The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications

In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a resette.


Introduction
The local and global geometry of fronts and coherent tangent bundles, which are natural generalizations of fronts, has been recently very carefully studied in [18,28,29,34,35,36,37]. In particular in [34,35] the results of M. Kossowski ([19,20]) and R. Langevin, G. Levitt, H. Rosenberg ( [22]) were generalized to the following Gauss-Bonnet type formulas for the singular coherent tangent bundle E over a compact surface M whose set of singular points Σ admits at most peaks: In the above formulas K is the Gaussian curvature, κ s is the singular curvature, dτ is the arc length measure on Σ, dÂ (respectively dA) is the signed (respectively unsigned) area form, M + (respectively M − ) is the set of regular points in M , where dÂ = dA (respectively dÂ = −dA), P + (respectively P − ) is the set of positive (respectively negative) peaks (see [34] and Section 2 for details). K. Saji, M. Umeraha and K. Yamada also found several interesting applications of the above formulas (see especially [36]).
The classical Gauss-Bonnet theorem was formulated for compact oriented surfaces with boundary. Therefore it is natural to find the analogous Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces with boundary (see Theorem 2.20). Coherent tangent bundles over compact oriented surfaces with boundary also appear in many problems. In this paper we apply the Gauss-Bonnet formulas to study smooth maps between compact oriented surfaces with boundary and affine extended wave fronts of the planar non-singular hedgehogs (rosettes). As a result, we obtain a new proof of Fukuda-Ishikawa's theorem ( [11]) and we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front of a rosette.. This leads to a relation between the integrals of the function of the width of the rosette, in particular of the width of an oval (see Theorem 5.22 and Conjecture 5.26).
In Section 2 we briefly sketch the theory of coherent tangent bundles and state the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary (Theorem 2.20), which is the main result of this paper. The proof of Theorem 2.20 is presented in Section 3. We apply this theorem to study the global properties of maps between compact oriented surfaces with boundary in Section 4. The last section contains the results on the geometry of the affine extended wave fronts of rosettes.

The Gauss-Bonnet theorem
In this section we formulate the Gauss-Bonnet type theorem for coherent tangent bundles over compact oriented surfaces with boundary. The proof of this theorem is presented in the next section. Coherent tangent bundles are intrinsic formulation of wave fronts. The theory of coherent tangent bundles were introduced and developed in [34,35,36]. We recall basic definitions and facts of this theory (for details see [34,36]). The pull-back metric ds 2 := ψ * ·, · is called the first fundamental form on M . Let E p denote the fiber of E at a point p ∈ M . If ψ p := ψ| TpM : T p M → E p is not a bijection at a point p ∈ M , then p is called a singular point. Let Σ denote the set of singular points on M . If a point p ∈ M is not a singular point, then p is called a regular point. Let us notice that the first fundamental form on M is positive definite at regular points and it is not positive definite at singular points.
Let µ ∈ Sec(E * ∧ E * ) be a smooth non-vanishing skew-symmetric bilinear section such that for any orthonormal frame {e 1 , e 2 } on E µ(e 1 , e 2 ) = ±1. The existence of such µ is a consequence of the assumption that E is orientable. A co-orientation of the coherent tangent bundle is a choice of µ. An orthonormal frame {e 1 , e 2 } such that µ(e 1 , e 2 ) = 1 (respectively µ(e 1 , e 2 ) = −1) is called positive (respectively negative) with respect to the co-orientation µ.
From now on, we fix a co-orientation µ on the coherent tangent bundle.
Definition 2.2. Let (U ; u, v) be a positively oriented local coordinate system on M . Then dÂ := ψ * µ = λ ψ du ∧ dv (respectively dA := |λ ψ |du ∧ dv) is called the signed area form (respectively the unsigned area form), where The function λ ψ is called the signed area density function on U .
