremal point. Then the rotation index of c is

rot.c/ D ˙1 C p; (3.2) pX2S where the term ˙1 is positive or negative depending, respectively, on whether c is positively or negatively oriented with respect to its initial point.

This theorem shows that the rotation index is independent of the parametrization and of the initial point, as long as the initial point is a smooth extremal point. We sometimes write rot.C/ instead of rot.c/.

Example 3.3. For the curve C in Figure 0.1,

rot.C/ D1 C pi D 1 C .1 C 1 1 C 1/ D 3: XiD1 Example 3.4 (Curve with a triple point). For the curve C with a triple point in Figure 2.3,

rot.C/ D 1 C .p C q C r / D 1 C .3 1 1/ D 2: THEROTATIONINDEXOFAPLANECURVE 7

4. THE SECANT ANGLE FUNCTION In this section, cW Œ0;` ! R2 is a smooth unit-speed closed curve with ﬁnitely many self- intersections such that the initial point c.0/ is not a self-intersection point. Denote by T the closed triangular region (Figure 4.1)

T Df.t1; t2/ 2 Œ0;` Œ0;` j t1 t2g: (4.1) Let be the diagonal Df.t;t/ 2 T g: and P the set P Df.t1; t2/ 2 T j t1 < t2 and c.t1/ D c.t2/g: (4.2) The set P parametrizes the pairs of times at which a self-intersection of C occurs: a double point p D c.t1/ D c.t2/ of C corresponds to a unique .t1; t2/ in P ; an m-fold point of C ,

p D c.t1/ D D c.tm/; m corresponds to the 2 points .ti ; tj /, 1 i < j m, in P . By hypothesis, P is a ﬁnite set. ! Any pair .t1; t2/ 2 T . [ P/ deﬁnes a unit secant vector c.t1/c.t2/. Let ˛.t1; t2/ be the ! ! angle that the secant c.t1/c.t2/ makes relative to the horizontal: if u D @=@x, then

! c.t2/ c.t1/ 1 ! ! c.t1/c.t2/ D and ˛.tN 1; t2/ D cos c.t1/c.t2/ u : kc.t / c.t /k 2 1 Since ˛.tN 1; t2/ is well defined up to an integer multiple of 2, it is a function from T . [ P/ to ! R=2Z. When .t1; t2/ approaches a point .t;t/ in the diagonal , the unit secant vector c.t1/c.t2/ approaches the unit tangent vector at c.t/, and the secant angle ˛.tN 1; t2/ approaches the tangent angle .t/N defined earlier. Therefore, the secant angle function ˛.tN 1; t2/, which a priori is defined and C 1 only on the interior of T P , can be extended to a continuous function ˛NW T P ! R=2Z. We call ˛NW T P ! R=2Z the secant angle function of the curve c, and a point .t1; t2/ 2 P a pole of the secant angle function.

T 0 T

FIGURE 4.1. An open neighborhood T 0 of the triangular region T .

Since the domain of the smooth function cW Œ0;` ! R2 can be enlarged to an open interval ;` C Œ containing Œ0;`, the domain of the secant angle function can be extended smoothly to an open set T 0 P in R2 containing T P (Figure 4.1). Although ˛N is not a real-valued function (it is R=2Z-valued), locally on any simply connected subset of T 0 P it can be represented by real-valued functions that differ by a constant integer 0 multiple of 2. Let fUi g be an open cover of T P by simply connected open sets. For any representative ˛1 on U1 and ˛2 on U2, we have d˛1 D d˛2 on U1 \ U2. Hence, the forms d˛i piece together to give well-deﬁned global form ! on T 0 P . The form ! is exact on any simply connected open subset of T 0 P , but not necessarily exact on T 0 P . Being locally exact, ! is a closed form. We will call ! the secant angle form of the curve c. 8 AARON W. BROWN AND LORING W. TU

5. THE LOCAL INDEXASAN INTEGRAL In this section we show that the local index at a self-intersection can be computed as an integral of the secant angle form !.

