Topology and its Applications 234 (2018) 209–219 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Virtual Special Issue – Real and Complex Singularities and their applications in Geometry and Topology Geometry of cuspidal edges with boundary Luciana F. Martins a, Kentaro Saji b,∗,1 a Departamento de Matemática, Instituto de Biociências, Letras e Ciências Exatas, UNESP – Univ Estadual Paulista, Câmpus de São José do Rio Preto, SP, Brazil b Department of Mathematics, Graduate School of Science, Kobe University, Rokkodai 1-1, Nada, Kobe 657-8501, Japan a r t i c l e i n f o a b s t r a c t Article history: Received 21 February 2016 Received in revised form 2 June 2017 Accepted 14 September 2017 Available online 26 November 2017 Dedicated to Professor Takashi Nishimura on the occasion of his sixtieth birthday MSC: 53A05 58K05 58K50 Keywords: Cuspidal edge Map-germs with boundary We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invariants. © 2017 Elsevier B.V. All rights reserved. 1. Maps from manifolds with boundary There are several studies for C∞ map-germs f : (Rm, 0) → (Rn, 0) with A-equivalence. Two map-germs f, g : (Rm, 0) → (Rn, 0) are A-equivalent if there exist diffeomorphisms ϕ : (Rm, 0) → (Rm, 0) and Φ : (Rm, 0) → (Rm, 0) such that g ◦ ϕ = Φ ◦ f . There is also several studies for the case that the source space has a boundary. In [2], map-germs from 2-dimensional manifolds with boundaries into R2 are classified, and in [9], map-germs from 3-dimensional manifolds with boundaries into R2 are considered. Let W ⊂ (Rm, 0) be a closed submanifold-germ such that 0 ∈ ∂W and dimW = m. We call f |W a map-germ with boundary, and we call interior points of W interior domain of f |W . Since ∂W is an (m −1)-dimensional * Corresponding author. E-mail addresses: lmartins@ibilce.unesp.br (L.F. Martins), saji@math.kobe-u.ac.jp (K. Saji). 1 Partly supported by the Japan Society for the Promotion of Science (JSPS) and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior under the Japan–Brazil research cooperative program and the JSPS KAKENHI Grant Number 26400087. https://doi.org/10.1016/j.topol.2017.11.024 0166-8641/© 2017 Elsevier B.V. All rights reserved. https://doi.org/10.1016/j.topol.2017.11.024 http://www.ScienceDirect.com/ http://www.elsevier.com/locate/topol mailto:lmartins@ibilce.unesp.br mailto:saji@math.kobe-u.ac.jp https://doi.org/10.1016/j.topol.2017.11.024 http://crossmark.crossref.org/dialog/?doi=10.1016/j.topol.2017.11.024&domain=pdf 210 L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 submanifold, regarding ∂W = B, map-germs from manifolds with boundaries can be treated as a map-germ f : (Rm, 0) → (Rn, 0) with a codimension one oriented submanifold B ⊂ (Rm, 0). We consider that (Rm, 0) has an orientation and the submanifold B such that 0 ∈ B. We define the interior domain of such a map-germ f as the component of (Rm, 0) \ B such that the positively oriented normal vectors of B point into it. With this terminology, an equivalent relation for map-germs with boundary is the following. Let f, g : (Rm, 0) → (Rn, 0) be map-germs with codimension one submanifolds B, B′ ⊂ (Rn, 0) respectively, which contain 0. Then f and g are B-equivalent if there exist an orientation preserving diffeomorphism ϕ : (Rm, 0) → (Rm, 0) such that ϕ(B) = B′, and a diffeomorphism Φ : (Rn, 0) → (Rn, 0) that satisfies g ◦ ϕ = Φ ◦ f. A map-germ f : (R2, 0) → (R3, 0) is a cuspidal edge if f is A-equivalent to the map-germ (u, v) �→ (u, v2, v3) at the origin. We say that f is a cuspidal edge with boundary B ⊂ (R2, 0) if B is a codimension one oriented submanifold, that is, there exists a parametrization b : (R, 0) → (R2, 0) to B satisfying b′(0) �= (0, 