Integrating curvature: from Umlaufsatz to invariant.
Abstract.
Hopf’s Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some nontrivial densities on a curve, such that the corresponding integrals are (possibly after some corrections) also invariant under regular homotopies of the curve in the class of generic immersions. We construct a family of such densities using indices of points relative to the curve. This family depends on a formal parameter and may be considered as a quantization of the total curvature. The linear term in the Taylor expansion at coincides, up to a normalization, with Arnold’s invariant. This leads to an integral expression for .
Key words and phrases:
plane curves, curvature, rotation number, regular homotopy2010 Mathematics Subject Classification:
53A04, 57R42Let be a closed oriented immersed plane curve . One of the fundamental notions related to is its curvature . Another important notion is that of a rotation number (or Whitney winding number) , i.e. the number of turns made by the tangent vector as we follow along its orientation.
Hopf’s Umlaufsatz [2] is one of the simplest versions of the GaussBonnet theorem and one of the fundamental theorems in the theory of plane curves. It relates two different types of data: local geometric characteristic of a plane curve – its curvature – and a global topological characteristic – its rotation number . Although the curvature of a plane curve changes under local deformations, the theorem states that its average (integral) over a closed curve is invariant under homotopies in the class of immersed curves:
Theorem 1 (Hopf’s Umlaufsatz).
(1) 
A natural question is whether one can find some natural densities on such that the average is (possibly after some corrections) also invariant under local deformations of . Since the rotation number is (up to normalization) the only invariant of in the class of immersed curves, we cannot expect such an expression to remain invariant under arbitrary homotopies. We can hope, however, that the result is invariant under regular homotopies in the class of generic immersions, i.e. immersions with a finite set of transversal double points as the only singularities. Invariants of such a type were originally introduced by Arnold [1] and include the celebrated and invariants (see [1] for details).
We construct a family of such densities using the index of a point relative to . Given , we define as the number of turns made by the vector pointing from to , as we follow along its orientation. This defines a locallyconstant function on . See Figure 1a. Suppose that is generic. Then we can extend to a valued function on . To define for , average its values on the regions adjacent to – two regions if is a regular point of , and four regions if is a double point of . See Figure 1b.
For each double point , define as the (nonoriented) angle between two tangent vectors and . For , define by
(2) 
Theorem 2.
is invariant under regular homotopies of in the class of generic immersions.
Proof.
Note that we can generalize all above notions and formulas to the case of a multicomponent curve (by a summation of indices relative to all components of ).
Let us smooth the original curve in each double point respecting the orientation to get a multicomponent curve without double points. Denote by the index of a point relative to . Note that values of on and differ by an easily computable factor (which depends only on the regular homotopy class of in the class of generic immersions). Indeed, consider a small neighborhood of a double point of index , see Figure 1c. Under smoothing of , the total curvature of differs from that of by for the fragment with index , see Figure 1c. Thus the integral part of changes by . Also, the double point contributes to . Smoothing removes , so this summand disappears from . Thus, the total change of under smoothing of equals . Hence
Since is invariant under regular homotopies of in the class of generic immersions, it remains to prove the invariance of .
Note that is constant on each component of , so
and by Umlaufsatz (1) we get , depending on . Thus, is invariant under regular homotopies of . But a regular homotopy of in the class of generic immersions induces a regular homotopy of and the theorem follows.
∎
Any two immersions with the same rotation number can be connected by regular homotopy in the class of generic immersions and a finite sequence of selftangency and triplepoint modifications, shown in Figure 2.
Depending on orientations and indices of adjacent regions, one can distinguish several types of these modifications. Selftangencies can be separated into direct (or dangerous) and opposite (or safe), shown in Figure 3a and 3b respectively. An index of a selftangency modification is the index of two newborn double points (e.g., modifications in Figure 3 are of index ). Triplepoint modifications can be separated into weak (or acyclic) and strong (or cyclic), shown in Figure 4a and 4b respectively. An index of a triplepoint modification^{1}^{1}1Our indices of modifications differ from the ones of [3] by an shift. is the minimum of indices of double points involved in this modification (e.g., modifications in Figure 4 are of index ).
Invariants of regular homotopy classes of generic immersions are uniquely determined by their behavior under these modifications, together with normalizations on standard curves of , shown in Figure 5.
Basic invariants and of (regular homotopy classes of) generic plane curves were introduced axiomatically by Arnold [1]. In particular, is uniquely determined by the following axioms:

does not change under an opposite selftangency or triplepoint modifications.

Under a direct selftangency modification which increases the number of double points, jumps by 2.

On the standard curves we have and for .
In a similar way, is uniquely determined by the following
Theorem 3.
The invariant satisfies the following properties:

does not change under opposite selftangencies.

Under direct selftangencies of index , the invariant jumps by .

Under (both weak and strong) triplepoint modifications of index , jumps by .

We have , where denotes with the opposite orientation.

On the standard curves we have and for
Proof.
A straightforward computation verifies both the behavior of under selftangencies and triplepoint modifications and its values on the curves . To verify the behavior of under an orientation reversal, note that , which corresponds to the involution in terms and of (2). Also, both terms in (2) change signs: the integral due to the change of parametrization, and the sum over double points due to the equality . ∎
Substituting into (2), we readily obtain and recover the classical Hopf Umlaufsatz, see Theorem 1. In this sense, invariant may be considered as a quantization of the total curvature (1). Let us study the next term of the Taylor expansion of at .
Proposition 4.
is related to Arnold’s invariant by
Proof.
Note that by Theorem 2, is invariant under regular homotopies of in the class of generic immersions. Differentiating at expressions for jumps of in Theorem 3 we immediately conclude that is invariant under opposite tangencies and triplepoint modifications. Moreover, under direct tangencies, jumps by . Thus its behavior under all modifications is the same as that of (up to an additive constant depending on ). A straightforward computation shows that takes values and for on the standard curves and the proposition follows. ∎
Corollary 5.
The following integral expression for holds:
Remark 6.
An infinite family of invariants, called “momenta of index” together with their generating function were introduced by Viro in [3, Section 5]. A careful check of their behavior under selftangencies and triplepoint modifications, together with their values on the standard curves , allow one to relate to as follows:
Remark 7.
Our choice of the function in the integral part of (2) was motivated by considerations of conciseness and convenience. In fact, one can use an arbitrary function of instead of (with an appropriate change of the correction term) to produce an invariant. Namely, repeating the proof of Theorem 2, one can show that
is an invariant of regular homotopy in the class of generic immersions, which does not change under opposite selftangencies. Under direct selftangencies of index , jumps by . Under triplepoint modifications of index , it jumps by .
References
 [1] V. I. Arnold, Topological invariants of plane curves and caustics, University Lecture Series 5, Providence, RI (1994).
 [2] H. Hopf, Über die Drehung der Tangenten und Sehenen ebener Kurven, Compos. Math., 2, 50–62 (1935).
 [3] O. Viro, Generic immersions of the circle to surfaces and the complex topology of real algebraic curves, Topology of real algebraic varieties and related topics, Amer. Math. Soc. Transl., Ser. 2, Vol. 173, 231–252 (1996).