Theorem 5.1 (Integral formula for the local index of a double point). Let cW Œ0;` ! R2 be a smooth, unit-speed, closed curve with ﬁnitely many self-intersections such that the initial point c.0/ is not a self-intersection. Suppose p D c.e1/ D c.e2/ is a double point. Let T be the closed triangular region of the previous section and a counterclockwise loop in T enclosing .e1; e2/ but no other poles of ˛. Then the local index at p is 1 p D !: (5.1) 2 Z

PROOF. By the hypothesis that the initial point c.0/ is not a self-intersection, the poles of the secant angle function ˛N are all in the interior of the closed triangular region T . By Stokes’s theorem, since ! is a closed form, the integral in (5.1) is independent of the counterclockwise loop , as long as .e1; e2/ is the only pole of ˛N enclosed by . To prove the theorem, it sufﬁces to show that for each of the three types of double points in (2.1), the integral .1=2/ ! for an appropriate loop gives the correct local index. R ! BF

b b F D c.f / A D c.a/ ı ı3 2 ! ! b BD AF b p ı 4 ı1 b D D c.d/ B D c.b/ b ! AD

FIGURE 5.1. Local index at p.

Consider now the self-intersection point p D c.e1/ D c.e2/ in Figure 5.1. Pick times a and b near e1, and d and f near e2 such that

a < e1

f e2 b p d

a e1 b t1

FIGURE 5.2. A small rectangular loop about p.

Similarly,

! D ı2.counterclockwise/; Z.b;d/.b;f /

! D ı3.counterclockwise/ Z.b;f /.a;f /

! D ı4.counterclockwise/: Z.a;f /.a;d/

From Figure 5.1, we see that the sum of ı1, ı2, ı3, ı4 is 2. Hence, 1 1 ! D ! C ! C ! C ! 2 Z 2 Z.a;d/.b;d/ Z.b;d/.b;f / Z.b;f /.a;f / Z.a;f /.a;d/ 1 1 D .ı1 C ı2 C ı3 C ı4/ D .2/ D 1: 2 2 The computation of q in Figure 5.3 differs from that of p in that the relative disposition of the four points A;B;D;F are different: ! BD

b b D D c.d/ A D c.a/ ı ı1 2 ! ! b BF AD b q ı 4 ı3 b F D c.f / B D c.b/ b ! AF

FIGURE 5.3. Local index at q.

This time, 1 1 ! D .ı1 C ı2 C ı3 C ı4/ 2 Z 2 1 D .2/ D1: 2 10 AARON W. BROWN AND LORING W. TU

When the two branches do not cross each other, there are four cases for the relative disposition ! of the four points A;B;D;F as in Figure 2.2. In case (i), the change in the secant angle from AD ! ! ! to BD to BF to AF is (Figure 5.4) 1 1 ! D .ı1 C ı2 C ı3 C ı4/ D 0: 2 Z 2 ! ! BF AD ! ! BD b b AF F D b b

B b b A

FIGURE 5.4. Local index at a nontransversal double point.

The other three cases (ii), (iii), (iv) are similar and all give 0 as the local index at r.

Theorem 5.2 (Integral formula for the local index of an m-fold point). Let cW Œ0;` ! R2 be a smooth, unit-speed, closed curve with ﬁnitely many self-intersections such that the initial point c.0/ is not a self-intersection. Suppose p D c.e1/ D D c.em/ is an m-fold point. For 1 i < j m, let ij be a counterclockwise loop in T enclosing .ei ; ej / but no other poles of ˛. Then the local index at p is 1 p D !: 2 Z 1i

PROOF. At p, the curve has m branches, labelled chronologically 1;:::;m. By deﬁnition, the index ij is the local index of the double point deﬁned by branches i and j . By Theorem 5.1, 1 ij D !: 2 Z ij

Hence, by the deﬁnition of p, 1 p D ij D !: 2 Z 1i

6. PROOF OF THE ROTATION INDEX THEOREM In this section we prove the rotation index theorem for a closed smooth plane curve cW Œ0;` ! R2 with self-intersections. Note that a rotation or a translation of the plane leaves the rotation index as well as the local indices p of the curve C invariant. By an appropriate rotation, one may assume that the tangent line Tc.0/.C/ is horizontal. Assume now that c is positively oriented with respect to c.0/. Then the initial vector c0.0/ points to the right and the angle of the initial vector c0.0/ is 0 (see Figure 6.1). Let O D .0; 0/, L D .`;`/, and Q D .0;`/ be the three vertices of the triangular region T (Figure 6.2), and let T 0 be an open neighborhood of T as in Figure 4.1. THEROTATIONINDEXOFAPLANECURVE 11

b c.0/ c0.0/

FIGURE 6.1. Horizontal initial vector at c.0/.