0). In this case, the domain which lies the left hand side of b with respect to the velocity direction is the interior domain of f . In this note, we will consider differential geometric properties of cuspidal edges with boundaries. In order to do this, we first construct a normal form (Proposition 2.1) of it. It can be seen that all the coefficients of the normal form are differential geometric invariants. We give geometric meanings of these invariants. See [4,6,8,11,12,15–17,22], for example, for geometry of cuspidal edges itself. An application of this study is given by considering flat extensions of flat ruled surfaces with boundaries. See [13] for singularities of the flat ruled surfaces, and see [14] for flat extensions of flat ruled surfaces with boundaries. See [3] for flat extensions from general surfaces. The authors would like to thank Takashi Nishimura for his constant encouragement. 2. Normal form of cuspidal edge with boundary Now we look for a normal form of cuspidal edges with boundary. Let f : (R2, 0) → (R3, 0) be a cuspidal edge with boundary b : (R, 0) → (R2, 0), b′(0) �= (0, 0). One can take a local coordinate system (u, v) on (R2, 0) and an isometry Φ on (R3, 0) satisfying that Φ ◦ f(u, v) = ( u, a20 2 u2 + a30 6 u3 + 1 2v 2, b20 2 u2 + b30 6 u3 + b12 2 uv2 + b03 6 v3 ) + h(u, v), (2.1) where b03 �= 0, b20 ≥ 0, and h(u, v) = ( 0, u4h1(u), u4h2(u) + u2v2h3(u) + uv3h4(u) + v4h5(u, v) ) , with h1(u), h2(u), h3(u), h4(u), h5(u, v) smooth functions. See [11] for details. Now we consider b(t) = (b1(t), b2(t)). We have two cases. (1) b′1(0) �= 0, (2) b′1(0) = 0, b′2(0) �= 0. In the case (1), one can take u for the parameter of b and parametrized by b(u) = ( εu, 3∑ ck k!u k + u4c(u) ) (ε = ±1). (2.2) k=1 L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 211 Fig. 1. Cuspidal edges with boundary. The boundaries are drawn by thick lines, and the exteriors of the surfaces are drawn by thin colors. Left to right, b(t) = (t, t), b(t) = (t, t2), b(t) = (t, −t2), b(t) = (t2, t). In the case (2), one can take v for the parameter of b and parametrized by b(v) = ( 3∑ k=2 dk k! v k + v4d(v), εv ) (ε = ±1). (2.3) In summary, we have the following proposition. Proposition 2.1. For any cuspidal edge f : (R2, 0) → (R3, 0) with boundary b : (R, 0) → (R2, 0), there exists a coordinate system on (R2, 0) and an isometry Φ : (R3, 0) → (R3, 0) such that Φ ◦ f(u, v) has the form (2.1) and b is parameterized by (2.2) (respectively, (2.3)) if b′(0) /∈ ker df0 (respectively, b′(0) ∈ ker df0). By this proposition, all coefficients c1, c2, c3, d2, d3 are geometric invariants of the cuspidal edge with boundary. In fact, one can easily verify that all coefficients of (2.1), (2.2) and (2.3) are uniquely determined. It means that any cuspidal edge with boundary has this form using only coordinate changes on the source and isometries of R3. See Fig. 1. Let f : (R2, 0) → (R3, 0) be a cuspidal edge. Then there exists a unit vector field ν along f satisfying 〈dfp(X), ν(p)〉 = 0 for any X ∈ TpR 2 and p ∈ (R2, 0), where 〈 , 〉 stands for the Euclidean inner product of R3. We call ν unit normal vector of f . Moreover, we see that the pair of the map-germs (f, ν) : (R2, 0) → (R3 × S2, (0, ν(0)) is an immersion. Thus a cuspidal edge is a front in the sense of [1]. See also [17]. 3. Differential geometric information Several geometric invariants on cuspidal edges are defined and studied. See [11,12,17] for details. Co- efficients of (2.1) are invariants and, according to [11], it is known that a20 coincides with the singular curvature κs, b20 coincides with the absolute value of the limiting normal curvature κν, b03 coincides with the cuspidal curvature κc and b12 coincides with the cusp-directional torsion κt at the origin. In what follows, we consider the geometry of the boundary. Let f : (R2, 0) → (R3, 0) be a cuspidal edge with boundary, γ : (R, 0) → (R2, 0) a parametrization of its singular set S(f), and b : (R, 0) → (R2, 0) a parametrization of the boundary. We set γ̂(t) = f ◦ γ(t) and b̂(s) = f ◦ b(s). 