The set of singular points on U is expressed as Let us notice that the signed and unsigned area forms, dÂ and dA, give globally defined 2-forms on M and they are independent of the choice of positively oriented local coordinate system (u, v). Let us define We say that a singular point p ∈ Σ is non-degenerate if dλ ψ does not vanish at p. Let p be a non-degenerate singular point. There exists a neighborhood U of p such that the set Σ ∩ U is a regular curve, which is called the singular curve. The singular direction is the tangential direction of the singular curve. Since p is non-degenerate, the rank of ψ p is 1. The null direction is the direction of the kernel of ψ p . Let η(t) be the smooth (non-vanishing) vector field along the singular curve σ(t) which gives the null direction.
Let ∧ be the exterior product on T M . Definition 2.3. Let p ∈ M be a non-degenerate singular point and let σ(t) be a singular curve such that σ(0) = p. The point p is called an A 2 -point (or an intrinsic cuspidal edge) if the null direction at p (i.e. η(0)) is transversal to the singular direction at p (i.e.σ(0) := dσ dt t=0 ). The point p is called an A 3 -point (or an intrinsic swallowtail ) if the point p is not an A 2 -point and d dt (σ(t) ∧ η(t))| t=0 = 0.
Definition 2.4. Let p be a singular point p ∈ M which is not an A 2 -point. The point p is called a peak if there exists a coordinate neighborhood (U ; u, v) of p such that: (ii) the rank of the linear map ψ p : T p M → E p at p is equal to 1; (iii) the set Σ ∩ U consists of finitely many C 1 -regular curves emanating from p.
A peak is a non-degenerate if it is a non-degenerate singular point. From now one we assume that the set of singular points Σ admits at most peaks, i.e. Σ consists of A 2 -points and peaks.
Furthermore let us fix a Riemannian metric g on M . Since the first fundamental form ds 2 degenerates on Σ, there exists a (1, 1)-tensor field I on M such that for smooth vector fields X, Y on M . We fix a singular point p ∈ Σ. Since Σ admits at most peaks, the point p is an A 2 -point or a peak. Let λ 1 (p), λ 2 (p) be the eigenvalues of I p := I TpM : T p M → T p M . Since the kernel of ψ p is one dimensional, the only one of λ 1 (p), λ 2 (p) vanishes. Thus there exists a neighborhood V of p such that for every point q ∈ V the map I q has eigenvalues λ 1 (q), λ 2 (q), such that 0 λ 1 (q) < λ 2 (q). Furthermore there exists a coordinate neighborhood (U ; u, v) of p such that U is a subset of V and the u -curves (respectively vcurves) give the λ 1 -eigendirections (respectively λ 2 -eigendirections). Such a local coordinate system (U ; u, v) is called a g-coordinate system at p. Definition 2.5. Let γ(t) (0 ≤ t < 1) be a C 1 -regular curve on M such that γ(0) = p. The E-initial vector of γ at p is the following limit Remark 2.6. If p is a regular point of M then the E -initial vector of γ at p is the unit tangent vector of γ at p with respect to the first fundamental form ds 2 .
Proposition 2.7 (Proposition 2.6 in [34]). Let γ be a C 1 -regular curve emanating from an A 2 -point or a peak p such thatγ(0) is a not a null vector or γ is a singular curve. Then the E -initial vector of γ at p exists.
Since we study coherent tangent bundles over surfaces with boundary, we also consider a curve γ on the boundary which is tangent to the null direction at a singular point p on the boundary. We prove that in this case the E-initial vector of γ at p exists if the singular direction is transversal to the boundary at p. Proposition 2.8. Let (E, ·, · , D, ψ) be a coherent tangent bundle over an compact oriented surface M with boundary. Let p be an A 2 -point in the boundary ∂M . If the boundary ∂M is transversal to Σ at p and γ : (−ε, ε) → ∂M is a C 2 -regular curve such that γ(0) = p, γ (−ε, ε) ∩ Σ = {p} andγ(0) ∈ T p ∂M is a null direction, then the E-initial vector Ψ γ of γ at p exists, D d dt ψ γ(t) t=0 = 0, and Proof. Let σ : [0, ε) → Σ be a singular curve such that σ(0) = p. Let (U ; u, v) be a g-coordinate system at p i.e. the null direction at σ(t) is spanned by ∂ ∂u . Since λ ψ (σ(t)) = 0, we get that since the vectorsσ(0) andγ(0) span the space T p M and dλ ψ p = 0.
