Title. Author(s)IZUMIYA, SHYUICHI; PEI, DONGHE; TAKAHASHI, MASATOMO. CitationProceedings of the Edinburgh Mathematical Society, 4.

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1 Title SINGULARITIES OF EVOLUTES OF HYPERSURFACES IN HYPERB Authors)IZUMIYA, SHYUICHI; PEI, DONGHE; TAKAHASHI, MASATOMO CitationProceedings of the Edinburgh Mathematical Society, 4 Issue Date 2004 Doc URL Rights Copyright 2004 Cambridge University Press Type article File Information PEMS47pdf Instructions for use Hokkaido University Collection of Scholarly and Aca

2 Proceedings of the Edinburgh Mathematical Society 2004) 47, c DOI:101017/S Printed in the United Kingdom SINGULARITIES OF EVOLUTES OF HYPERSURFACES IN HYPERBOLIC SPACE SHYUICHI IZUMIYA 1, DONGHE PEI 2 AND MASATOMO TAKAHASHI 3 1 Department of Mathematics, Hokkaido University, Sapporo , Japan izumiya@mathscihokudaiacjp) 2 Department of Mathematics, Northeast Normal University, Changchun , People s Republic of China northlcd@publicccjlcn; pei@mathscihokudaiacjp) 3 Department of Mathematics, Hokkaido University, Sapporo , Japan takahashi@mathscihokudaiacjp) Received 23 April 2003) Abstract We study the differential geometry of hypersurfaces in hyperbolic space As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes Keywords: hypersphere; equidistant hyperplane; hyperbolic de Sitter) evolute; hyperbolic timelike spacelike) height function; Lagrangian singularities 2000 Mathematics subject classification: Primary 53A35 Secondary 58C27 1 Introduction In this paper we study the differential geometry of hypersurfaces in hyperbolic space from a contact viewpoint as an application of singularity theory There are several articles [9,12 15] concerning the contact of submanifolds in Euclidean space with hyperplanes or hyperspheres Such hypersurfaces are known as totally umbilic hypersurfaces in Euclidean space A singular point of the Gauss map of a hypersurface ie a parabolic point) is a point at which the tangent hyperplane has degenerate contact with the hypersurface cf [2, 3, 9]) Therefore, we might say that the theory of singularities for Gauss maps describes the contact of hypersurfaces with hyperplanes For the contact of hypersurfaces with hyperspheres, the evolute of a hypersurface plays a role similar to that of a Gauss map On the other hand, the basic notions and tools for the study of the differential geometry of hypersurfaces in hyperbolic space has recently been established in [6 8] The hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space has been explicitly described and the contact of hypersurfaces with hyperhorospheres has been systematically studied 131

3 132 S Izumiya, D Pei and M Takahashi as an application of singularity theory to the hyperbolic Gauss indicatrix In hyperbolic space there are four kinds of totally umbilic hypersurfaces cf 2) The hyperhorosphere is one of the totally umbilic hypersurfaces in hyperbolic space We have already studied the contact of hypersurfaces with hyperhorospheres in [6] Therefore, we study the contact of hypersurfaces with totally umbilic hypersurfaces other than hyperhorospheres in this paper In 2 we review the basic notions and concepts in hyperbolic differential geometry on hypersurfaces We adopt the model of hyperbolic space in Minkowski space, which is quite natural for the study of hypersurfaces from the contact viewpoint We introduce the notion of hyperbolic respectively, de Sitter) evolutes of hypersurfaces whose singularities describe the contact of hypersurfaces with hyperspheres respectively, equidistant hyperplanes) in 3 We also introduce the notion of timelike ridge points respectively, spacelike ridge points) at which the hypersurface has A k 3 -type contact with a hypersphere respectively, an equidistant hyperplane) The ridges of surfaces in Euclidean 3-space were originally introduced by Porteous [15] as the sets of points at which the surface has a higher-order contact with some of their focal spheres It is deeply related to the singularities of the distance-squared function on the surface We define the analogous notion of hypersurfaces in hyperbolic space For the study of their geometric meanings, we investigate hyperbolic timelike respectively, spacelike) height functions on hypersurfaces In 4 we show that the hyperbolic respectively, de Sitter) evolute of a hypersurface is a caustic of a certain Lagrangian submanifold in the cotangent bundle of the hyperbolic n-space whose generating family is the hyperbolic timelike respectively, spacelike) height function In 5 we apply the theory of Lagrangian singularities and interpret a singularity of a hyperbolic respectively, de Sitter) evolute as describing not only the contact of the hypersurface with a hypersphere respectively, equidistant hyperplane) but also the contact of the hypersurface with a family of hypersurfaces respectively, equidistant hyperplanes) This study leads us to the osculating spherical respectively, equidistant planar) foliations In 6 we study generic properties, and we give a classification for n =3in 7 We shall assume throughout the paper that all the maps and manifolds are C unless the contrary is explicitly stated 2 Basic concepts and notions In this section we review basic notions and concepts on the differential geometry of hypersurfaces in hyperbolic space We adopt the model of hyperbolic space in Minkowski space Let R n+1 = {x 0,x 1,,x n ) x i R i =0, 1,,n)} be an n + 1)-dimensional vector space For any vectors x = x 0,x 1,,x n ), y = y 0,y 1,,y n ) in R n+1, the pseudo-scalar product of x and y is defined to be x, y = x 0 y 0 + n i=1 x iy i We call R n+1,, ) the n + 1)-dimensional Minkowski space We use R n+1 1 instead of R n+1,, ) We say that a vector x R n+1 1 \{0} is spacelike, lightlike or timelike if x, x > 0, = 0 or < 0, respectively The norm of the vector x R n+1 is defined by x = x, x For any vector v R n+1 and a real number c, we define the hyperplane with pseudo-normal