3.1. The case (1) We assume that b′(0) /∈ ker df0 and, by this assumption, γ̂ = f ◦ γ and b̂ = f ◦ b are both regular curves and they are tangent each other at 0. Hence we have l �= 0 such that d dt γ̂ ∣∣ t=0 = l d ds b̂ ∣∣ s=0. (3.1) We take a parametrization of s by t as s = s(t). By the assumption (3.1), s′(0) = l. Let d(t) be the curve given by 212 L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 d(t) = γ̂(t) − b̂(s(t)) l . Then we define the approaching ratio of boundary to cuspidal edge (or shortly approaching ratio) by α = ∣∣∣∣ 1 |γ̂′(0)|3 det ( γ̂′(0), d′′(0), ν(0, 0) )∣∣∣∣ 1/2 , where ′ = d dt , and ν is the unit normal vector of f . Lemma 3.1. The number α does not depend on the choice of the parameter t and the function s(t). Proof. Since d′(t) = γ̂′(t) − 1 l d ds b̂(s(t))s′(t) d′′(t) = γ̂′′(t) − 1 l ( d2 ds2 b̂(s(t))(s ′(t))2 − d ds b̂(s(t))s′′(t) ) , and (d/ds)b̂ ∣∣ s=0 is parallel to γ̂′(0), s′(0) = l, we have α = ∣∣∣∣ 1 |γ̂′(0)|3 det ( γ̂′(0), γ̂′′(0) − b̂ss(0)l, ν(0, 0) )∣∣∣∣ 1/2 . Thus α does not depend on s(t). We next assume t = t(x) (t(0) = 0 and s(0) = 0) for a parameter x, and denote (·)x = (d/dx)(·), (·)s = (d/ds)(·). By (3.1), γ̂′(0) and b̂s(0) are linearly dependent, and by s′(0) = l, we have det ( (γ̂(t(x)))x , (d(t(x)))xx , ν0 ) |γ̂(t(x))x|3 ∣∣∣∣∣∣ x=0 = det ( γ̂′(t(x))tx(x), γ̂′′(t(x))(tx(x))2 − b̂ss(s(t(x)))Q, ν0 ) |γ̂′(t(x))tx(x)|3 ∣∣∣∣∣∣ x=0 = det ( γ̂′(t(0)), γ̂′′(t(0)) − b̂ss(s(t(0)))l, ν0 ) |γ̂′(t(0))|3 , where Q = (s′(t(x)))2(tx(x))2l−1, and ν0 = ν(0, 0). This proves the assertion. � Since the image b̂ of the boundary is a curve in R3, its curvature κ and torsion τ as a curve in R3 are invariants. Moreover, b̂ is a curve on the surface f . Thus the normal curvature κnb and the geodesic curvature κgb of b are invariants. We have the following proposition for these invariants. Proposition 3.2. It holds that (1) κ(0) = √ b220 + (c21 + a20)2, (2) κ′(0) = b20(b03c31 + 3εb12c21 + εb30) + (c21 + a20)(3c1c2 + εa30)√ b220 + (c21 + a20)2 , (3) τ(0) = (c21 + a20)(εb03c31 + 3b12c21 + b30) − b20(3εc1c2 + a30) 2 2 2 , b20 + (c1 + a20) L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 213 (4) κnb(0) = b20, (5) κ′ nb(0) = b03c 3 1 2 + 2εb12c21 − a20b03c1 2 + εb30 − εa20b12, (6) κgb(0) = −ε(c21 + a20), (7) κ′ gb(0) = −c1 ( εb03b20 2 + 3εc2 ) − a30 − b12b20, (8) α = |c1|. Proof. Since b̂(u) = ( εu, c21 + a20 2 u2 + 3c1c2 + εa30 6 u3, b20 2 u2 + b03c 3 1 + ε(3b12c21 + b30) 6 u3 ) + O(4), (3.2) where O(i) stands for the terms whose degrees are higher or equal to i (i = 1, 2, . . .), we have the assertions (1)–(3). Next, since ν(b(u)) = ( −b20εu, ( −b03c1 2 − b12ε ) u, 1 ) + O(2), we have the assertions (4)–(7). The assertion (8) is obvious by (3.2) and γ̂(u) = (u, a20u 2/2, b20u2/2) + O(3). � Since α = |c1|, and b(u) = (εu, c1u + O(2)), the invariant α measures the difference of boundary. We can give a geometric interpretation of α by using the curvature parabola given by [10] as follows. Let f : (R2, 0) → (R3, 0) be a map-germ satisfying rank df0 = 1, and set N0f = {Y ∈ R3 ; 〈Y, df0(X)〉 = 0 for all X ∈ T0R 2}, where we identify T0R 3 with R3. By this identification, N0f is a normal plane of df0(X) passing through 0. The curvature parabola Δ0 is defined by Δ0 = {a2f⊥ uu(0) + 2abf⊥ uv(0) + b2f⊥ vv(0) ∈ N0f ; a, b ∈ R, a2E(0) + 2abF (0) + b2G(0) = 1}, where E(0) = 