The Gauss-Bonnet theorem

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

Wojciech Domitrz, Michał Zwierzyński Faculty of Mathematics and Information Science
Warsaw University of Technology
ul. Koszykowa 75
00-662 Warszawa
Abstract.

In [34, 35, 36] 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.

Key words and phrases:
coherent tangent bundle, wave front, Gauss-Bonnet formula
2010 Mathematics Subject Classification:
Primary 57R45, Secondary 53A05
The work of W. Domitrz and M. Zwierzyński was partially supported by NCN grant no. DEC-2013/11/B/ST1/03080.

1. 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 over a compact surface whose set of singular points admits at most peaks:

 (1.1) 2πχ(M) =∫MKdA+2∫Σκsdτ, (1.2) 12π∫MKd^A =χ(M+)−χ(M−)+#P+−#P−.

In the above formulas is the Gaussian curvature, is the singular curvature, is the arc length measure on , (respectively ) is the signed (respectively unsigned) area form, (respectively ) is the set of regular points in , where (respectively ), (respectively ) 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.

2. 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]).

Definition 2.1.

Let be a -dimensional compact oriented surface (possibly with boundary). A coherent tangent bundle over is a -tuple , where is an orientable vector bundle over of rank , is a metric, is a metric connection on and is a bundle homomorphism

 ψ:TM→E,

such that for any smooth vector fields , on

 (2.1) DXψ(Y)−DYψ(X)=ψ([X,Y]).

The pull-back metric is called the first fundamental form on . Let denote the fiber of at a point . If is not a bijection at a point , then is called a singular point. Let denote the set of singular points on . If a point is not a singular point, then is called a regular point. Let us notice that the first fundamental form on is positive definite at regular points and it is not positive definite at singular points.

Let be a smooth non-vanishing skew-symmetric bilinear section such that for any orthonormal frame on . The existence of such is a consequence of the assumption that is orientable. A co-orientation of the coherent tangent bundle is a choice of . An orthonormal frame such that (respectively ) is called positive (respectively negative) with respect to the co-orientation .

From now on, we fix a co-orientation on the coherent tangent bundle.

Definition 2.2.

Let be a positively oriented local coordinate system on . Then (respectively ) is called the signed area form (respectively the unsigned area form), where

 λψ:=μ(ψu,ψv),ψu:=ψ(∂∂u),ψv:=ψ(∂∂v).

The function is called the signed area density function on .

The set of singular points on is expressed as

 Σ∩U:={p∈U:λψ(p)=0}.

Let us notice that the signed and unsigned area forms, and , give globally defined -forms on and they are independent of the choice of positively oriented local coordinate system . Let us define

 M+:={p∈M∖Σ ∣∣ d^Ap=dAp},   M−:={p∈M∖Σ ∣∣ d^Ap=−dAp}.

We say that a singular point is non-degenerate if does not vanish at . Let be a non-degenerate singular point. There exists a neighborhood of such that the set is a regular curve, which is called the singular curve. The singular direction is the tangential direction of the singular curve. Since is non-degenerate, the rank of is . The null direction is the direction of the kernel of . Let be the smooth (non-vanishing) vector field along the singular curve which gives the null direction.

Let be the exterior product on .

Definition 2.3.

Let be a non-degenerate singular point and let be a singular curve such that . The point is called an -point (or an intrinsic cuspidal edge) if the null direction at (i.e. ) is transversal to the singular direction at (i.e. ). The point is called an -point (or an intrinsic swallowtail) if the point is not an -point and

 ddt(˙σ(t)∧η(t))|t=0≠0.
Definition 2.4.

Let be a singular point which is not an -point. The point is called a peak if there exists a coordinate neighborhood of such that:

1. if then is an -point;

2. the rank of the linear map at is equal to ;

3. the set consists of finitely many -regular curves emanating from .

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 admits at most peaks, i.e. consists of -points and peaks.