4 Singularities of evolutes of hypersurfaces in hyperbolic space 133 v by HPv,c)={x R n+1 1 x, v = c} We call HPv,c)aspacelike hyperplane, atimelike hyperplane or a lightlike hyperplane when v is timelike, spacelike or lightlike, respectively We now define the hyperbolic n-space by H+ 1) n = {x R n+1 1 x, x = 1, x 0 > 0} and the de Sitter n-space by S1 n = {x R n+1 1 x, x =1} We also define H 1) n = {x R n+1 1 x, x = 1, x 0 < 0} For any a 1, a 2,,a n R n+1 1, we define a vector a 1 a 2 a n by e 1 e 2 e n+1 a 1 0 a 1 1 a 1 n a 1 a 2 a n = a 2 0 a 2 1 a 2 n, a n 0 a n 1 a n n where e 1, e 2,,e n+1 is the canonical basis of R n+1 1 and a i =a i 0,a i 1,,a i n) We can easily check that a, a 1 a 2 a n = deta, a 1,,a n ), so that a 1 a 2 a n is pseudo-orthogonal to any a i i =1,,n) We also define a set LC c = {x R n+1 1 x c, x c =0}, which is called a closed lightcone with the vertex c We define LC + = {x =x 0,x 1 x n ) LC 0 x 0 > 0} and we call it the future lightcone at the origin If x =x 0,x 1,,x n ) is a non-zero lightlike vector, then x 0 0 Therefore, we have x = 1, x 1,, x ) n S+ n 1 = {x =x 0,x 1,,x n ) x, x =0,x 0 =1} x 0 x 0 Here, we call S+ n 1 the lightcone n 1)-sphere Let x : U H+ 1) n be an embedding, where U R n 1 is an open subset We define M = xu) and identify M and U by the embedding x For any p = xu) M H+ 1), n we have xu), xu) = 1 It follows that x ui u), xu) =0 i =1, 2,,n 1), where u =u 1,u 2,,u n 1 ) and x ui u) = x/ u i u) =x 0ui u),x 1ui u),,x nui u)) Hence the tangent space of M at p is T p M = x u1 u), x u2 u),,x un 1 u) R

