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The Gauss-Bonnet type formulas for surfaces with singular points

Masaaki Umehara (Osaka University)

最近の筆者と

佐治健太郎氏(岐阜大),

山田光太郎氏(東工大)

との共同研究に関連した内容.

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1. Gaussian curvature K

Figure 1. (Surfaces of K < 0 and K > 0

Lp(r) =the length of the geod. circle of radius r at p

K(p) = lim r→0 3 π  2πr − Lp(r) r3  .

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2. The Gauss-Bonnet formula (local version)

B

C A



ABC KdA = ∠A + ∠B + ∠C − π,

∠A + ∠B + ∠C < π (if K < 0), ∠A + ∠B + ∠C > π (if K > 0).

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3. The Gauss-Bonnet formula (global version)

Polygonal division of closed surfaces.

(The Gauss-Bonnet formula) (3.1)  M2 KdA = 2πχ(M2), where χ(M2) = V − E + F is the Euler number of the surface M2.

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4. Parallel surfaces An immersion

f = f (u, v) : (U ; u, v) → R3 (U ⊂ R2),

the unit normal vector ν(u, v) := fu(u, v) × fv(u, v)

|fu(u, v) × fv(u, v)|

.

For each real number t,

ft(u, v) = f (u, v) + tν(u, v)

is called a parallel surface of f .

p is a singular pt of ft ⇐⇒ (ft)u(p) × (ft)v(p) = 0.

Figure 2. a cuspidal edge and a swallowtail

Cuspidal edges and swallowtails are generic

singular points appeared on parallel surfaces.

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An ellipsoid: x2 + y 2 4 + z2 9 = 1.

Parallel surfaces of the ellipsoid are given as follows:

Figure 3. the cases of t = 0 and t = 1.2

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5. Singular curvature κs

Figure 5. Surfaces of K < 0 and K > 0

The image of cuspidal edges consists of regular curves in R3. We denote it by γ(s), where s is the arclength parameter.

Figure 6. a (−)-cuspidal edge and a (+)-cuspidal edge

the singular curvature κs(s) = ε(s)(geodesic curvature)

= ε(s)| det(ν(s), γ(s), γ(s))| , where ε(s) = ⎧ ⎨ ⎩

1 (if the surface is bounded by a plane at γ(s)),

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6. Generic cuspidal edges A generic cuspidal edge:

the osculating plane = the tangent plane. In this case, K → ±∞ at the same time.

Saji-Yamada-U. [7] K ≥ 0 =⇒ κs ≤ 0. -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -2 -1 0 1 2 1 -0.5 0 0.5

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7. Gauss-Bonnet formula of surfaces with singularity

M2; a compact oriented 2-manifold

f : M2 → R3; a C-map

         having only cuspidal edges and swalowtails

ν : M2 → S2; the unit normal vector field

(U ; u, v); a (+)-oriented local coordinate M2.

the area density function λ := det(fu, fv, ν)

λ(p) = 0 ⇐⇒ p is a singular point

area element dA := |λ|du ∧ dv, signed area element d ˆA := λdu ∧ dv.

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8. Two Gauss-Bonnet formulas

Figure 7. a positive swallowtail

Formulas given by Kossowski(02)and Langevin-Levitt-Rosenberg(95)

M2f KdA + 2  Σf κsds = 2πχ(M2), 2deg(ν) = 1  M2f Kd ˆA = χ(M+) − χ(M) + #SW+ − #SW, M+ := {p ∈ M2 ; dAp = d ˆAp}, M := {p ∈ M2 ; dAp = −d ˆAp}.

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9. singular points of a map between 2-manifolds

Maps between planes

R2  (u, v) → f(u, v) = (x(u, v), y(u, v)) ∈ R2,

Singular points of f ⇐⇒ det 

xu(u, v) xv(u, v)

yu(u, v) yv(u, v)

= 0.

Generic singular points

a fold R2  (u, v) −→ (u2, v) ∈ R2, the singular set u = 0, f (0, v) = (0, v)

a cusp R2  (u, v) −→ (uv + v3, u) ∈ R2, the singular set u = −3v2, f (−3v2, v) = (−2v3,−3v2)

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f = f (u, v)

Singular curves of cuspidal edges and folds

Figure 9. a fold and a cuspidal edge

Singular curves of swallowtails and cusps

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10. Singularities of Gauss maps M2:an oriented compact manifold.

f : M2 → R3,an immersion. Singular points of ν : M2 → S2 ⇐⇒ Kf = 0. ft = 1 t(f + tν), t ∈ R. Then lim t→∞ ft = ν, 2deg(ν) = χ(M+t ) − χ(Mt ) + #SW+t − #SWt . In particular, 2deg(ν) = χ(M2) = χ(M+t ) + χ(Mt ). Hence, 2χ(Mt ) = #SW+t − #SWt .

Taking the limit t → ∞, we have that

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The Gauss map ν satisfies

2χ(M∞) = #SW+ − #SW∞.

If t → ∞, then the cuspidal edge collapses to a fold, and a swallowtail collapses to a cusp. In particular,

#SW+ := #{(+)-cusps of ν}, #SW := #{(−)-cusps of ν}.

Since d ˆAν = KfdAf, dAν = |Kf|dAf, it holds that

M = {p ∈ M2 ; Kf(p) < 0}. Thus (the Bleecker and Wilson formula [1])

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Figure 11. a symmetric torus and its perturbation A deformation of the rotationally symmetric torus

fa(u, v) = (cos v(2 + ε(v) cos u),

sin v(2 + ε(v) cos u), ε(v) sin u), where

ε(v) := 1 + a cos v. a = 0 : the original torus

a = 4/5:χ({K < 0}) = 1

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11. A similar application The following identity holds for f t : M2 → R3.