Furthermore let us fix a Riemannian metric on . Since the first fundamental form degenerates on , there exists a -tensor field on such that

 ds2(X,Y)=g(IX,Y),

for smooth vector fields on . We fix a singular point . Since admits at most peaks, the point is an - point or a peak. Let , be the eigenvalues of . Since the kernel of is one dimensional, the only one of , vanishes. Thus there exists a neighborhood of such that for every point the map has eigenvalues , , such that . Furthermore there exists a coordinate neighborhood of such that is a subset of and the - curves (respectively - curves) give the - eigendirections (respectively - eigendirections). Such a local coordinate system is called a -coordinate system at .

Definition 2.5.

Let be a -regular curve on such that . The -initial vector of at is the following limit

 (2.2) Ψγ:=limt→0+ψ(˙γ(t))|ψ(˙γ(t))|∈Ep

if it exists.

Remark 2.6.

If is a regular point of then the - initial vector of at is the unit tangent vector of at with respect to the first fundamental form .

Proposition 2.7 (Proposition 2.6 in [34]).

Let be a - regular curve emanating from an - point or a peak such that is a not a null vector or is a singular curve. Then the - initial vector of at exists.

Since we study coherent tangent bundles over surfaces with boundary, we also consider a curve on the boundary which is tangent to the null direction at a singular point on the boundary. We prove that in this case the -initial vector of at exists if the singular direction is transversal to the boundary at .

Proposition 2.8.

Let be a coherent tangent bundle over an compact oriented surface with boundary. Let be an -point in the boundary . If the boundary is transversal to at and is a -regular curve such that , and is a null direction, then the -initial vector of at exists, , and

 (2.3) Ψγ=Dddt(ψ(˙γ(t)))∣∣t=0∣∣∣Dddt(ψ(˙γ(t)))∣∣t=0∣∣∣∈Ep.
Proof.

Let be a singular curve such that . Let be a -coordinate system at i.e. the null direction at is spanned by . Since , we get that

 (2.4) ddt(λψ(σ(t)))∣∣t=0=dλψ∣∣p⋅˙σ(0)=0.

Let us notice that

 (2.5) ddt(λψ(γ(t)))∣∣t=0=dλψ∣∣p⋅˙γ(0)≠0

since the vectors and span the space and .

On the other hand, since and , we get the following:

 (2.6) =μ(Dddt(ψu(γ(t))∣∣t=0),ψv(γ(0)))

By (2.5) and (2.6) we get that , are linearly independent.

The vector field can be written in the following form , where , , and , are some functions such that . Similarly since and we can write , where and .

Now we will prove the formula (2.3).

 limt→0+ψ(˙γ(t))∣∣ψ(˙γ(t))∣∣ =limt→0+˙u(t)ψu(γ(t))+˙v(t)ψv(γ(t))∣∣˙u(t)ψu(γ(t))+˙v(t)ψv(γ(t))∣∣ =limt→0+t((a+h(t)+t˙h(t))ξ(t)+(2g(t)+t˙g(t))ψv(γ(t)))t∣∣(a+h(t)+t˙h(t))ξ(t)+(2g(t)+t˙g(t))ψv(γ(t))∣∣ =aξ(0)+2g(0)ψv(γ(0))|aξ(0)+2g(0)ψv(γ(0))|,

where the expression is non-zero since the vectors and are linearly independent.

Since , the equality (2.3) holds. ∎

Proposition 2.9.

Under the assumptions of Proposition 2.8, if , then

 (2.7) Ψγ=Ψ¯¯¯γ.
Proof.

Since , we get that and in particular . Since

 =Dddt(−ψ(˙γ(−t)))=−Dddt(ψ(˙γ(−t)))=D−ddt(ψ(˙γ(−t))),

the equality (2.7) holds. ∎

Definition 2.10.

Let and be two -regular curves emanating from such that -initial vectors of and at exist. Then the angle

 arccos(⟨Ψγ1,Ψγ2⟩)∈[0,π]

is called the angle between the initial vectors of and at .

We generalize the definition of singular sectors from [34] to the case of coherent tangent bundles over surfaces with boundary.