5 134 S Izumiya, D Pei and M Takahashi Let N p M be the normal space of M at p = xu) inr n+1 1, then N p M is a Lorentz plane Define a spacelike unit vector eu) = Since xu) N p M,wehave xu) x u 1 u) x un 1 u) xu) x u1 u) x un 1 u) S1 N p M T p H n + 1)) N p M = xu), eu) R A map E : U S1 n defined by Eu) =eu) is called the de Sitter Gauss indicatrix of xu) =M We construct an extrinsic differential geometry on x by using the unit normal e as the unit normal of a hypersurface in Euclidean space In this case, the de Sitter Gauss indicatrix of a hypersurface plays a role similar to that of the Gauss map for a hypersurface in Euclidean space We can easily show that D v e T p M for any p = xu 0 ) M and v T p M Here D v denotes the covariant derivative with respect to the tangent vector v We call the linear transformation A p = de : T p M T p M the de Sitter) shape operator of M = xu) atp = xu 0 ) We denote the eigenvalue of A p by κ p, which we call a de Sitter) principal curvature We call the eigenvector of A p the de Sitter) principal direction By definition, κ p is a de Sitter) principal curvature if and only if deta p κ p I) = 0 We now define the notion of de Sitter) Gauss Kronecker curvatures as follows The de Sitter) Gauss Kronecker curvature of M = xu) atp = xu 0 )is defined to be K d u 0 ) = det A p We say that a point p = xu 0 ) M is an umbilic point if A p = k p id TpM We also say that M is totally umbilic if all points of M are umbilic A hypersurface given by the intersection of H+ 1) n and a spacelike hyperplane, a timelike hyperplane or a lightlike hyperplane is, respectively, called a hypersphere, anequidistant hyperplane or a hyperhorosphere Moreover, if the hypersurface is given by the intersection of H+ 1) n and a timelike hyperplane through the origin of R n+1 1, the equidistant hyperplane is simply called a hyperplane Then the following proposition is a well-known result Proposition 21 Suppose that M = xu) is totally umbilic, then κp) is constant κ Under this condition, we have the following classification 1) Suppose that κ 2 1 a) If κ 0and κ 2 < 1, then M is part of an equidistant hyperplane b) If κ 0and κ 2 > 1, then M is part of a hypersphere c) If κ =0, then M is part of a hyperplane 2) If κ 2 =1, then M is part of a hyperhorosphere Since x ui i =1,,n 1) are spacelike vectors, we induce the Riemannian metric first fundamental form) ds 2 = n 1 i=1 g ijdu i du j on M = xu), where g ij u) = x ui u), x uj u) for any u U We define the de Sitter second fundamental invariant by h ij u) = E ui u), x uj u) for any u U By arguments similar to those of differential geometry

6 Singularities of evolutes of hypersurfaces in hyperbolic space 135 on hypersurfaces in Euclidean space, we can show that the de Sitter Gauss Kronecker curvature is given by K d = deth ij) detg αβ ) For a hypersurface x : U H+ 1), n we say that a point u 0 U or p = xu 0 )is ade Sitter) flat point if h ij u 0 ) = 0 for all i, j Therefore, p = xu 0 ) is a de Sitter) flat point if and only if p is an umbilic point with the vanishing de Sitter) principal curvature We now introduce the notion of evolutes of hypersurfaces in hyperbolic space We say that a point p = xu 0 ) M is a horoparabolic point if one of the de Sitter principal curvatures satisfies the condition that κ 2 u 0 ) = 1 For a hypersurface x : U H+ 1), n we define the total evolute of xu) =M by TE ± M {± = κu) κ2 u) 1 xu)+ 1 ) κu) eu) κu) is a de Sitter principal } curvature at p = xu), u U For a hypersurface as above, we have the following decomposition of the total evolute: TE ± M u) =HE± M SE± M, where HE ± M {± = κu) xu)+ 1 ) κ2 u) 1 κu) eu) κu) is a de Sitter principal curvature with κ 2 u) > 1atp = xu), u U and SE ± M {± = κu) xu)+ 1 ) κ2 u) 1 κu) eu) κu) is a de Sitter principal curvature with κ 2 u) < 1atp = xu), u U }, } We can show that HE ± M Hn + 1) H 1) n and SE ± M Sn 1 Ifx H 1), n then x H+ 1) n It follows that HE + M Hn + 1) or HE M Hn + 1) We consider the component of HE ± M located on Hn + 1) Therefore, we call HE ± M respectively, SE± M ) the hyperbolic evolute respectively, de Sitter evolute) ofxu) =M We define a smooth mapping HE ± κ : U H+ 1) n by HE ± κu) κ u) =± xu)+ 1 ) κ2 u) 1 κu) eu), where we fix a de Sitter principal curvature κu) onu at u with κ 2 u) > 1 We can also define a smooth mapping SE ± κ : U S n 1 in a similar way with κ 2 u) < 1 We have the following proposition

7 136 S Izumiya, D Pei and M Takahashi Proposition 22 Let M = xu) be a hypersurface in H n + 1) without horoparabolic points or de Sitter flat points A) The following are equivalent 1) M is totally umbilic with κ 2 > 1 2) HE ± M isapointinhn + 1) 3) M is part of a hypersphere B) The following are equivalent 1) M is totally umbilic with 0 <κ 2 < 1 2) SE ± M isapointinsn 1 3) M is part of an equidistant hyperplane Proof A) We assume that condition 1) holds Then the de Sitter) principal curvature κu) =κ is constant and κ 2 > 1 Therefore, we have HE ± κ κ u) =± x ui u)+ 1 ) u i κ2 1 κ e u i u) for any u U By the definition of the de Sitter) principal curvature, e ui = κx ui for i =1,,n 1 It follows that HE ± κ / u i u) =0fori =1,,n 1 We conclude that HE ± κ u) is a point Conversely, for any u U and a de Sitter principal curvature κu), we assume that HE ± κu) κ u) =± xu)+ 1 ) κ2 u) 1 κu) eu) is a point We calculate that HE ± κ u i u) = κ ui u) xu)+ 1 κ 2 u) 1) 3/2 ± ) κu) eu) κu) κ2 u) 1 x ui u) κ u i u) κ 2 u) eu)+ 1 κu) e u i u) Since e ui x u1,,x un 1 R and {x, x u1,,x un 1, e} are linearly independent, HE ± κ / u i u) = 0 if and only if κ i u) =0fori =1,,n 1 Therefore, κu) =κ is constant and κ 2 > 1 Moreover, we assume that there exists another de Sitter) principal curvature κ Since HE ± κ u) =HE ± κ u) is a point, we have κ = κ This means that M is totally umbilic Since a hypersphere is totally umbilic, conditions 1) and 3) are equivalent by Proposition 21 This completes the proof of A) The proof of B) is also given by straightforward calculations like those for the proof of A) )

