The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

In Saji et al. (J Math 62:259–280, 2008; Ann Math 169:491–529, 2009; J Geom Anal 222):383–409, 2012) 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 a special version of 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 rosette.


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 [19,29,30,[35][36][37][38]. In particular in [35,36] the results of Kossowski [20,21] and Langevin et al. [24] were The work of W. Domitrz  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 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 , dÂ (respectively d A) 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 [35] and Sect. 2 for details). Saji et al. also found several interesting applications of the above formulas (see especially [37]). 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 affineextended wave fronts of the planar non-singular hedgehogs (rosettes). As a result, we obtain a new proof of a special version of Fukuda-Ishikawa's theorem [12] 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 5.24 and Conjecture 5.28).
In Sect. 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 Sect. 3. We apply this theorem to study the global properties of maps between compact-oriented surfaces with boundary in Sect. 4. The last section contains the results on the geometry of the affine-extended wave fronts of rosettes.
E is an orientable vector bundle over M of rank 2, ·, · is a metric, D is a metric connection on (E, ·, · ) and ψ is a bundle homomorphism ψ : T M → E, such that for any smooth vector fields X , Y on M (2.1) 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. (iii) the set ∩ 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 on, we assume that the set of singular points admits at most peaks, i.e. 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 , 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 ∈ . Since 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 T p M : 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. Let us assume that λ 1 ( p) = 0. Then λ 2 ( p) > 0. Thus, there exists a neighborhood V of p such that for every point q ∈ V the map I q has two distinct 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 v-curves) give the λ 1 -eigendirections (respectively λ 2 -eigendirections), because the eigenvectors of eigenvalues λ 1 (q), λ 2 (q) depends smoothly on q. Such a local coordinate system (U ; u, v) is called a g-coordinate system at p.
The E-initial vector of γ at p is the following limit if it exists.

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 [35]). 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 at p and γ : since the vectorsσ (0) andγ (0) span the space T p M and dλ ψ p = 0. On the other hand, since λ ψ γ (t) = μ ψ u γ (t) , ψ v γ (t) and ψ u γ (0) = 0, we get the following: By (2.5) and (2.6) we get that D d , ψ v γ (0) are linearly independent.
The vector fieldγ can be written in the following formγ (t) =u(t) ∂ ∂u +v(t) ∂ ∂v , where u(t) = t(a + h(t)), v(t) = t 2 g(t), a = 0 and h, g are some functions such that Now, we will prove the formula (2.3).

Definition 2.10
Let γ 1 and γ 2 be two C 1 -regular curves emanating from p such that E-initial vectors of γ 1 and γ 2 at p exist. Then the angle is called the angle between the initial vectors of γ 1 and γ 2 at p.
We generalize the definition of singular sectors from [35] 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 is called a singular sector at p if it satisfies the following conditions (i) the boundary of ∩ U consists of σ 1 , σ 2 and the boundary of U . (ii) ∩ = ∅.
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 ∩ = {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 γ j−1 and γ j is not greater than π .
Let 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: 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 .
The interior angle of the singular sector is n j=1 If is a singular sector at a singular point p then is contained in M + or M − . The singular sector is called positive (respectively negative) if ⊂ M + (respectively ⊂ 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.
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 [35]. 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) ∈ 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( d dτ 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 { d dτ 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 [35] 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 [35] 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.
Then, the geodesic curvature measureκ g dτ is continuous on [0, ε) , where dτ is the arclength measure with respect to the first fundamental form ds 2 .
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 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 [35,36]). 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 admits at most peaks. If the set of singular points is transversal to the boundary ∂ M, then
where dτ is the arc length measure, P + (respectively P − ) is the set of positive (respec- 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 [35]. 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 is contained in the set of singular points and the set σ ((a, b)) does not contain a peak.
the set σ ((a, b)) does not contain a singular point and the tangent vectorσ (t) Let U be a domain in M. [35]) Let T ⊂ U be the closure of a simply connected domain T which is bounded by three admissible arcs γ 1 , γ 2 , γ 3 . Let A, B and C be the distinct three boundary points of T which are intersections of these three arcs. Then T is called an admissible triangle on the surface with boundary if it satisfies the following conditions:

Definition 3.3 (See Definition 3.1 in
(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 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.