Let be a (sufficiently small) neighborhood of a singular point . Let and be curves in starting at such that both are singular curves or one of them is a singular curve and the other one is in . A domain is called a singular sector at if it satisfies the following conditions

1. the boundary of consists of , and the boundary of .

2. .

If the peak is an isolated singular point than the domain is a singular sector at , where is a neighborhood of such that . We assume that singular direction is transversal to the boundary of . Therefore there are no isolated singular points on the boundary.

We define the interior angle of a singular sector. If is in , then the interior angle of a singular sector at is the angle of the initial vectors of and at .

While the interior angle of a singular sector may take value greater than if , we can choose for inside the singular sector in a such way that the angel between and is not greater than .

Let be a singular sector at the peak . Then there exists a positive integer and -regular curves starting at satisfying the assumptions of Proposition 2.7 and the following conditions:

1. if then in ,

2. for each there exists a sector domain such that is bounded by and and for ,

3. if the vectors , are linearly independent and form a positively oriented frame for .

If the peak is an isolated singular point then there exist curves satisfying the above assumptions and conditions (i)-(iii). We also put .

The interior angle of the singular sector is

 n∑j=1arccos(⟨Ψγj−1,Ψγj⟩).

If is a singular sector at a singular point then is contained in or . The singular sector is called positive (respectively negative) if (respectively ).

Definition 2.11.

Let be a singular point. Then (respectively ) is the sum of all interior angles of positive (respectively negative) singular sectors at .

Proposition 2.12 (Theorem A in [34]).

Let be a peak. The sum of all interior angles of positive singular sectors at and the sum of all interior angles of negative singular sectors at satisfy

 α+(p)+α−(p) =2π, α+(p)−α−(p) ∈{−2π,0,2π}.
Theorem 2.13.

Let be a singular point. We assume that the singular direction is transversal to the boundary at .

If the null direction is transversal to the boundary at , then

 α+(p)+α−(p) =π, α+(p)−α−(p) ∈{−π,π}.

If the null direction is tangent to the boundary at , then

 α+(p)=α−(p).
Proof.

The first part of this theorem follows from Proposition 2.15 in [34]. By Proposition 2.9 we get the second part. ∎

Definition 2.14.

A peak in is called positive (null, negative, respectively) if (, , respectively).

Definition 2.15.

A singular point in is called positive (null, negative, respectively) if (, , respectively).

Remark 2.16.

It is easy to see that a peak in is not null if is transversal to the singular direction at and an singular point in is null if the null vector at is tangent to .

Definition 2.17.

Let be a peak in . We say that is in the positive boundary (respectively in the negative boundary) if there exists a neighborhood in of such that (respectively ).

Let be a -regular curve on . We assume that if then is transversal to the null direction at . Then the image does not vanish. Thus we take a parameter of such that

 ⟨ψ(˙σ(τ)),ψ(˙σ(τ))⟩≡1,

where .

Definition 2.18.

Let be a section of along such that is a positive orthonormal frame. Then

 ^κg(τ):=⟨Dddτψ(˙σ(τ)),n(τ)⟩=μ(ψ(˙σ(τ)),Dddτψ(˙σ(τ)))

is called the -geodesic curvature of , which gives the geodesic curvature of the curve with respect to the orientation of .

We assume that the curve is a singular curve consisting of -points. Take a null vector field along such that is a positively oriented field along for each . Then the singular curvature function is defined by

 κs(τ):=sgn(dλψ(η(τ)))⋅^κg(τ),

where denotes the sign of the function at . In a general parameterization of , the singular curvature function is defined as follows

 κs(t)=sgn(dλψ(η(t)))⋅μ(ψ(˙σ(t)),Dddtψ(˙σ(t)))|ψ(˙σ(t))|3,

where , .

By Proposition 1.7 in [34] the singular curvature function does not depend on the orientation of , the orientation on , nor the parameter of the singular curve .

By Proposition 2.11 in [34] the singular curvature measure is bounded on any singular curve, where is the arclength measure of this curve with respect to the first fundamental form . Now we prove the following proposition concerning the geodesic curvature measure on the boundary of .