〈fu(0), fu(0)〉, F (0) = 〈fu(0), fv(0)〉, G(0) = 〈fv(0), fv(0)〉 and, given w ∈ T0R 3, w⊥ is the orthogonal projection of w to N0f . The curvature parabola is a usual parabola if and only if f is a cross cap, and otherwise, Δ0 is a line, a half-line or a point. If f is a cuspidal edge, then as we will see, Δ0 degenerates in a half-line. Let be the line which contains Δ0. In this case, the umbilic curvature is defined in [10] by the distance from the origin to and it is equal to the limiting normal curvature defined in [17] up to sign (see also [10,11]). Let n be the principal normal vector of b̂ as a space curve. Since b̂ is tangent to γ̂ at 0, the vector n(0) lies in N0f . Lemma 3.3. Under the above setting, if the limiting normal curvature of the cuspidal edge f is non zero, then 0 /∈ , and and n(0) are not parallel. 214 L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 Fig. 2. Situation of Proposition 3.4. Proof. Without loss of generality, we can take the normal form for f as in (2.1). Then since E(0) = 1, F (0) = G(0) = 0, and f⊥ uu(0) = (a20, b20), f⊥ uv(0) = (0, 0), f⊥ vv(0) = (1, 0), we see that Δ0 is a half-line: Δ0 = {(0, a20 + t2, b20) ; t ∈ R}, where the normal plane is N0f = {(0, y, z) ; y, z ∈ R}. This means that = {(0, a20, b20) +t(0, 1, 0) ; t ∈ R}. On the other hand, we see that n(0) = (0, c21+a20, b20)/ √ (c21 + a20)2 + b220. Since b20 �= 0, two vectors (0, 1, 0) and (0, c21 + a20, b20) are not parallel. This proves the assertion. � Since Δ0 is a half-line, let us set that V be the endpoint of Δ0. For instance, for f given as in (2.1), V = (0, a20, b20). By Lemma 3.3, if the limiting normal curvature of f is non zero, there exists a intersection point P of lines containing n and Δ0. Proposition 3.4. If the limiting normal curvature of f is non zero, then the distance between V and P coincides with c21. Proof. Like as the proof of Lemma 3.3, we take the normal form for f . Then P = (0, c21 + a20, b20) which proves the assertion. � We illustrate the situation in N0f of Proposition 3.4 in Fig. 2. 3.2. The case (2) We assume that b′(0) ∈ ker df0, and set b̂ = f ◦ b. Then we see that b̂′(0) = 0 and b̂′′(0) �= 0. Thus we define the angle between boundary and cuspidal edge by β = 〈 b̂′′(0), γ′(0) 〉 |b̂′′(0)||γ′(0)| . One can easily check that β does not depend on the choice of parameters of b and γ. If f is given by the normal form (2.1) with (2.3), we have β = d2. On the other hand, since b̂ has a singularity, the curvature and torsion may diverge. So we have to prepare curvature and torsion for a singular curve. We denote by κsing (respectively, τsing) the cuspidal curvature (respectively, the cuspidal torsion) defined in Appendix A for a singular curve. Then the following proposition holds. Proposition 3.5. The cuspidal curvature and the cuspidal torsion of b̂ satisfy that κsing = √ b203(1 + d2 2) + d2 3 (1 + d2 2)5/4 , τsing = −3εa20b03d 3 2 + 3b20d2 2d3 + 6b12d2d3 − h5(0, 0)d3 + εb03d4( b203(1 + d2 2) + d2 3 )3/4 √ 1 + d2 2. L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 215 4. Singularities of flat extension of a flat surface In this section, as an application of the study on cuspidal edges with boundary, we consider flat extensions of a flat ruled surface with boundary. Let γ : I → R3 be a smooth curve satisfying γ′(t) �= 0 for any t ∈ I, where I is an open interval and 0 ∈ I. Let δ : I → S2 be a smooth curve satisfying δ′(t) �= 0 for any t ∈ I, where S2 is the unit sphere in R3. Then the map F : I × (−ε, ε) → R3 F (t, v) = F(γ,δ)(t, v) = γ(t) + vδ(t), (4.1) where ε > 0, is called a ruled surface. It is known that F is flat if and only if det(γ′, δ, δ′) identically vanishes. (See [7, Proposition 2.2], for