Proposition 3.4 Let ABC be an admissible triangle on the surface with boundary such that A is an A 2 -point, AB ⊂ ∂ M and ABC \ AC lies in M + or in M − . Suppose that the boundary ∂ M is transversal to 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 ⊂ or the interior angle ∠B AC with respect to the metric g is greater than Fig. 1 A decomposition of the triangle ABC into admissible triangles π 2 , we decompose the triangle ABC into admissible triangles AB D and ADC such that the interior angle ∠B AD 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 [35], so it is enough to prove the formula (3.1) for the triangle AB D.
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 2regular arcs starting at D. Since the interior angle ∠B AD is in (0, π 2 ) and B D, AD are transversal at D, restricting the image of this family to the triangle AB D, we obtain a family of C 2 -regular curves where ε ∈ [0, 1] and: Since A ε B D for ε > 0 is an admissible triangle, then by Theorem 3.3 in [35] we get that Since AB D is admissible andκ g is bounded on both AB and AD, by taking the limit as ε → 0 + , we have that This completes the proof.
Proof By the definition of Euler's characteristic we get that Furthermore, it is easy to verify that By Lemma 3.6 we get that Hence we get the following: Similarly we get that where It is easy to see that Proof We know that 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

is the set of positive (respectively negative) singular points in ∩ ∂ 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 ∪ ∂ 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 [35] we get that

Applications of the Gauss-Bonnet Formulas to Maps
As a corollary of Theorem 2.20 we get a special version of Fukuda-Ishikawa's theorem (Theorem 1.1 in [12], see also [22]), which is the generalization of Quine's formula (Theorem 1 in [33]) for surfaces with boundary (see also Proposition 3.6 in [37]). We assume that the set of singular points of a map is transversal to the boundary of a surface.

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, d f ) is a coherent tangent bundle on M (see [37]). 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 12 is a curvature 2-form. Furthermore, it is well known that M f * 12 = deg( f ) N 12 (see for instance Remark 1 in [11] page 111). On the other hand, we have N 12 = N K N d A, where K N is the Gaussian curvature of N . By the Gauss-Bonnet theorem for N we get We can also get easily the generalization of Proposition 3.7 in [37] by the Gauss-Bonnet formulas. (N , h) be an oriented Riemannian 2-manifold, let M be a compact oriented 2-manifold with boundary. Let f : M → N be a C ∞ -smooth map whose set of singular points consists of folds and cusps and is transversal to ∂ M. Then the total singular curvature κ s dτ with respect to the length element dτ (with respect to h) on the set of singular points is bounded, and satisfies the following identity

Proposition 4.2 Let
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 * d A 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 -regular parameterization of the boundary ∂ M in the neighborhood of p and σ is a parameterization of 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.
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 closed 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. pair of points a, b ∈ C (a = b) is called a parallel pair if the tangent lines to C at a and b are parallel.

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.
A chord passing through a parallel pair a, b ∈ C is the following set:

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 [1,13,15,17], the number of cusps of the Wigner caustic and the centre symmetry set of C are odd and not smaller than 3 [1,13], the number of cusps of CSS(C) is not smaller than the number of cusps of E 1 2 (C) [5] and the number of cusps of E λ (C) is even for a generic λ = 1 2 [10]. 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 [42][43][44]. 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 [39,40].

Definition 5.4
The extended affine space is the space R 3 e = R × R 2 with coordinate λ ∈ R (called the affine time) on the first factor and a projection on the second factor denoted by π : R 3 e (λ, x) → x ∈ R 2 .
, each embedded into its own slice of the extended affine space.
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 [2,44]. 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 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 − 1 2 cos 2θ + sin 3θ . We also Fig. 3 The affine extended wave front of an oval present the following curves: Let be a set of singular points of E. It is well known that π( (E(R 1 ))) = CSS(R 1 ) and the map (E(R 1 )) p → π( p) ∈ CSS(R 1 ) is the double covering of CSS(R 1 ).
Let E k (R m ) for k = 1, . . . , m be a branch of E(R m ) which has the following parameterization We use the following notation: In Figs. 4 and 5 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 . Directly by Definition 5.5 we get the following proposition.

Proposition 5.6 Every branch of E(R m ) is a ruled surface.
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.
We use (5.4) 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 (5.6). By Fact 1.5 in [36] we get (5.7) and (5.8). [15] there exists an open and dense subset of the space of rosettes such that the affine extended wave front E(R m ) has only A 2 and A 3 singularities (cuspidal edges and swallowtails) for any rosette R m in this subset. Thus, by Proposition 5.7 a rosette R m is called generic if there do not exist θ and k ∈ {1, · · · , m} such that

Remark 5.8 By Theorem 3.3 in
By direct calculation we get the following proposition (see also Definition 2.2).