8 3 Height functions Singularities of evolutes of hypersurfaces in hyperbolic space 137 In this section we consider two kinds of families of height functions on a hypersurface in hyperbolic space in order to describe the hyperbolic and the de Sitter evolute of a hypersurface For this purpose we need some concepts and results in the theory of unfoldings of function germs We give a brief review of the theory in the appendix We now define two families of functions H T : U H n + 1) \ M) R by H T u, v) = xu), v and H S : U S1 n R by H S u, v) = xu), v We call H T respectively, H S )ahyperbolic timelike height function respectively, hyperbolic spacelike height function) onx : U H+ 1) n We define h T v u) =H T u, v) respectively, h S vu) =H S u, v)) The following proposition is a standard result Proposition 31 Let x : U H+ 1) n be a hypersurface Then 1) h T v / u i )u) =0i =1,,n 1) if and only if there exist real numbers λ, µ such that v = λxu)+µeu), λ 2 µ 2 =1; and 2) h S v/ u i )u) =0i =1,,n 1) if and only if there exist real numbers λ, µ such that v = λxu)+µeu), λ 2 µ 2 = 1 Since v / M, we have that µ 0 in case 1) By Proposition 31, we can detect both of the catastrophe sets cf the appendix) of H T and H S as follows: CH T )={u, v) U H+ 1) n \ M) v = λxu)+µeu)}, CH S )={u, v) U S1 n v = λxu)+µeu)} We also calculate that 2 H T u, v) = x uiu u i u j u), v = λg ij + µh ij j on CH T ) and 2 H S u i u j u, v) = x uiu j u), v = λg ij + µh ij on CH S ) Therefore, dethh T v )u)) = det 2 H T / u i u j )u, v) =0 respectively, dethh S v)u)) = 0) if and only if κu) = λ/µ is a de Sitter principal curvature Since v H+ 1) n respectively, v S1 n ) and κu) = λ/µ is a de Sitter principal curvature with κ 2 u) > 1 respectively, κ 2 u) < 1), we have B H T =HE + M HE M respectively, B H S =SE+ M SE M )