example.) Since δ′ �= 0, one can assume that the parameter t is the arc-length. Then {δ, δ′, δ × δ′} forms an orthonormal frame along δ, and δ′′(t) = −δ(t) + κδ(t)δ(t) × δ′(t). The function κδ is called the geodesic curvature of δ, and δ is determined by κδ with an initial condition. On the other hand, we set γ′(t) = x(t)δ(t) + y(t)δ′(t) + z(t)δ(t) × δ′(t). (4.2) Then γ is determined by {x(t), y(t), z(t)} with an initial condition. Then F is flat if and only if z(t) identically vanishes. Moreover, setting S(F ) the singular set of F , so S(F ) ∩ (I× [−ε, ε]) = ∅ if and only if |y| > ε since (t, v) is a singular point of F if and only if y(t) + v = 0 as we will see. Thus we set the space of flat ruled surface FR as FR = {(x, y, κδ) ∈ C∞(I,R× (R \ [−ε, ε]) ×R)} ×X, where X = {(δ0, δ1) ∈ S2 × S2 ; 〈δ0, δ1〉 = 0} represents the initial conditions δ(0) = δ0 and δ′(0) = δ1. Let us assume that a ruled surface F = F(γ,δ) satisfies S(F ) ∩ (I × {0}) = ∅. Then consider extensions of F for v ∈ (−M, M) (M > ε) by the same formula (4.1). We call singular points (t, v) of F the birth of singularities of extension of F if t is a minimal value of y(t), since (t, v) is a singular point of F if and only if y(t) + u = 0. We have the following result. Proposition 4.1. Let I be an open interval. Then the set O = {( (x, y, κδ), (δ0, δ1) ) ∈ FR ; all birth of singularities of the extensions of F(γ,δ) are cuspidal edges whose c1 vanishes and c2 �= 0, where c1, c2 are given by (2.2), γ is defined by (4.2), and δ is defined by the curvature κδ with the initial condition δ0, δ1. } is open and dense in FR with respect to the Whitney C∞ topology. To prove this proposition, we show the following lemma. 216 L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 Lemma 4.2. For a flat ruled surface F as in (4.1), • (t, v) is a singular point of F if and only if y(t) + v = 0. • F is a cuspidal edge at (t, v) ∈ S(F ) if and only if y′(t) − x(t) �= 0, κδ(t) �= 0. Proof. Since F ′ = γ′ + vδ′ = xδ + (y + v)δ′ and Fv = δ, where we omit (t) and ′ = ∂/∂t, (·)v = ∂/∂v, we see the first assertion. Moreover, we see that ker dF(t,v) = 〈∂t − x∂v〉R for (t, v) ∈ S(F ), and δ × δ′ gives a unit normal vector of F . Set η = ∂t − x∂v. Thus we see that η(δ × δ′) = κδ, and η(y + v) = y′ − x. By the well-known criteria for cuspidal edge ([18, Corollary 2.5], see also [8, Proposition 1.3]), we see the second assertion. � Proof of Proposition 4.1. We define subsets of the 2-jet space J2(I, R× (R \ [−ε, ε]) ×R) as follows: C1 = {j2(x, y, κδ)(t, v) ; κδ(t) = 0}, C2 = {j2(x, y, κδ)(t, v) ; y′(t) − x(t) = 0}, C3 = {j2(x, y, κδ)(t, v) ; y′(t) = 0}, C4 = {j2(x, y, κδ)(t, v) ; y′′(t) = 0} (4.3) Since a coordinate system of J2(I, R× (R \ [−ε, ε]) ×R) is given by (t, x, y, κδ, x ′, y′, κ′ δ, x ′′, y′′, κ′′ δ ), we see that these subsets are closed submanifolds with codimension 1, and Ci ∩ C3 (i = 1, 2, 4) are closed submanifolds with codimension 2. By the Thom jet transversality theorem, the set O′ ={((x, y, κδ), (δ0, δ1)) ∈ FR ; j2(x, y, κδ) : I �→ J2(I,R× (R \ [−ε, ε]) ×R) is transverse to C1, C2, C3, C4 and Ci ∩ C3 (i = 1, 2, 4)} is a residual subset of FR. Let ((x, y, κδ), (δ0, δ1)) ∈ O′ and assume that (t0, v0) is a birth of singularity of F . Since (t0, v0) is a birth of singularity, and S(F ) = {y(t0) − v0 = 0}, we see y′(t0) = 0. Since y′(t0) = 0 and (x, y, κδ) ∈ O′, F at (t0, v0) is a cuspidal edge by Lemma 4.2. Moreover, we have y′′(t0) �= 0. This implies that the contact of S(F ) and the t-curve {(t, v) ; v = v0} is of second degree. On the other hand, the condition c1 = 0 and c2 �= 0 as in (2.2) implies that the contact of S(f) (the v-axis) and b is of second degree. Since the degrees of contact of