We generalize the definition of singular sectors from [34] to the case of coherent tangent bundles over surfaces with boundary.
Let U be a (sufficiently small) neighborhood of a singular point p. Let σ 1 and σ 2 be curves in U starting at p such that both are singular curves or one of them is a singular curve and the other one is in ∂M . A domain Ω is called a singular sector at p if it satisfies the following conditions (i) the boundary of Ω ∩ U consists of σ 1 , σ 2 and the boundary of U .
(ii) Ω ∩ Σ = ∅. If the peak p ∈ M \ ∂M is an isolated singular point than the domain U \ {p} is a singular sector at p, where U is a neighborhood of p such that U ∩ Σ = {p}. We assume that singular direction is transversal to the boundary of M . Therefore there are no isolated singular points on the boundary.
We define the interior angle of a singular sector. If p is in ∂M , then the interior angle of a singular sector at p is the angle of the initial vectors of σ 1 and σ 2 at p.
While the interior angle of a singular sector may take value greater than π if p ∈ M \ ∂M , we can choose γ j for j = 0, . . . , n inside the singular sector in a such way that the angel between Ψ γj−1 and Ψ γj is not greater than π.
Let Ω be a singular sector at the peak p. Then there exists a positive integer n and C 1 -regular curves starting at p γ 0 = σ 0 , γ 1 , · · · , γ n = σ 1 satisfying the assumptions of Proposition 2.7 and the following conditions: (i) if i = j then γ i ∩ γ j = ∅ in Ω, (ii) for each j = 1, . . . , n there exists a sector domain ω j ⊂ Ω such that ω j is bounded by γ j−1 and γ j and ω j ∩ γ i = ∅ for i = j − 1, j, (iii) if n 2 the vectorsγ j−1 (0),γ j (0)) are linearly independent and form a positively oriented frame for j = 1, . . . , n. If the peak p is an isolated singular point then there exist curves γ 0 , γ 1 , γ 2 satisfying the above assumptions and conditions (i)-(iii). We also put γ 3 = γ 0 .
If Ω is a singular sector at a singular point p then Ω is contained in M + or M − . The singular sector Ω is called positive (respectively negative) if Ω ⊂ M + (respectively Ω ⊂ M − ).
Definition 2.11. Let p be a singular point. Then α + (p) (respectively α − (p)) is the sum of all interior angles of positive (respectively negative) singular sectors at p. Proposition 2.12 (Theorem A in [34]). Let p ∈ M \ ∂M be a peak. The sum α + (p) of all interior angles of positive singular sectors at p and the sum α − (p) of all interior angles of negative singular sectors at p satisfy Theorem 2.13. Let p ∈ ∂M be a singular point. We assume that the singular direction is transversal to the boundary ∂M at p.
If the null direction is transversal to the boundary ∂M at p, then If the null direction is tangent to the boundary ∂M at p, then Proof. The first part of this theorem follows from Proposition 2.15 in [34]. By Proposition 2.9 we get the second part.
Remark 2.16. It is easy to see that a peak p in ∂M is not null if ∂M is transversal to the singular direction at p and an A 2 singular point p in ∂M is null if the null vector at p is tangent to ∂M .
Definition 2.17. Let p be a peak in ∂M . We say that p is in the positive boundary Let σ(t) (t ∈ (a; b)) be a C 2 -regular curve on M . We assume that if σ(t) ∈ Σ thenσ(t) is transversal to the null direction at σ(t). Then the image ψ (σ(t)) does not vanish. Thus we take a parameter τ of σ such that Definition 2.18. Let n(τ ) be a section of E along σ(τ ) such that {ψ(σ(τ )), n(τ )} is a positive orthonormal frame. Then is called the E-geodesic curvature of σ, which gives the geodesic curvature of the curve σ with respect to the orientation of E.