9 138 S Izumiya, D Pei and M Takahashi Proposition 32 We assume that p = xu 0 ) is not a de Sitter flat point of xu) =M, then we have the following assertions 1) p is an umbilic point with κ 2 p) > 1 if and only if there exists v 0 H n + 1) \ M such that u 0 is a singular point of h T v 0 and rank Hh T v 0 )u 0 )=0 2) p is an umbilic point with 0 <κ 2 p) < 1 if and only if there exists v 0 S n 1 such that u 0 is a singular point of h S v 0 and rank Hh S v 0 )u 0 )=0 Proof 1) Since p is an umbilic point, A p = κ p id TpM There exists an orthogonal matrix Q such that t Qh) j i )Q = κ pi Hence, we may consider the case h) j i = κ pi, so that h ij )=κ p g ij ) Then we put v 0 = λxu 0 )+µeu 0 ) H n + 1) \ M, where λ = ± κ pu 0 ) κ 2 p u 0 ) 1, µ = ± 1 κ 2 p u 0 ) 1 In this case the Hessian matrix Hh T v 0 )u 0 )= λg ij + µh ij )= λ + µκ p u 0 ))g ij )=0 On the other hand, if λg ij + µh ij = 0 for all i, j, then h ij )=κ p g ij )κ p = λ/µ) This is equivalent to the condition h) j i )=κ pi The proof of 2) is also given by direct calculation similar to 1) We say that u 0 is a timelike ridge point respectively, spacelike ridge point) ifh T v respectively, h S v) has the A k 3 -type singular point at u 0, where v B H T respectively, v B H S) For a function germ f :R n 1, ũ 0 ) R, f has A k -type singular point at ũ 0 if f is R + -equivalent to the germ u k+1 1 ± u 2 2 ± ± u 2 n 1 We say that two function germs f i :R n 1, ũ i ) R i =1, 2) are R + -equivalent if there exists a diffeomorphism germ Φ :R n 1, ũ 1 ) R n 1, ũ 2 ) and a real number c such that f 2 Φu) =f 2 u)+c We now consider the geometric meaning of timelike ridge points Let F : H+ 1) n R be a submersion and let x : U H+ 1) n be a hypersurface We say that x and F 1 0) have a corank-r contact at p 0 = xu 0 ) if the Hessian of the function gu) =F xu) has corank r at u 0 We also say that x and F 1 0) have an A k -type contact at p 0 = xu 0 ) if the function gu) =F xu) has the A k -type singularity at u 0 By definition, if x and F 1 0) have an A k -type contact at p 0 = xu 0 ), then these have a corank-1 contact For any r R and a 0 H+ 1) n respectively, a 0 S1 n ), we consider a function F : H+ 1) n R defined by F u) = u, a 0 r We define PS n 1 a 0,r)=F 1 0) = {u H n + 1) u, a 0 = r} It follows that PS n 1 a 0,r) is a hypersphere respectively, equidistant hyperplane) with centre a 0 if a 0 is in H+ 1) n respectively, S1 n ) We put a 0 =HE ± κ u 0 ) respectively, a 0 =SE ± κ u 0 )) and κu 0 ) r 0 = κ2 u 0 ) 1,

10 Singularities of evolutes of hypersurfaces in hyperbolic space 139 where we fix a de Sitter principal curvature κu) onu at u 0 We then have the following simple proposition Proposition 33 With the above notation, there exists an integer l with 1 l n 1 such that xu) =M and PS n 1 a 0,r 0 ) have corank-l contact at u 0 In the above proposition, PS n 1 a 0,r 0 ) is called an osculating hypersphere respectively, osculating equidistant hyperplane) ofm = xu) ifa 0 H n + 1) respectively, a 0 S n 1 ) We also call a 0 the centre of de Sitter principal curvature κu 0 ) By Proposition 32, xu) = M and the osculating hypersphere respectively, equidistant hyperplane) has corank-n 1) contact at an umbilic point Therefore, the hyperbolic respectively, de Sitter) ridge point is not an umbilic point By the general theory of unfoldings of function germs, the bifurcation set B F is nonsingular at the origin if and only if the function f = F R n {0} has the A 2 -type singularity ie the fold-type singularity) Therefore, we have the following proposition Proposition 34 With the same notation as in the previous proposition, the total evolute TE ± M is non-singular at a 0 =TE ± κ u 0 ) if and only if xu) =M and PS n 1 a 0,r 0 ) have A 2 -type contact at u 0 Here, TE ± κ u 0 )=HE ± κ u 0 ) if a 0 H n + 1) and TE ± κ u 0 )=SE ± κ u 0 ) if a 0 S n 1 4 Evolutes as caustics In this section we naturally interpret the hyperbolic de Sitter) evolute of hypersurface in hyperbolic space as a caustic in the framework of symplectic geometry and consider the geometric meaning of singularities In the appendix we give a brief survey of the theory of Lagrangian singularities For notions and basic results on the theory of Lagrangian singularities, please refer to the appendix For a hypersurface x : U H n + 1), we consider the hyperbolic timelike height function H T and the hyperbolic spacelike height function H S cf 3) We have the following propositions Proposition 41 Both the hyperbolic timelike height function H T : U H n + 1) R and the hyperbolic spacelike height function H S : U S n 1 R on x are Morse families Proof First we consider the hyperbolic timelike height function For any v =v 0,v 1,v n ) H+ 1), n we have v 0 = v v2 n + 1, so that H T u, v) = x 0 u) v v2 n +1+x 1 u)v x n u)v n, where xu) =x 0 u),,x n u)) We will prove that the mapping H H T T H T ) =,, u 1 u n 1