Proposition 5.9
If R m is generic then every singular point of E(R m ) is non-degenerate.

Remark 5.10
In [6,10,44] we study in details the geometry of affine λ-equidistants of rosettes. We show among other things that there exist m branches of E 1 2m − 1 branches of E λ (R m ) for λ = 0, 1 2 , 1. Let E 1 2 ,k (R m ) for k = 1, 2, . . . , m denote different branches of E 1 2 (R m ) and let E λ,k (R m ) for k = 1, 2, . . . , 2m − 1 denote different branches of E λ (R m ) for λ = 0, 1 2 , 1. Then, the support function of E 1 2 ,k (R m ) for k = 1, . . . , m is in the form (5.10), the support function of E λ,k (R m ) for k = 1, 2, . . . , m (respectively k = m + 1, m + 2, . . . , 2m − 1) in the form (5.11) (respectively in the form (5.12)), where Let γ λ,k denote the parameterization of E λ,k in terms of the support function accordingly to (5.10), (5.11) and (5.12), respectively. Furthermore each branch of E λ (R m ), except E 1 2 ,m (R m ), has the rotation number equal to m. The rotation number of . . , m + 2 1 2 m − 1 can admit cusp singularities and branches E 1 2 ,k (R m ) for k = 1, 3, . . . , 2 1 2 m − 1 has cusp singularities. By [13] we known that if a, b is parallel pair of R m and R m is parameterized at a and b in different directions and κ(a), κ(b) denote the signed curvatures of R m at a and b, respectively, then the point aκ 1 +bκ 2 κ 1 +κ 2 , which is lying on the line between a and b, belongs to CSS(R m ). Proof It is a consequence of Remark 5.10.
Let CSS k (R m ) for k = 1, 3, . . . , 2 1 2 m − 1 denote a branch of CSS(R m ). Then, the parameterization of CSS k (R m ) is in the following form 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 shown in Fig. 6. If m is an integer, then the number of cusps of C is even, otherwise is odd. In Figs. 4 and 5 we present two branches of E(R 2 ): E 1 (R 2 ) and E 2 (R 2 ), respectively.

Proposition 5.15 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 (5.4).
At a non-singular point (λ, θ ) the Gaussian curvature K of E k is equal to (5.14) 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 k is parameterized by (λ k (θ ), θ ), where λ k (θ ) = κ(θ) κ(θ)+κ(θ+kπ) . 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.
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 [10]).
By Theorem 1.6 in [36] 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 (5.19)-(5.22) can be seen as integrals on f k ( k ) and f k ({λ} × S 1 ) = {λ} × E k,λ (R m ) since the arclength measure, the singular curvature and E f k -geodesic 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 .
where the orientations of S 1 in the integrals on the left hand sides and the right-hand sides are opposite in the above formulas, C = k ∩ ({ 1 2 } × S 1 ), dτ is the arclength measure and is a front. It is easy to see that χ(M + λ ) = 0 and χ(M − λ ) = # P + − # P − is the number of cusps of E f k M λ (that is # k ∩ ({λ} × S 1 ) ). Since every point p ∈ k ∩ ∂ M λ is a null singular point, by Theorem 2.20 (see (2.9)) we get (5.20).
The angle between initial vectors (see Definition 2.5) of the singular curve at p and of the boundary curve at p is α + ( p) (see Theorem 2.20). By Proposition 2.7 and Proposition 2.8 we get (5.23).
Furthermore, directly by (2.9) we get the following proposition.

Proposition 5.22
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 5.20 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.

Theorem 5.24
Let k be an odd number and let R m be a generic m-rosette. Then 2mπ 0 w k (θ ) dθ.

Remark 5.25
Since w k (θ ) = sinh(C 1 θ + C 2 ) for C 1 , C 2 ∈ R is the general solution of w k (θ ) , (5.26) the only periodic solution of (5.26) is a constant function. Therefore, the relation (5.25) is naively fulfilled only for rosettes of constant k-width.

Remark 5.26
The condition that w is C 2 -smooth cannot be omitted. We can consider the function w(θ) = 1 + |x − π | 3 and the interval [0, 2π ]. One can check that relation (5.25) does not hold.

Remark 5.27
By (5.15) 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 (5.25).