Proposition 2.19.

Let be a -regular curve such that is an -point and the vector is the null vector at . Then the geodesic curvature measure is continuous on , where is the arclength measure with respect to the first fundamental form .

Proof.

The point is a null -point. By Proposition 2.8 we can write that for for sufficiently small , where and . The geodesic curvature in a general parameterization has the following form

 ^κg(t)=μ(ψ(˙γ(t)),Dddtψ(˙γ(t)))|ψ(˙γ(t))|3.

Thus the geodesic curvature measure

 ^κg(τ)dτ=^κg(t)|ψ(˙γ(t))|dt=μ(ζ(t),Dddtζ(t))|ζ(t)|2dt

is bounded and continuous on . It implies that the geodesic curvature measure is continuous on since . ∎

Let be a domain and let be a positive orthonormal frame field on defined on . Since is a metric connection, there exists a unique -form on such that

 DXe1=−ω(X)e2,DXe2=ω(X)e1,

for any smooth vector field on . The form is called the connection form with respect to the frame . It is easy to check that does not depend on the choice of a frame and gives a globally defined -form on . Since is a metric connection and it satisfies (2.1) we have

 dω=Kd^A={KdA on M+,−KdA on M−,

where is the Gaussian curvature of the first fundamental form (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 be a coherent tangent bundle on a smooth compact oriented surface with boundary whose set of singular points admits at most peaks. If the set of singular points is transversal to the boundary , then

 (2.8) 2πχ(M) =∫MKdA+2∫Σκsdτ +∫∂M∩M+^κgdτ−∫∂M∩M−^κgdτ−∑p∈null(Σ∩∂M)(2α+(p)−π), (2.9) ∫MKd^A +∫∂M^κgdτ=2π(χ(M+)−χ(M−))+2π(#P+−#P−) +π(#(Σ∩∂M)+−#(Σ∩∂M)−)+π(#P∂M+−#P∂M−),

where is the arc length measure, (respectively ) is the set of positive (respectively negative) peaks in , (respectively , ) is the set of positive (respectively negative, null) singular points in , (respectively ) is the set of peaks in the positive (respectively negative) boundary.

3. 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.

Definition 3.1.

A curve () is admissible on the surface with boundary if it satisfies one of the following conditions:

1. is a - regular curve such that does not contain a peak, and the tangent vector () is transversal to the singular direction, the null direction if and is transversal to the boundary if .

2. is a - regular curve such that the set is contained in the set of singular points and the set does not contain a peak.

3. is - regular curve such that the set is contained in the boundary , the set does not contain a singular point and the tangent vector () is transversal to the singular direction if .

Remark 3.2.

A curve is admissible in the sense of Definition 2.12 in [34] if it satisfies conditions (1) or (2) in Definition 3.1. For the purpose of this paper we add (3) in Definition 3.1 and the transversality of the admissible curve to the boundary in (1).

Let be a domain in .

Definition 3.3 (see Definition 3.1 in [34]).

Let be the closure of a simply connected domain which is bounded by three admissible arcs , , . Let , and be the distinct three boundary points of which are intersections of these three arcs. Then is called an admissible triangle on the surface with boundary if it satisfies the following conditions:

1. admits at most one peak on .

2. the three interior angles at , and with respect to the metric are all less than .

3. if for is not a singular curve, it is -regular, namely it is a restriction of a certain open -regular arc.

We write and we denote by

 ¯¯¯¯¯¯¯¯BC:=γ1, ¯¯¯¯¯¯¯¯CA:=γ2, ¯AB:=γ3

the regular arcs whose boundary points are , respectively.

We give the orientation of compatible with respect to the orientation of . We denote by the interior angles (with respect to the first fundamental form ) of the piecewise smooth boundary of at and , respectively if and are regular points.

If is a singular point and is a -coordinate system at , then we set (see Proposition 2.15 in [34])

 ∠A:={πif the u−% curve passing through A separates ¯¯¯¯¯¯¯¯AB and ¯¯¯¯¯¯¯¯AC,0otherwise.