M2ft

KtdAt + 2



Σft

κsds = 2πχ(M2).

Taking t → ∞, we have that (Saji-Yamada-U. [10])

1  {K<0} KfdAf =  Σν κsds, where

κs := the singular curvature of ν = ±the geodesic curvature of ν

⎨ ⎩

> 0 if ν points Im(ν),

< 0 otherwise. To prove the formula, we apply

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The intrinsic formulation of wave fronts (Saji-Yamada-U. [10])

The definition (E, ,  , D, ϕ) of coherent tangent bundle on Mn: (1) E is a vector bundle of rank n over Mn,

(2) E has a inner product  , ,

(3) D is a metric connection of (E, , ),

(4) ϕ : T Mn → E is a bundle homomorphism s.t.

DXϕ(Y ) − DYϕ(X) = ϕ([X, Y ]),

where X, Y are vector fields on Mn. The pull-back of the metric  , 

ds2ϕ := ϕ∗ ,  is called the first fundamental form of ϕ.

p ∈ Mn; ϕ-singular point ⇐⇒ Ker(ϕp: TpMn → Ep) = {0}.

coherent tangent bundle = generalized Riemannian manifold

When (Mn, g) is a Riemannian manifold, then

E = T Mn, ,  := g, D = ∇g, ϕ = id .

If f : M2 → R3 is a wave front, then

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M2; a compact oriented 2-manifold,

(E,,  , D, ϕ); an orientable coherent tangent bundle,

∃μ ∈ Sec(E∗ ∧ E∗ \ {0})

such that μ(e1, e2) = 1 for (+)-frame {e1, e2}.

The intrinsic definition of the singular curvature

κs := sgn(dλ(η(t)))μ(Dγ n(t), ϕ(γ

))

|ϕ(γ)|3 , where n(t) ∈ Eγ(t) is the unit vector perpendicular to ϕ(γ) on E.

(u, v)a (+)-local coordinate on M2

d ˆA = λdu ∧ dv, dA = |λ|du ∧ dv, λ := μ  ϕ( ∂u), ϕ( ∂v)  . E =) 1  M2 Kd ˆA = χ(M+) − χ(M) + SW+ − SW,  M2 KdA + 2  Σϕ κsdτ = 2πχ(M2). where Σϕ; ϕ-singular set, p ∈ Σϕ is non-degenerate ⇔ dλ(p) = 0,

p ∈ Σϕ; A2-pt (intrinsic cuspidal edge) ⇔ η  γ(0) at p,

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Examples of coherent tangent bundle:

(1) Wave fronts as a hypersurface of Riem. manifold, (2) Smooth maps between n-manifolds

Mn; an orientable manifold,

(Nn, g); an orientable Riemannian manifold

f : Mn → (Nn, g); C∞-map, Ef := f∗T Nn,  ,  := g|E

f,

D; induced connection.

ϕ := df : T Mn −→ Ef := f∗T Nn,

Figure 13. a fold and a cuspidal edge

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An application of the intrinsic G-B formula

f : M2 → R3; a strictly convex surface,

ξ : M2 → R3; the affine normal map. ∇XY = DXY + h(X, Y )ξ,

DXξ = −α(X),

where α : T M2 → T M2.We set

M2 := {p ∈ M2 ; det(αp) < 0}, then (Saji-Yamada-U. [9])

(21)

[1] D. Bleecker and L. Wilson, Stability of Gauss maps, Illinois J. of Math. 22 (1978) 279–289.

[2] M. Kossowski, The Boy-Gauss-Bonnet theorems forC∞-singular surfaces with limiting tangent bundle, Annals of Global Analysis

and Geometry21 (2002), 19–29.

[3] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic 3-space, Pacific J. Math.221 (2005), 303–351.

[4] R. Langevin, G. Levitt and H. Rosenberg, Classes d’homotopie de surfaces avec rebroussements et queues d’aronde dans R3, Canad. J. Math.47 (1995), 544–572.

[5] K. Saji, Criteria for singularities of smooth maps from the plane into the plane and their applications, preprint.

[6] K. Saji, M. Umehara and K. Yamada, Behavior of corank one singular points on wave fronts, Kyushu Journal of Mathematics

62 (2008), 259–280.

[7] K. Saji, M. Umehara and K. Yamada, Geometry of fronts, Ann. of Math.169 (2009), 491-529.

[8] K. Saji, M. Umehara and K. Yamada,Aksingularities of wave fronts, Mathematical Proceedings of the Cambridge Philosophical

Society,146 (2009), 731-746.

[9] K. Saji, M. Umehara and K. Yamada, Singularities of Blaschke normal maps of convex surfaces, C.R. Acad. Sxi. Paris. Ser. I 348 (2010), 665-668.

[10] K. Saji, M. Umehara and K. Yamada, The intrinsic duality on wave fronts, preprint, arXiv:0910.3456.

[11] 梅原雅顕,特異点をもつ曲線と曲面の幾何学,慶應大学数理科学セミナー・ノート38 (2009). (書店マテマティカTel 03-3816-3724, fax: 03-3816-3717)から直接購入可.)

[12] 梅原雅顕・山田光太郎,曲線と曲面,—微分幾何的アプローチ—,裳華房(2002).

Figure 1. (Surfaces of K &lt; 0 and K &gt; 0 )
Figure 2. a cuspidal edge and a swallowtail
Figure 3. the cases of t = 0 and t = 1.2
Figure 7. a positive swallowtail
+5

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