11 140 S Izumiya, D Pei and M Takahashi is non-singular at any point The Jacobian matrix of H T is given as follows: x u1u 1, v x u1u n 1, v x un 1u 1, v x un 1u n 1, v v 1 v n x 0u1 + x 1u1 x 0u1 + x nu1 v 0 v 0 v 1 v n, x 0un 1 + x 1un 1 x 0un 1 + x nun 1 v 0 v 0 where x uiu j = 2 x/ u i u j u) We will show that the rank of the matrix X = v 1 v n x 0u1 + x 1u1 x 0u1 + x nu1 v 0 v 0 x 0un 1 v 1 v 0 + x 1un 1 x 0un 1 v n v 0 + x nun 1 is n 1atu, v) CH T ) It is enough to show that the rank of the matrix v 1 v n x 0 + x 1 x 0 + x n 0 v 0 1 v n x 0u1 + x 1u1 x 0u1 + x nu1 v A = 0 v 0 v 1 v n x 0un 1 + x 1un 1 x 0un 1 + x nun 1 v 0 v 0 is n at u, v) CH T ) We define a i = x i x iu1 x iun 1 for i =0,,n Then we have and A = ) v 1 v n a 0 + a 1,, a 0 + a n v 0 v 0 det A = v 0 v 0 deta 1,,a n ) v 1 v 0 deta 0, a 2,,a n ) v n v 0 deta 1,,a n 1, a 0 )

12 Singularities of evolutes of hypersurfaces in hyperbolic space 141 On the other hand, we have x x u1 x un 1 = deta 1,,a n ), deta 0, a 2,,a n ),, 1) n deta 0,,a n 1 )) Therefore, we have det A = v0,, v n ), x x u1 x un 1 v 0 v 0 = 1 v 0 λx + µe, x x u1 x un 1 e = 1 v 0 x x u1 x un 1 µ 0 for u, v) CH T ) Next we consider the hyperbolic spacelike height function The proof is also given by direct calculation but a bit more carefully than in the previous case We use the same notation as in the previous case eg x and a i, etc) For any v S1 n, we have v0 2 + v vn 2 = 1 Without loss of the generality we might assume that v n 0 We have v n = ± 1+v0 2 v2 1 v2 n 1, so that H S u, v) = x 0 u)v 0 + x 1 u)v x n 1 u)v n 1 ± x n u) 1+v0 2 v2 1 v2 n 1 We also prove that the mapping H H S S =,, u 1 H S u n 1 is non-singular at any point The Jacobian matrix of H S is given as follows: x u1u 1, v x u1u n 1, v x un 1u 1, v x un 1u n 1, v v 0 v n 1 x 0u1 + x nu1 x n 1u1 x nu1 v n v n v 0 v n 1 x 0un 1 + x nun 1 x n 1un 1 x nun 1 v n v n )

13 142 S Izumiya, D Pei and M Takahashi We will also show that the rank of the matrix v 0 v 1 v n 1 x 0u1 + x nu1 x 1u1 x nu1 x n 1u1 x nu1 v n v n v n X = v 0 v 1 v n 1 x 0un 1 + x nun 1 x 1un 1 x nun 1 x n 1un 1 x nun 1 v n v n v n is n 1atu, v) CH S ) It can be proved that the rank of the matrix ) v 0 v 1 v n 1 à = a 0 + a n, a 1 a n,,a n 1 a n v n v n v n is n at u, v) CH S ) Therefore, we have det à = 1)n 1 { v0 v n deta 1,,a n ) v 1 v n deta 0, a 2,,a n ) = 1) n 1 v0,, v n ), x x u1 x un 1 v n v n = 1)n 1 λx + µe, x x u1 x un 1 e v n = 1)n 1 x x u1 x un 1 µ 0 v n + + 1) n v } n deta 0,,a n 1 ) v n for u, v) CH S ) This completes the proof of the proposition By the method for constructing the Lagrangian immersion germ from the Morse family see the appendix), we can define a Lagrangian immersion germ whose generating family is the hyperbolic timelike height function or the hyperbolic spacelike height function of M = xu) as follows For a hypersurface x : U H n + 1), xu) =x 0 u),,x n u)), we define a smooth mapping LH T ):CH T ) T H n + 1) by LH T )u, v) = v, x 0 u) v 1 + x 1 u),, x 0 u) v ) n + x n u), v 0 v 0 where v =v 0,,v n ) H+ 1) n and v 0 = v v2 n + 1 Here we have used the triviality of the cotangent bundle T H+ 1) n For the de Sitter space S1 n, we consider the local coordinate U i = {v =v 0,,v n ) S1 n v i 0} Since T S1 n U i is a trivial bundle, we define a map L i H S ):CH S ) T S n 1 U i i =0, 1,,n)