We assume that the curve σ is a singular curve consisting of A 2 -points. Take a null vector field η(τ ) along σ(τ ) such that {σ(τ ), η(τ )} is a positively oriented field along σ(τ ) for each τ . Then the singular curvature function is defined by where sgn(dλ ψ (η(τ ))) denotes the sign of the function dλ ψ (η) at τ . In a general parameterization of σ = σ(t), the singular curvature function is defined as follows By Proposition 1.7 in [34] the singular curvature function does not depend on the orientation of M , the orientation on E, nor the parameter t of the singular curve σ(t).
By Proposition 2.11 in [34] the singular curvature measure κ s dτ is bounded on any singular curve, where dτ is the arclength measure of this curve with respect to the first fundamental form ds 2 . Now we prove the following proposition concerning the geodesic curvature measure on the boundary of M . Proof. The point γ(0) ∈ ∂M is a null A 2 -point. By Proposition 2.8 we can write that Ψ(γ(t)) = tζ(t) for t ∈ [0,ε) for sufficiently smallε ≤ ε, where ζ(t) ∈ E γ(t) and The geodesic curvature in a general parameterization has the following formκ Thus the geodesic curvature measurê is bounded and continuous on [0,ε). It implies that the geodesic curvature measure is continuous Let U ⊂ M be a domain and let {e 1 , e 2 } be a positive orthonormal frame field on E defined on U . Since D is a metric connection, there exists a unique 1-form ω on U such that for any smooth vector field X on U . The form ω is called the connection form with respect to the frame {e 1 , e 2 }. It is easy to check that dω does not depend on the choice of a frame {e 1 , e 2 } and gives a globally defined 2-form on M . Since D is a metric connection and it satisfies (2.1) we have where K is the Gaussian curvature of the first fundamental form ds 2 (see [34,35]).
The next theorem is a generalization of the Gauss-Bonnet theorem for coherent tangent bundles over smooth compact oriented surfaces with boundary.
Theorem 2.20 (The Gauss-Bonnet type formulas). Let E be a coherent tangent bundle on a smooth compact oriented surface M with boundary whose set of singular points Σ admits at most peaks. If the set of singular points Σ is transversal to the boundary ∂M , then where dτ is the arc length measure, P + (respectively P − ) is the set of positive (respectively negative) peaks in M \ ∂M , (Σ ∩ ∂M ) + (respectively (Σ ∩ ∂M ) − , null(Σ ∩ ∂M )) is the set of positive (respectively negative, null) singular points in Σ ∩ ∂M , P ∂M + (respectively P ∂M − ) is the set of peaks in the positive (respectively negative) boundary.

The proof of Theorem 2.20
We use the method presented in the proof of Theorem B in [34]. First we formulate the local Gauss-Bonnet theorem for admissible triangles.
is admissible on the surface with boundary if it satisfies one of the following conditions: (1) σ is a C 2 -regular curve such that σ((a, b)) does not contain a peak, and the tangent vectorσ is contained in the set of singular points Σ and the set σ((a, b)) does not contain a peak.  (1) T admits at most one peak on {A, B, C}.
(2) the three interior angles at A, B and C with respect to the metric g are all less than π. (3) if γ j for j = 1, 2, 3 is not a singular curve, it is C 2 -regular, namely it is a restriction of a certain open C 2 -regular arc.
We write ∆ABC := T and we denote by the regular arcs whose boundary points are {B, C}, {C, A}, {A, B}, respectively. We give the orientation of ∂∆ABC compatible with respect to the orientation of M . We denote by ∠A, ∠B, ∠C the interior angles (with respect to the first fundamental form ds 2 ) of the piecewise smooth boundary of ∆ABC at A, B and C, respectively if A, B and C are regular points.
If A ∈ M \ ∂M is a singular point and (U ; u, v) is a g-coordinate system at A, then we set (see Proposition 2.15 in [34]) ∠A := π if the u − curve passing through A separates AB and AC, 0 otherwise.