Q D .0;`/ L D .`;`/

b b

O D .0; 0/

FIGURE 6.2. Small loops about the poles.

In a simply connected neighborhood U of the line segment OL in T 0, the secant angle function ˛N W U ! R=2Z has a unique lift ˛W U ! R with a given initial value ˛.0; 0/. On U , the secant angle form ! is exact: ! D d˛. Hence, the rotation angle of C is

.`/ .0/ D ˛.`;`/ ˛.0; 0/ D !: ZOL

Around each pole e of the 1-form !, we draw a small counterclockwise circle e. Let Be be the interior of e. Then T Be is a manifold with boundary S @ T Be D OL C LQ C QO e: e[2P e[2P

Since ! is a closed 1-form on T Be, by Stokes’s theorem, S ! C ! C ! D !: (6.1) Z Z Z Z OL LQ QO eX2P e The vertex O D .0; 0/ of the triangular region T corresponds to the tangent vector c0.0/, the vertex Q D .0;`/ to the tangent vector c0.0/, and the vertex L D .`;`/ to the tangent vector c0.`/ D c0.0/. Since the line segment OQ is simply connected, along OQ the form ! is exact with 0 0 OQ ! being the change in the angle from the tangent vector c .0/ to the tangent vector c .0/. RHence, ! D 0 D : (6.2) ZOQ 12 AARON W. BROWN AND LORING W. TU

Similarly, along QL the secant moves from c0.0/ to c0.`/ and the secant angle changes from to 2, so ! D 2 D : ZQL Equation (6.1) becomes

! ! ! D ! D !; Z Z Z Z Z OL QL OQ OL eX2P e which gives 1 1 rot.c/ D ! D 1 C ! 2 Z 2 Z OL eX2P e

D 1 C p (by Theorem 5.2). pX2S If the curve c is negatively oriented with respect to c.0/, then the secant angle along OQ goes from to 0 and the secant angle along QL goes from 0 to . A computation similar to the above then gives

rot.c/ D1 C p: pX2S

p D1

b p

b c.0/

FIGURE 6.3. Nonextremal initial point.

Remark 6.1 (Dependence on the initial point). In the rotation index theorem in the plane (The- orem 3.2), the initial point must be an extremal point. When the initial point is not an extremal point, for example, as in In Figure 6.3, Equation (6.2) fails and the rotation index formula (3.2) also fails. Indeed, the correct formula for the rotation index in Figure 6.3 is instead

rot.C/ D 3 C p D 3 C .1/ D 2:

7. PIECEWISE SMOOTH PLANE CURVES We now extend the proof of the rotation index theorem to the piecewise smooth case. Let 2 cW Œ0;` ! R be a piecewise smooth closed curve with singularities at 1 < 2 < < r in the open interval 0;`Œ. Assume further the initial point c.0/ is not a singularity and that the self-intersections of c occur away from the singularities. Define the triangular region T and the set P as in (4.1) and (4.2) respectively. The secant angle ˛ is defined and continuous on T P , but it is not smooth whenever t1 or t2 equals one of the i ’s. Thus, ! is defined and smooth on the complement in T of the union of P and the vertical and horizontal lines ti D j (Figure 7.1). Since the self-intersection points of c are not singularities of c, the poles of ! do not meet the 1 lines t1 D j or t2 D j and so ! is defined and C on a punctured neighborhood of each THEROTATIONINDEXOFAPLANECURVE 13

t2 1 2

t1 FIGURE 7.1. The domain of ! for a piecewise smooth curve.