14 Singularities of evolutes of hypersurfaces in hyperbolic space 143 by L i H S )u, v) = v, x 0 u)+x i u) v 0,x 1 u) x i u) v 1,, x i u) x i u) v i,,x n u) x i u) v ) n, v i v i v i where v =v 0,,v n ) S n 1 and we define x 0,,ˆx i,,x n )asapointinn-dimensional space such that the ith component x i is removed We can show that if U i U j for i j, then L i H S ) and L j H S ) are Lagrangian equivalent such that the corresponding Lagrangian equivalence is given by the local coordinate change of S n 1 and the Lagrangian lift of it Indeed, we define the local coordinate change of S n 1 for i<j; ϕ ij : U i U j,by ϕ ij v 0,,ˆv i,,v n ) = v 0,,v i = 1+v 20 v21 ˆv2i v2n,,ˆv j,,v n ), and ϕ ij : T S1 n T S1 n are Lagrangian lifts of ϕ ij that are defined by ϕ ij ξ) =ϕ 1 ij ) ξ Then ϕ ij are symplectic diffeomorphism germs cf [1]) Also we define diffeomorphism germs ˆσ ij : U U i U U j by ˆσ ij u, v) =u, ϕ ij v)) and σ ij =ˆσ ij CHS ), then ϕ ij L i H S )=L j H S ) σ ij and ϕ ij π = π ϕ ij Therefore, we can define a global Lagrangian immersion: LH S ):CH S ) T S1 n By definition, we have the following corollary of the above proposition Corollary 42 With the above notation, LH T )respectively, LH S )) is a Lagrangian immersion such that the hyperbolic timelike height function H T : U H n + 1) R respectively, hyperbolic spacelike height function H S : U S n 1 R) of x is a generating family of LH T )respectively, LH S )) Therefore, we have the Lagrangian immersion LH T ) respectively, LH S )) whose caustic is the hyperbolic evolute respectively, de Sitter evolute) of x We call LH T ) respectively, LH S )) thelagrangian lift of the hyperbolic evolute respectively, de Sitter evolute) of x v i 5 Contact with families of hyperspheres and equidistant hyperplanes Before we start to consider the contact between a hypersurface and a family of hyperspheres or equidistant hyperplanes, we briefly describe the theory of contact with foliations Montaldi [13] considered that the relationship between the contact of submanifolds and singularity type more precisely, the K-class; cf [11]) of maps Here we consider the relationship between the contact of submanifolds with foliations and the R + -class of functions Let X i i =1, 2) be submanifolds of R n with dim X 1 = dim X 2, let g i :X i, x i ) R n, ȳ i ) be immersion germs, and let f i :R n, ȳ i ) R, 0) be submersion germs For a submersion germ f :R n, 0) R, 0), we let F f be the regular foliation defined by f, ie F f = {f 1 c) c R, 0)} We say that the contact of X 1 with the regular foliation F f1 at ȳ 1 is the same type as the contact of X 2 with the regular foliation F f2

15 144 S Izumiya, D Pei and M Takahashi at ȳ 2 if there is a diffeomorphism germ Φ :R n, ȳ 1 ) R n, ȳ 2 ) such that ΦX 1 )=X 2 and ΦY 1 c)) = Y 2 c), where Y i c) =f 1 i c) for each c R, 0) In this case we write KX 1, F f1 ;ȳ 1 )=KX 2, F f2 ;ȳ 2 ) It is clear that in the definition R n could be replaced by any manifold We apply the method of Goryunov [5] to the case for R + -equivalences among function germs, so that we have the following Proposition 51 see the appendix in [5]) Let X i i =1, 2) be submanifolds of R n with dim X 1 = dim X 2 = n 1ie hypersurfaces), let g i :X i, x i ) R n, ȳ i ) be immersion germs, and let f i :R n, ȳ i ) R, 0) be submersion germs We assume that x i are singularities of function germs f i g i :X i, x i ) R, 0) Then KX 1, F f1 ;ȳ 1 )= KX 2, F f2 ;ȳ 2 ) if and only if f 1 g 1 and f 2 g 2 are R + -equivalent On the other hand, Golubitsky and Guillemin [4] have given an algebraic characterization for the R + -equivalence among function germs We denote by C0 X) the set of function germs X, 0) R Let J f be the Jacobian ideal in C0 X) ie J f = f/ x 1,, f/ x n C 0 X)) Let R k f) =C0 X)/Jf k and let f be the image of f in this local ring We say that f satisfies the Milnor Condition if dim R R 1 f) < Proposition 52 see Proposition 41 in [4]) Let f and g be germs of functions at 0 in X satisfying the Milnor Condition with f/ x i 0) = g/ x i 0)=0i =1,,n) Then f and g are R + -equivalent if 1) the rank and signature of the Hessians Hf)0) and Hg)0) are equal; and 2) there is an isomorphism γ : R 2 f) R 2 g) such that γ f) =ḡ We consider a function H T : H n + 1) H n + 1)\M) R defined by H T x, v) = x, v For any v 0 H n + 1) \ M, we define h T v 0 x) =H T x, v 0 ) and we have a hypersphere h T v 0 ) 1 λ) = HPv 0,λ) H n + 1)=PS n 1 v 0,λ) It is easy to show that h T v 0 is a submersion For any ū 0 U, we consider a timelike vector ie in hyperbolic n-space) v 0 = λxū 0 )+µeū 0 ) H+ 1), n then we have h T v 0 xū 0 )=H x id H n + 1))ū 0, v 0 )=λ, and h T v 0 x ū 0 )= HT ū 0, v 0 )=0, u i u i for i =1,,n 1 This means that the hypersphere h T v 0 ) 1 λ) =PS n 1 v 0,λ) is tangent to M = xu)atp = xū 0 ) In this case we call p = xū 0 ) and PS n 1 v 0,λ)atangent hypersphere with the centre v 0 However, there are infinitely many tangent hyperspheres at a general point p = xū 0 ) depending on the real number λifv 0 is a point of the hyperbolic evolute, the tangent hypersphere with the centre v 0 is called the osculating hypersphere at p = xū 0 ), which is uniquely determined Let x i :U, ū i ) H+ 1), n x i ū i )) i = 1, 2) be hypersurface germs We consider hyperbolic timelike height functions Hi T : U H+ 1), n ū i, v i )) R of x i, where v i are points of hyperbolic evolutes of x i, respectively We define h T i,v i u) =Hi Tu, v i), then we have h T i,v i u) =h T v i x i u) Then we have the following theorem