two curves do not depend on the diffeomorphism, the cuspidal edge F at (t0, v0) has the property c1 = 0 and c2 �= 0. This proves the assertion. � We remark that singularities of flat surfaces with boundaries are studied in [13], and the flat extensions of flat surfaces are studied in [14]. Flat extensions of generic surfaces with boundaries are studied in [3]. In [5], flat ruled surfaces approximating regular surfaces are studied. Appendix A. Curvature and torsion of space curves with singularities In the case (2) in Section 2, the image of the boundary of a cuspidal edge with boundary has a singularity. Thus we need differential geometry of space curves with singularities. In this appendix we give curvature and torsion for space curves with singularities. It should be mentioned that the discussions here are quite analogous to the study for the case of plane curves given by Shiba and Umehara [20], and we follow their discussions in the following. L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 217 Let γ : (R, 0) → (R3, 0) be a smooth curve and assume that γ′(0) = (0, 0, 0). We say that 0 is called A-type if γ′′(0) �= (0, 0, 0), and 0 is called (2, 3)-type if γ′′(0) × γ′′′(0) �= (0, 0, 0). Let 0 be a A-type singular point of γ, then we define κsing = |γ′′(0) × γ′′′(0)| |γ′′(0)|5/2 . We call κsing the cuspidal curvature of γ. This definition is analogous to the cuspidal curvature for (2, 3)-cusp of plane curve introduced in [21]. See [19] for detail. Moreover, let 0 be a (2, 3)-type singular point of γ, then we define τsing = √ |γ′′(0)| det(γ′′(0), γ′′′(0), γ′′′′(0)) |γ′′(0) × γ′′′(0)|2 . We call τsing the cuspidal torsion of γ. By a direct calculation, one can show that κsing and τsing do not depend on the choice of parameter. Furthermore, we have the following. Let sg be the arc-length function sg(t) = ∫ t 0 |γ′(t)| dt, and let κ and τ be curvature and torsion of the space curve γ defined on regular set. Fact A.1. ([20, Theorem 1.1, Lemma 2.1]) The functions sgn(t) √ |sg(t)| and √ |sg(t)|κ(t) are C∞-differentiable, and lim t→0 √ |sg(t)|κ(t) = 1 2 √ 2 κsing. By this fact, sgn(t) √ |sg(t)| can be taken as a local coordinate of the curve γ at t = 0. It is called half-arclength parameter. We have an analogous claim for the torsion. Proposition A.2. The function sgn(t) √ |sg(t)|τ(t) is C∞ differentiable, and lim t→0 sgn(t) √ |sg|τ(t) = 2 3 √ 2 τsing. Proof. By L’Hôpital’s rule, we see lim t→0 |γ′ × γ′′|2 t4 = 6|γ′′(0) × γ′′′(0)| 4! , lim t→0 det(γ′, γ′′, γ′′′) t3 = det(γ′′(0), γ′′′(0), γ′′′′(0)) 3! . Thus these two functions are C∞-differentiable at t = 0. Moreover, lim t→0 tτ(t) = lim t→0 det(γ′, γ′′, γ′′′) t3 t4 |γ′ × γ′′|2 = det(γ′′(0), γ′′′(0), γ′′′′(0)) 3! 4! |γ′′(0) × γ′′′(0)|2 shows that tτ(t) is C∞-differentiable. On the other hand, by L’Hôpital’s rule, we have 218 L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 lim t→0 ∣∣∣∣sg(t)t2 ∣∣∣∣ = lim t→0 |γ′(t)| |2t| = |γ′′(0)| 2 . (A.1) Thus lim t→0 √ |sg(t)| |t| = √ |γ′′(0)|√ 2 . Hence lim t→0 sgn(t)|t| √ |γ′′(0)|√ 2 τ = √ |γ′′(0)|√ 2 lim t→0 det(γ′, γ′′, γ′′′) t3 t4 |γ′ × γ′′|2 = 2 3 √ 2 √ |γ′′(0)| det(γ′′(0), γ′′′(0), γ′′′′(0)) |γ′′(0) × γ′′′(0)|2 which shows the assertion. � We remark that this proof is analogous to that of [20, Lemma 2.1]. Thus κsing (respectively, τsing) is a geometric invariant of A-type (respectively, (2, 3)-type) singular space curve, and it can be regarded as a natural limit of usual curvature (respectively, torsion). We also remark that an A-type space curve-germ γ : (R, 0) → (R3, 0) at 0 is (2, 3)-type if and only if κsing �= 0. By (A.1), a parametrization t of the A-type space curve-germ γ is the half-arclength parameter if and only if |γ′(t)| = 