Let σ(t) be an admissible curve. We define a geometric curvatureκ g (t) in the following way:κ whereκ g is the geodesic curvature with respect to the orientation of M and κ s is the singular curvature. Suppose that the boundary ∂M is transversal to Σ at A and let T A ∂M be a null direction at A. Then Proof. Without loss of generality, let us assume that ∆ABC \ AC lies in M + . If the arc AC ⊂ Σ or the interior angle ∠BAC with respect to the metric g is greater than π 2 , we decompose the triangle ∆ABC into admissible triangles ∆ABD and ∆ADC such that the interior angle ∠BAD with respect to the metric g is in the interval (0, π 2 ) and the arc AD is transversal to the arc BC at D, see Fig. 1. The formula (3.1) for ∆ADC follows from Theorem 3.3 in [34], so it is enough to prove the formula (3.1) for the triangle ∆ABD. We can take the arc AD and rotate it around D with respect to the canonical metric du 2 + dv 2 on the uv-plane. Then we obtain a smooth one-parameter family of C 2 -regular arcs starting at D. Since the interior angle ∠BAD is in (0, π 2 ) and BD, AD are transversal at D, restricting the image of this family to the triangle ∆ABD, we obtain a family of C 2 -regular curves where ε ∈ [0, 1] and: (i) γ 0 parameterizes AD and γ 0 (0) = A, γ 0 (1) = D, (ii) γ ε (1) = D for all ε ∈ [0, 1], (iii) the correspondence σ : ε → γ ε (0) gives a subarc of AB. We set A ε = γ ε (0), where A 0 = A. Since ∆A ε BD for ε > 0 is an admissible triangle, then by Theorem 3.3 in [34] we get that KdA.
Since ∆ABD is admissible andκ g is bounded on both AB and AD, by taking the limit as ε → 0 + , we have that By Proposition 2.8 we have This completes the proof. Lemma 3.6. The following relation holds: Proof. By the definition of Euler's characteristic we get that Furthermore, it is easy to verify that Then By Lemma 3.6 we get that Since ∂M + is an Eulerian graph, the number deg ∂M + (v) is even and let us write that Hence we get the following: Similarly we get that It is easy to see that if p is a peak in the positive boundary, −1 if p is a peak in the negative boundary, 0 otherwise. (3.11) Lemma 3.7. The Euler characteristic of Σ is equal to Proof. We know that If p ∈ P \ ∂M then deg Σ (p) = deg Σ∪∂M (p) and if p ∈ Σ ∩ ∂M then deg Σ (p) = deg Σ∪∂M (p) − 2. By (3.9) and (3.10) we get that Lemma 3.8. The following equality holds: Lemma 3.9. The following equality holds: where P + (respectively P − ) is the set of positive (respectively negative) peaks in M \ ∂M , (Σ ∩ ∂M ) + (respectively (Σ ∩ ∂M ) − ) is the set of positive (respectively negative) singular points in Σ ∩ ∂M , P ∂M + (respectively P ∂M − ) is the set of peaks in the positive (respectively negative) boundary.
Proof. It is a consequence of (3.6), (3.7), Lemma 3.7 and Theorem 2.13 and the fact that Since the integration of the geometric curvature on curves which are not included in Σ∪∂M are canceled by opposite integrations and the singular curvature does not depend on the orientation of the singular curve, by Proposition 3.4 and Theorem 3.3 in [34] we get that

Applications of the Gauss-Bonnet formulas to maps
As a corollary of Theorem 2.20 we get Fukuda-Ishikawa's theorem [11] (see also [21]), which is the generalization of Quine's formula ( [32]) for surfaces with boundary (see also Proposition 3.6. in [36]).
where M + f (respectively M − f ) is the set of regular points at which f preserves (respectively reverses) the orientation, S + f (respectively S − f ) is the number of positive cusps (respectively the number of negative cusps).
