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1 The Gauss-Bonnet type formulas for surfaces with singular points Masaaki Umehara Osaka University 1

2 2 1. Gaussian curvature K Figure 1. (Surfaces of K<0andK>0 L p (r) =the length of the geod. circle of radius r at p 3 K(p) = lim r 0 π ( ) 2πr Lp (r). r 3

3 3 2. The Gauss-Bonnet formula (local version) A C B ABC KdA = A + B + C π, A + B + C <π A + B + C >π (if K<0), (if K>0).

4 4 3. The Gauss-Bonnet formula (global version) Polygonal division of closed surfaces. (The Gauss-Bonnet formula) (3.1) KdA =2πχ(M 2 ), M 2 where χ(m 2 )=V E + F is the Euler number of the surface M 2.

5 4. Parallel surfaces 5 An immersion f = f(u, v) :(U; u, v) R 3 (U R 2 ), the unit normal vector ν(u, v) := f u(u, v) f v (u, v) f u (u, v) f v (u, v). For each real number t, f t (u, v) =f(u, v)+tν(u, v) is called a parallel surface of f. p is a singular pt of f t (f t ) u (p) (f t ) v (p) =0. Figure 2. a cuspidal edge and a swallowtail Cuspidal edges and swallowtails are generic singular points appeared on parallel surfaces. f C =(u 2,u 3,v), f S =(3u 4 + u 2 v, 4u 3 +2uv, v).

6 6 An ellipsoid: x 2 + y2 4 + z2 9 =1. Parallel surfaces of the ellipsoid are given as follows: Figure 3. the cases of t =0andt =1.2 Figure 4. the cases of t =2andt =3.2

7 7 5. Singular curvature κ s Figure 5. Surfaces of K<0 and K>0 The image of cuspidal edges consists of regular curves in R 3.Wedenoteitbyγ(s), where s is the arclength parameter. Figure 6. a( )-cuspidal edge and a (+)-cuspidal edge the singular curvature κ s (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)), 1 (otherwise).

8 8 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 = κ s

vector field (U; u, v); a (+)-oriented local coordinate M 2 the area density function λ :=

9 7. Gauss-Bonnet formula of surfaces with singularity 9 M 2 ; a compact oriented 2-manifold f : M 2 R 3 ; a C -map having only cuspidal edges and swalowtails ν : M 2 S 2 ; the unit normal vector field (U; u, v); a (+)-oriented local coordinate M 2 the area density function λ := det(f u,f v,ν) λ(p) =0 p is a singular point area element da := λ du dv, signed area element dâ := λdu dv.

10 10 8. Two Gauss-Bonnet formulas Figure 7. a positive swallowtail Formulas given by Kossowski(02)and Langevin-Levitt-Rosenberg(95) KdA +2 κ s ds =2πχ(M 2 ), M 2 \Σ f Σ f 2deg(ν) = 1 KdÂ 2π M 2 \Σ f = χ(m + ) χ(m )+#SW + #SW, M + := {p M 2 ; da p = dâp}, M := {p M 2 ; da p = dâp}.

11 11 9. singular points of a map between 2-manifolds Maps between planes R 2 (u, v) f(u, v) =(x(u, v),y(u, v)) R 2, ( ) Singular points of f det x u (u, v) x v (u, v) y u (u, v) y v (u, v) =0. Generic singular points afold R 2 (u, v) (u 2,v) R 2, the singular set u =0, f(0,v)=(0,v) acusp R 2 (u, v) (uv + v 3,u) R 2, the singular set u = 3v 2, f( 3v 2,v)=( 2v 3, 3v 2 ) Figure 8. afoldandacusp

12 12 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 Figure 10. a cusp and a swallowtail

13 Singularities of Gauss maps M 2 :an oriented compact manifold f : M 2 R 3 an immersion Singular points of ν : M 2 S 2 K f =0. Then f t = 1 (f + tν), t R. t lim t f t = ν, 2deg(ν) =χ(m t +) χ(m t )+#SW t + #SW t. In particular, Hence, 2deg(ν) =χ(m 2 )=χ(m t +)+χ(m t ). 2χ(M t )=#SW t + #SW t. Taking the limit t,wehavethat 2χ(M )=#SW + #SW.

14 14 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 dâν = K f da f, da ν = K f da f, it holds that M = {p M 2 ; K f (p) < 0}. Thus (the Bleecker and Wilson formula [1]) 2χ({K f < 0}) = #positive cusps #negative cusps.