2|t| (see [20, Remark 2.2]). We have the following proposition. Proposition A.3. Let α, β : (R, 0) → R be C∞-functions satisfying α > 0. Then there exists a unique (2, 3)-type curve-germ γ : (R, 0) → (R3, 0) up to orientation preserving isometric transformations in R3 such that √ |sg(t)|κ(t) = α(t) and √ |sg(t)|τ(t) = β(t) (A.2) and t is the half-arclength parameter. Proof. Let us consider an ordinary differential equation A′(t) = 2A(t) ( 0 −α(t) 0 α(t) 0 −β(t) 0 β(t) 0 ) . Then we see that A(t) is an orthonormal matrix under an initial condition and A(0) is the identity matrix. Set A(t) = (e(t), n(t), b(t)) and set γ(t) = 2 ∫ t 0 te(t) dt. Then |γ′(t)| = 2|t| and which shows that t is the half-arclength parameter. One can easily see that γ(t) satisfies (A.2). � We remark that this proof is analogous to that of [20, Theorem 1.1]. For a space curve-germ γ of A-type, one can easily see that there exist a parameter t and an isometry A such that A ◦ γ(t) = ( t2 2 , l∑ i=3 1 i!γ2it i, l∑ i=4 1 i!γ3it i ) + (0, O(l + 1), O(l + 1)), (A.3) where O(l+1) stands for the terms whose degrees are greater than l+1, and γji ∈ R (j = 2, 3, i = 2, . . . , l). If γ is of (2, 3)-type, then γ23 �= 0, and we see that L.F. Martins, K. Saji / Topology and its Applications 234 (2018) 209–219 219 κsing = |γ23| 2 √ 2 , τsing = γ34 γ23 . We set κ′ sing = d dt (√ |sg(t)|κ(t) )∣∣∣∣ t=0 . Then κ′ sing = (γ23 +4γ24)/(12 √ 2|γ23|). Hence we would like to say that κsing, κ′ sing, τsing are all invariants of (2, 3)-type singular space curve up to fourth degree. However, it is not easy to compute the differentiation of √ |sg(t)|κ(t) for a given curve. Thus we set σsing = 〈 γ′′(0) × γ′′′(0), γ′′(0) × γ(4)(0) 〉 − 2 |γ ′′(0) × γ′′′(0)|2 〈γ′′(0), γ′′′(0)〉 〈γ′′(0), γ′′(0)〉 〈γ′′(0), γ′′(0)〉11/4 . Then this is independent of the choice of the parameter, and σsing = γ23(γ24 − 2γ23) holds for γ of the form (A.3). Thus invariants {κsing, σsing, τsing} can be used instead of {κsing, κ′ sing, τsing} for (2, 3)-type singular space curve up to fourth degrees. References [1] V.I. Arnol’d, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, vol. 1, Monogr. Math., vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. [2] J.W. Bruce, P.J. Giblin, Projections of surfaces with boundary, Proc. Lond. Math. Soc. (3) 60 (2) (1990) 392–416. [3] G. Ishikawa, Singularities of flat extensions from generic surfaces with boundaries, Differ. Geom. Appl. 28 (3) (2010) 341–354. [4] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Mosc. Math. J. 9 (2) (2009) 325–357. [5] S. Izumiya, S. Otani, Flat approximations of surfaces along curves, Demonstr. Math. 48 (2) (2015) 217–241. [6] S. Izumiya, M.C. Romero Fuster, M.A.S. Ruas, F. Tari, Differential Geometry from Singularity Theory Viewpoint, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. [7] S. Izumiya, N. Takeuchi, Geometry of ruled surfaces, in: Applicable Math. in the Golden Age, Narosa Publ. House, New Delhi, 2002, pp. 305–338. [8] M. Kokubu, W. Rossman, K. Saji, M. Umehara, K. Yamada, Singularities of flat fronts in hyperbolic 3-space, Pac. J. Math. 221 (2) (2005) 303–351. [9] L.F. Martins, A.C. Nabarro, Projections of hypersurfaces in R4 with boundary to planes, Glasg. Math. J. 56 (1) (2014) 149–167. [10] L.F. Martins, J.J. Nuño-Ballesteros, Contact properties of surfaces in R3 with corank 1 singularities, Tohoku Math. J. (2) 67 (2015) 105–124. [11] L.F. Martins, K. Saji, Geometric invariants of cuspidal edges, Can. J. Math. 68 (2) (2016) 445–462. [12] L.F. Martins, K. Saji, M. Umehara, K. Yamada, Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts, in: Geometry and Topology of Manifold, in: Springer Proc. Math. Stat., 2016, pp. 