Proof. Let h be a Riemannian metric on N and let D be the Levi-Civita connection on (N, h). Then the tuple (f * T N, h, D, df ) is a coherent tangent bundle on M (see [36]). Since f (∂M ) ⊂ ∂N and the set of singular points of f is transversal to ∂M , there are no cusps in ∂M and all folds in ∂M are null singular points. Therefore by Theorem 2.20 we get that: The following identity holds where Ω 12 is a curvature 2-form.   We can also get easily the generalization of Proposition 3.7. in [36] by the Gauss-Bonnet formulas.
where M + f (respectively M − f ) is the set of regular points at which f preserves (respectively reverses) the orientation, K is the Gaussian curvature function on (N, h),κ g is a geodesic curvature, |f * dA h | is the pull-back of the Riemannian measure of (N, h) and where D is the Levi -Civita connection on N , γ is a C 2 -parameterization of the boundary ∂M in the neighborhood of p and σ is a parameterization of Σ in the neighborhood of p.

Geometry of the affine extended wave front
In this section we apply Theorem 2.20 to an affine extended wave front of a planar non-singular hedgehog. Fronts are examples of coherent tangent bundles (see [34]).
Planar hedgehogs are curves which can be parameterized using their Gauss map. A hedgehog can be also viewed as the Minkowski difference of convex bodies (see [23,24,25,26,27]). The non-singular hedgehogs are also known as the rosettes (see [2,30,43]).
Let C be a smooth parameterized curve on the affine plane R 2 , i.e. the image of the C ∞ smooth map from an interval to R 2 . We say that a smooth curve is closed if it is the image of a C ∞ smooth map from S 1 to R 2 . A smooth curve is regular if its velocity does not vanish. A regular curve is called an m-rosette if its signed curvature is positive and its rotation number is m. A convex curve is a 1-rosette. Definition 5.2. An affine λ-equidistant is the following set: The set E 1 2 (C) will be called the Wigner caustic of C. Definition 5.3. The Centre Symmetry Set of C, which we will denote as CSS(C), is the envelope of all chords passing through parallel pairs of C.
If C is a generic convex curve, then the Wigner caustic of C, E λ (C), for a generic λ, and CSS(C) are smooth closed curves with at most cusp singularities ( [1,12,15,16]), the number of cusps of the Wigner caustic and the Centre Symmetry Set of C are odd and not smaller than 3 ( [1,12]), the number of cusps of CSS(C) is not smaller than the number of cusps of E 1 2 (C) ( [7]) and the number of cusps of E λ (C) is even for a generic λ = 1 2 ( [9]). Moreover, cusp singularities of all E λ (C) are lying on smooth parts of CSS(C) ( [15]). In addition, if C is a convex curve, then the Wigner caustic is contained in a closure of the region bounded by the Centre Symmetry Set ( [3], see Fig. 2). The Wigner caustic also appears in one of the two constructions of bi-dimensional improper affine spheres. This construction can be generalized to higher even dimensions ( [4]). The oriented area of the Wigner caustic improves the classical planar isoperimetric inequality and gives the relation between the area and the perimeter of smooth convex bodies of constant width ( [41,42,43]).
Recently the properties of the middle hedgehog, which is a generalization of the Wigner caustic in the case of non-smooth convex bodies, were studied in [38,39].
Note that, when R m is a circle on the plane, then E(R m ) is the double cone, which is a smooth manifold with the nonsingular projection π everywhere, but at its singular point, which projects to the center of the circle (the center of symmetry).
We will study the geometry of E(R m ) through the support function of R m ( [2,43]). Take a point O as the origin of our frame. Let θ be the oriented angle from the positive x 1 -axis. Let p(θ) be the oriented perpendicular distance from O to the tangent line at a point on R m and let this ray and x 1 -axis form an angle θ. The function p is a single valued periodic function of θ with period 2mπ and the parameterization of R m in terms of θ and p(θ) is as follows Then, the radius of curvature ρ of R m is in the following form ρ(θ) = ds dθ = p(θ) + p (θ) > 0, (5.2) or equivalently, the curvature κ of R m is given by In Fig. 3 we illustrate (with different opacities) the surface E(R 1 ), where R 1 is an oval represented by the support function p(θ) = 11 − 0.5 cos 2θ + sin 3θ. We also present the following curves:
Corollary 5.7. Let R m be a generic m-rosette. Then CSS(R m ) which is created from singular points of E λ (R m ) for λ ∈ [0, 1] consists of exactly 2 0.5m − 1 branches.