15 15 Figure 11. a symmetric torus and its perturbation A deformation of the rotationally symmetric torus f a (u, v) = (cos v(2 + ε(v)cosu), sin v(2 + ε(v)cosu), ε(v)sinu), where ε(v) :=1+acos v. a = 0 : the original torus a =4/5: χ({k <0}) =1 Figure 12. A parallel surface of f 4/5

16 A similar application The following identity holds for f t : M 2 R 3 K t da t +2 κ s ds =2πχ(M 2 ). M 2 \Σ ft Σ ft Taking t, we have that (Saji-Yamada-U. [10]) 1 K f da f = κ s ds, 2π {K<0} Σ ν where To prove the formula, we apply κ s := the singular curvature of ν = ±the geodesic curvature of ν > 0 if ν points Im(ν), < 0 otherwise. K ν dâν = KdA ν, K ν A ν = K f da f.

17 17 The intrinsic formulation of wave fronts (Saji-Yamada-U. [10]) The definition (E,,,D,ϕ) of coherent tangent bundle on M n : (1) E is a vector bundle of rank n over M n, (2) E has a inner product,, (3) D is a metric connection of (E,, ), (4) ϕ: TM n E is a bundle homomorphism s.t. D X ϕ(y ) D Y ϕ(x) =ϕ([x, Y ]), where X, Y are vector fields on M n. The pull-back of the metric, ds 2 ϕ := ϕ, is called the first fundamental form of ϕ. p M n ; ϕ-singular point Ker(ϕ p : T p M n E p ) {0}. coherent tangent bundle = generalized Riemannian manifold When (M n,g) is a Riemannian manifold, then E = TM n,, := g, D = g, ϕ =id. If f : M 2 R 3 is a wave front, then E = ν,, := g R 3, D = T, ϕ = df, (ψ = dν).

18 18 M 2 ; a compact oriented 2-manifold, (E,,, D, ϕ); an orientable coherent tangent bundle, μ Sec(E E \{0}) such that μ(e 1,e 2 ) = 1 for (+)-frame {e 1,e 2 } The intrinsic definition of the singular curvature κ s := sgn(dλ(η(t))) μ(d γ n(t),ϕ(γ )) ϕ(γ ) 3, where n(t) E γ(t) is the unit vector perpendicular to ϕ(γ )one. (u, v) a (+)-local coordinate on M 2 where Σ ϕ ; ϕ-singular set, dâ = λdu dv, da = λ du dv, ( λ := μ ϕ( u ),ϕ( ) v ). (χ E =) 1 KdÂ 2π = χ(m +) χ(m )+SW + SW, M 2 KdA +2 κ s dτ =2πχ(M 2 ). M 2 Σ ϕ p Σ ϕ is non-degenerate dλ(p) 0, p Σ ϕ ; A 2 -pt (intrinsic cuspidal edge) η γ (0) at p, p Σ ϕ ; A 3 -pt (intrinsic swallowtail) det(η, γ ) = 0 and det(η, γ ) =0atp.

19 Examples of coherent tangent bundle: 19 (1) Wave fronts as a hypersurface of Riem. manifold, (2) Smooth maps between n-manifolds M n ; an orientable manifold (N n,g); an orientable Riemannian manifold f : M n (N n,g); C -map, E f := f TN n,, := g Ef, D; induced connection. ϕ := df : TM n E f := f TN n, Figure 13. a fold and a cuspidal edge Figure 14. a cusp and a swallowtail

20 20 An application of the intrinsic G-B formula f : M 2 R 3 ; a strictly convex surface, ξ : M 2 R 3 ; the affine normal map. X Y = D X Y + h(x, Y )ξ, D X ξ = α(x), where α : TM 2 TM 2 We set M 2 := {p M 2 ;det(α p ) < 0}, then (Saji-Yamada-U. [9]) 2χ(M )=#SW 2 + (ξ) #SW (ξ).

21 References [1] D. Bleecker and L. Wilson, Stability of Gauss maps, Illinois J. of Math. 22 (1978) [2] M. Kossowski, The Boy-Gauss-Bonnet theorems for C -singular surfaces with limiting tangent bundle, Annals of Global Analysis and Geometry 21 (2002), [3] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic 3-space, PacificJ. Math. 221 (2005), [4] R. Langevin, G. Levitt and H. Rosenberg, Classes d homotopie de surfaces avec rebroussements et queues d aronde dans R 3, Canad. J. Math. 47 (1995), [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), [7] K. Saji, M. Umehara and K. Yamada, Geometry of fronts, Ann. of Math. 169 (2009), [8] K. Saji, M. Umehara and K. Yamada, A k singularities of wave fronts, Mathematical Proceedings of the Cambridge Philosophical Society, 146 (2009), [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), [10] K. Saji, M. Umehara and K. Yamada, The intrinsic duality on wave fronts, preprint, arxiv: [11], 38 (2009). ( Tel , fax: ).) [12],, (2002). 21

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