247–282. [13] S. Murata, M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differ. Geom. 82 (2) (2009) 279–316. [14] K. Naokawa, Singularities of the asymptotic completion of developable Möbius strips, Osaka J. Math. 50 (2) (2013) 425–437. [15] K. Naokawa, M. Umehara, K. Yamada, Isometric deformations of cuspidal edges, Tohoku Math. J. 68 (2016) 73–90. [16] R. Oset Sinha, F. Tari, Flat geometry of cuspidal edges, arXiv:1610.08702. [17] K. Saji, M. Umehara, K. Yamada, The geometry of fronts, Ann. Math. 169 (2009) 491–529. [18] K. Saji, M. Umehara, K. Yamada, Ak singularities of wave fronts, Math. Proc. Camb. Philos. Soc. 146 (3) (2009) 731–746. [19] K. Saji, M. Umehara, K. Yamada, The duality between singular points and inflection points on wave fronts, Osaka J. Math. 47 (2) (2010) 591–607. [20] S. Shiba, M. Umehara, The behavior of curvature functions at cusps and inflection points, Differ. Geom. Appl. 30 (3) (2012) 285–299. [21] M. Umehara, Differential geometry on surfaces with singularities, in: H. Arai, T. Sunada, K. Ueno (Eds.), The World of Singularities, Nippon-Hyoron-sha Co., Ltd., 2005, pp. 50–64 (in Japanese). [22] K. Teramoto, Parallel and dual surfaces of cuspidal edges, Differ. Geom. Appl. 44 (2016) 52–62. http://refhub.elsevier.com/S0166-8641(17)30616-8/bib414756s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib414756s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4247s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib49s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib49s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib697As1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib494Fs1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib53684361724369644661426F6F6Bs1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib53684361724369644661426F6F6Bs1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib697A74s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib697A74s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6B72737579s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6B72737579s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4D4E61s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4D4E61s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6D6Es1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6D6Es1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4D53s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4D535559s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4D535559s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6D75s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4Es1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib4Es1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6E616F6B6177616574616Cs1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib6F7365747461726932s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib66726F6E74s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib73757933s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib7375796Fs1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib7375796Fs1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib7375s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib7375s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib75s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib75s1 http://refhub.elsevier.com/S0166-8641(17)30616-8/bib746572616D6F746Fs1 Geometry of cuspidal edges with boundary 1 Maps from manifolds with boundary 2 Normal form of cuspidal edge with boundary 3 Differential geometric information 3.1 The case (1) 3.2 The case (2) 4 Singularities of flat extension of a flat surface Appendix A Curvature and torsion of space curves with singularities References