Proof. It is a consequence of Remark 5.6.
Lemma 5.8. Let C be a closed smooth curve with at most cusp singularities and let the rotation number of C be m. If m is an integer, then the number of cusp singularities is even. If m is the form 0.5d, where d is an odd integer, then the number of cusp singularities is odd. Proof. A continuous normal vector field to the germ of a curve with the cusp singularity is directed outside the cusp on the one of two connected regular components and is directed inside the cusp on the other component as it is showed in Fig. 4. If m is an integer, then the number of cusps of C is even, otherwise is odd. Proposition 5.9. Let R m be a generic m-rosette. If k = m and m is an odd number, then the number of cusp singularities of CSS k (R m ) is odd and not smaller than the number of cusp singularities of E 0.5,k (R m ), otherwise the number of cusp singularities of CSS k (R m ) is even and not smaller than the number of cusp singularities of E 0.5,k (R m ), which is even and positive.
Proof. The parity of the number of cusp singularities of CSS k (R m ) is a consequence of (5.7) and Lemma 5.8.
Let m be even and k m or m be odd and k < m. By Theorem 2.9 in [43] we know that E 0.5,k (R m ) has at least 2 cusp singularities. Because the cusp in E 0.5 appears when κ(a) κ(b) = 1 and cusp in CSS appears when κ(a) where a, b is a parallel pair and is used to denote the derivative with respect to the parameter along the corresponding segment of a curve. Therefore by Roll's theorem we get that the number of cusp singularities of CSS k (R m ) is not smaller than the number of cusp singularities of E 0.5,k (R m ). The same arguments works when m is odd and k = m.
Let E k (R m ) for k = 1, . . . , m be a branch of E(R m ) which has the following parameterization f k (λ, θ) = (λ, λγ(θ) + (1 − λ)γ(θ + kπ)) . (5.8) We use the following notation: In Fig. 5 and Fig. 6 we illustrate (with different opacities) the branches E 1 (R 2 ) and E 2 (R 2 ), respectively, where R 2 is a 2 -rosette represented by the support function p(θ) = 11 + sin θ 2 − 7 cos 3θ 2 − 1 2 sin 2θ.  It is well known that the Gaussian curvature of a ruled surface at a non-singular point is non-positive. By direct calculation we get the following proposition. Proof. We use (5.8) as the parameterization of E k (R m ). Let us notice that f k is singular if and only if (f k ) λ × (f k ) θ = 0. This condition is equivalent to (5.10). By Fact 1.5 in [35] we get (5.11) and (5.12). By direct calculation we get (iv) (see Definition 2.2).
Corollary 5.12. Let R m be an m-rosette. Then the number of branches of E(R m ) is equal to m and a branch E k (R m ) is singular if and only if k is odd.
Proposition 5.13. Let R m be an m -rosette and let p be a non-singular point of E k (R m ). Then the Gaussian curvature of E k (R m ) at p is equal to 0.
Proof. The surface is parameterized by (5.8).
At a non-singular point (λ, θ) the Gaussian curvature K of E k is equal to Since (f k ) λλ = 0 and vectors (f k ) θ and (f k ) λθ are linearly dependent, the Gaussian curvature K k at a non-singular point of E k is equal to zero.
The map (f k , ν k ) is a front. Then the coherent tangent bundle E f k over M has the following fiber at p ∈ M The set of singular points Σ k is parameterized by (λ k (θ), θ), where . Let us notice that Furthermore, if the function λ k (θ) has a local minimum, then the point (λ k (θ), θ) is a negative peak and if λ k (θ) has a local maximum, then this point is a positive peak. See Fig. 7. .