Let be an admissible curve. We define a geometric curvature in the following way:

 ~κg(t)=⎧⎪⎨⎪⎩^κg(t)ifσ(t)∈M+,−^κg(t)ifσ(t)∈M−,κs(t)itσ(t)∈Σ,

where is the geodesic curvature with respect to the orientation of and is the singular curvature.

Proposition 3.4.

Let be an admissible triangle on the surface with boundary such that is an -point, and lies in or in . Suppose that the boundary is transversal to at and let be a null direction at . Then

 (3.1) ∠A+∠B+∠C−π=∫∂ΔABC~κgdτ+∫ΔABCKdA.
Proof.

Without loss of generality, let us assume that lies in . If the arc or the interior angle with respect to the metric is greater than , we decompose the triangle into admissible triangles and such that the interior angle with respect to the metric is in the interval and the arc is transversal to the arc at , see Fig. 1. The formula (3.1) for follows from Theorem 3.3 in [34], so it is enough to prove the formula (3.1) for the triangle .

We can take the arc and rotate it around with respect to the canonical metric on the -plane. Then we obtain a smooth one-parameter family of -regular arcs starting at . Since the interior angle is in and , are transversal at , restricting the image of this family to the triangle , we obtain a family of -regular curves

 γε:[0,1]→ΔABD,

where and:

1. parameterizes and , ,

2. for all ,

3. the correspondence gives a subarc of . We set , where .

Since for is an admissible triangle, then by Theorem 3.3 in [34] we get that

 ∠Aε+∠B+∠AεDB−π=∫∂ΔAεBD~κgdτ+∫ΔAεBDKdA.

Since is admissible and is bounded on both and , by taking the limit as , we have that

 limε→0+∠Aε+∠B+∠D−π=∫∂ΔABD~κgdτ+∫ΔABDKdA.

By Proposition 2.8 we have

 (3.2) limε→0+∠Aε =limε→0+arccos⟨ψ(dγε(t)dt)∣∣t=0,ψ(dσ(ε)dε)⟩∣∣ψ(dγε(t)dt)∣∣t=0∣∣⋅∣∣ψ(dσ(ε)dε)∣∣=arccos⟨Ψγ0,Ψσ⟩.

This completes the proof. ∎

Remark 3.5.

By Theorem 3.3 in [34] and Proposition 3.4 the equality (3.1) holds for any admissible triangle on a surface with boundary.

Let , , , respectively, denote the closure of a subset of , the interior of and the boundary of , respectively.

Let us triangulate by admissible triangles such that each point in the set is a vertex, where is the set all peaks in . Let , and , respectively, denote the set of all triangles, the set of all edges and the set all of vertices in the given triangulation, respectively.

Lemma 3.6.

The following relation holds:

 #{v∈V | v∈(M+)∘}=χ(¯¯¯¯¯¯M+)+12#{Δ∈T | Δ⊂¯¯¯¯¯¯M+}+ +12#{e∈E | e⊂∂M+}−#{v∈V | v∈∂M+∖P⋆}−#P⋆.
Proof.

By the definition of Euler’s characteristic we get that

 (3.3)

Furthermore, it is easy to verify that

 (3.4) #{e∈E | e⊂¯¯¯¯¯¯M+}=32#{Δ∈F | Δ⊂¯¯¯¯¯¯M+}+12#{e∈E | e⊂∂M+}

and

 (3.5) #{v∈V | v∈(M+)∘}= #{v∈V | v∈¯¯¯¯¯¯M+}

Combining together (3.3), (3.4) and (3.5) we end the proof. ∎

Let us define the sum by .

Then

 S+= 2π#{v∈V | v∈(M+)∘}+π#{v∈V | v∈∂M+∖P⋆} +∑p∈P⋆α+(p)−π#{Δ∈T | Δ⊂¯¯¯¯¯¯M+}.

By Lemma 3.6 we get that

 S+ =2πχ(¯¯¯¯¯¯M+)+π#{e∈E | e∈∂M+}−