c c

FIGURE 7.2. Smoothing a corner. pole. It follows that for a piecewise smooth closed curve c, the proofs above still apply so that Theorems 5.1 and 5.2 remain true for a piecewise smooth closed curve. When the curve c is piecewise smooth, by a process of smoothing corners (see [2, p. 161] and Figure 7.2), one can ﬁnd a nearby smooth curve cN with the same rotation angle as c. Moreover, since by hypothesis the singularities of c are not self-intersection points, cN and c have the same self-intersection points and the same local index at each self-intersection point. Thus, the rotation index formula (3.2) remains valid for a piecewise smooth curve.

8. GENERALIZATIONS We give two generalizations of the rotation index theorem (Theorem 3.2), ﬁrst to an arbitrary metric in the plane, and then to an oriented Riemannian manifold. In Theorem 3.2, the angle relative to the horizontal is measured relative to the horizontal with respect to the usual Euclidean 2 metric g0 on R . In fact, we may replace g0 by an arbitrary metric in the plane. Theorem 8.1. The formula in Theorem 3.2 remains valid if the angles are measured relative to the horizontal (i.e., @=@x) with respect to an arbitrary Riemannian metric on the plane.

2 PROOF. Let g1 be an arbitrary Riemannian metric on R . Then gs D .1s/g0 Csg1, s 2 Œ0; 1, is a continuous family of Riemannian metrics on R2 parametrized by the interval Œ0; 1. Clearly, the rotation index of the curve c with respect to the metric gs is a continuous function of s. Since it is integer-valued, it must be a constant. Similarly, the secant angle function ˛N, which depends on gs, is a continuous function of s. Therefore, the local index p D .1=2/ ! from Theorem 5.1 is also a continuous function of s and hence is constant as a function of sR. Setting s D 1 proves that Theorem 3.2 is true with respect to any Riemannian metric g1 on the plane.

Recall that a coordinate chart in the differentiable structure of a smooth 2-manifold M is an open set U M together with a homeomorphism W U ! .U/ of U between U and an 14 AARON W. BROWN AND LORING W. TU open subset .U/ of R2. With respect to the differentiable structure of M , the coordinate map W U ! .U/ is a diffeomorphism. We write x;y for the two components of . On .U;x;y/, the angle of a tangent vector at p 2 U is measured relative to @=@xjp using the Riemannian metric. Let .U;/ D .U;x;y/ be a coordinate chart on an oriented Riemannian 2-manifold. Suppose cW Œ0;` ! U is a piecewise smooth immersion that is closed (c.0/ D c.`/) and has finitely many self-intersections. Assume further that the self-intersections of c are not the singularities of c. The diffeomorphism W U ! .U/ defines a metric on .U/ R2 such that becomes an isometry. Using this isometry, we can transfer Theorem 8.1 to the coordinate chart U . Theorem 8.2 (Rotation index theorem for a coordinate chart). The rotation index formulas in Theorem 3.2 remains valid if c maps into a coordinate chart in a Riemannian 2-manifold. Remark 8.3 (Dependence on the frame). In Theorem 8.2, the angle is measured relative to the first coordinate vector @=@x of a coordinate frame @=@x, @=@y. The theorem is not true for an arbitrary frame. For example, if U is the punctured plane R2 f0g with the Euclidean metric, r and are polar coordinates, and the angle is measure with respect to @=@r of the frame .@=@r; @=@/, then the rotation index of the circle c.t/ D .cos t; sin t/ is zero, not ˙1. Note that .r;/ is not a coordinate system on U , since is not a continuous real-valued function on U .

REFERENCES [1] H. Hopf, Uber¨ die Drehung des Tangenten und Sehnen ebener Kurven, Compositio Mathematica, tome 2 (1935), pp. 50–62. [2] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Graduate Texts in Mathematics 176, Springer, New York, 1997. [3] R. S. Millman and G. D. Parker, Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1977.

DEPARTMENT OF MATHEMATICS,TUFTS UNIVERSITY, MEDFORD, MA 02155-7049 E-mail address: [email protected]