(5.15)
Proof. Let s k (λ, θ) := λρ(θ)−(1−λ)ρ(θ+kπ). Then (5.15) follows from the formulâ Proposition 5.17. Let R m be a generic m-rosette. Let k be an odd number. Then the singular curvature on a cuspidal edge at a point κ(θ) κ(θ) + κ(θ + kπ) , θ is equal to where κ CSS k (θ) is a the curvature of CSS k (R m ), which is given by the following formula: Proof. It is a direct consequence of the formula of the singular curvature and the formula of the curvature of the Centre Symmetry Set (see Lemma 2.6 in [9]). By Theorem 1.6 in [35] we know that the singular curvature does not depend on the orientation of the parameter θ, the orientation of M , the choice of ν, nor the orientation of the singular curve. The sign of the singular curvature have a geometric interpretation, if the singular curvature is positive (respectively negative) then the cuspidal edge is positively (respectively negatively) curved. See Fig. 8.
We find a formula which gives us the relation between the total singular curvature on set of singular points and the total geodesic curvature on the boundary of M . The integrals in (5.18)-(5.21) can be seen as integrals on f k (Σ k ) and f k ({λ}×S 1 ) = {λ} × E k,λ (R m ) since the arclength measure, the singular curvature and E f kgeodesic curvature are defined with respect to the first fundamental form ds 2 which is the pullback of metric ·, · on E k (R m ) ⊂ R 3 .
Furthermore directly by (2.9) we get the following proposition.
Proposition 5.20. Let k be an odd number. Let R m be a generic m -rosette. Let C + (respectively C − ) be a simple regular curve in M + (respectively M − ) which is smoothly homotopic to {1} × S 1 (respectively {0} × S 1 ). If the orientations of C + , C − are opposite then where dτ denote the arc length measure.
By Theorem 5.18 we can get the relation between integrals of the curvature of the Centre Symmetry Set, the curvature of the rosette and the width of the rosette.
Remark 5.24. The condition that w is of class C 2 (R) cannot be omitted. We can consider the function w(θ) = 1 + |x − π| 3 and the interval [0, 2π]. One can check that relation (5.24) does not hold.
Remark 5.25. By (5.14) the odd coefficients of the Fourier series of a width of an oval vanish. Thus a function w(θ) = 2 + sin 3θ is not a width of any oval but it satisfies the relation (5.24).

(5.28)
The cusp-directional torsion is defined by the formula κ t (t) = det (γ , f ηη (γ), (f ηη (γ)) ) In [35] it was shown that a point p is a generic cuspidal edge if and only if κ ν (p) does not vanish. The curvature κ c is exactly the cuspidal curvature of the cusp of the plane curve obtained as the intersection of the surface by the plane H, where H is orthogonal to the tangential direction at a given cuspidal edge ( [29]). For the geometrical meaning of the cusp-directional torsion (5.29) see Proposition 5.2 in [28] and for global properties see Appendix A in [28]. By straightforward calculations we obtain the following lemma.

(5.32)
Proposition 5.28. Let R m be a generic m-rosette. Let k be an odd number. Then (i) cuspidal edges of E k (R m ) are not generic, (ii) the mean curvature of E k (R m ) is not bounded, (iii) the total torsion of the image of singular curveγ k (θ) for θ ∈ [0, 2kπ] is equal to 2nπ for some integer n, i.e. γ k τ k (s)ds = 2nπ, (5.33) where γ k is the singular curve, τ k is a torsion ofγ k and s is the arc length parameter ofγ k .
Proof. (i) It is a consequence of (5.30).
(ii) Since κ k,c (p) = 0 for any cuspidal edge p ∈ Σ, then by Proposition 2.8 in [29] we get that the mean curvature of CSS k (R m ) is not bounded. (iii) From Appendix A in [28] we know that in our case there is the following equality γ k κ k,t (s)ds = γ k τ k (s)ds − 2nπ.
It is easy to see that γ k κ k,t (s)ds = 0. Hence (5.33) holds.
Remark 5.29. For the geometrical meaning of the number n in Corollary 5.28(iii) see Appendix A in [28]. In [31] authors show that the total torsion of a closed line of curvature on a surface (i.e. a closed curve on a surface whose tangents are always in the direction of a principal curvature) is lπ, where l is an integer. Furthermore they show that if the total torsion of a closed curve is lπ for an integer l, then this curve can appear as a line of curvature on a surface and if l is even, then it can appear as a line of curvature on a surface of genus 1.