16 Theorem 53 Let Singularities of evolutes of hypersurfaces in hyperbolic space 145 x i :U, ū i ) H n + 1), x i ū i )) be hypersurface germs such that the corresponding Lagrangian immersion germs LH T i ):CH T i ), ū i, v i )) T H n + 1), z i ) are Lagrangian stable, where v i are centres of the osculating hyperspheres of x i, respectively Then the following conditions are equivalent 1) Kx 1 U), F h T v1 ; xū 1 )) = Kx 2 U), F h T v2 ; xū 2 )) 2) h T 1,v 1 and h T 2,v 2 are R + -equivalent 3) H T 1 and H T 2 are P R + -equivalent 4) LH T 1 ) and LH T 2 ) are Lagrangian equivalent 5) a) The rank and signature of the Hh T 1,v 1 )ū 1 ) and Hh T 2,v 2 )ū 2 ) are equal b) There is an isomorphism γ : R 2 h T 1,v 1 ) R 2 h T 2,v 2 ) such that γh T 1,v 1 )=h T 2,v 2 Proof By Proposition 51, condition 1) is equivalent to condition 2) Since both of LHi T) are Lagrangian stable, both of HT i are R + -versal unfoldings of h T i,v i, respectively By the uniqueness theorem on the R + -versal unfolding of a function germ, condition 2) is equivalent to condition 3) By Proposition A 3, condition 3) is equivalent to condition 4) It also follows from Proposition A 3 that both of h T i satisfy the Milnor Condition Therefore, we can apply Proposition 52 to our situation, so that condition 2) is equivalent to condition 5) This completes the proof We remark that if LH T 1 ) and LH T 2 ) are Lagrangian equivalent, then the corresponding hyperbolic evolutes are diffeomorphic Since the hyperbolic evolute of a hypersurface xu) =M is considered to be the caustic of LH T ), the above theorem gives a symplectic interpretation for the contact of hypersurfaces with a family of hyperspheres cf the appendix) Similarly, we can construct the osculating equidistant hyperplane of a hypersurface x : U H n + 1) by using a function H S : H n + 1) S n 1 R defined by H S x, v) = x, v For any v 0 S n 1, we also define h S v 0 x) =H S x, v 0 ) and we have h S 0,v 0 u) =h S v 0 xu) Then we have the following theorem Theorem 54 Let x i :U, ū i ) H n + 1), x i ū i )) be hypersurface germs such that the corresponding Lagrangian immersion germs LH S i ):CH T i ), ū i, v i )) T H n + 1), z i ) are Lagrangian stable, where v i are centres of osculating equidistant hyperplanes of x i, respectively Then the following conditions are equivalent

17 146 S Izumiya, D Pei and M Takahashi 1) Kx 1 U), F h S v1 ; xū 1 )) = Kx 2 U), F h S v2 ; xū 2 )) 2) h S 1,v 1 and h S 2,v 2 are R + -equivalent 3) H S 1 and H S 2 are P R + -equivalent 4) LH S 1 ) and LH S 2 ) are Lagrangian equivalent 5) a) The rank and signature of the Hh S 1,v 1 )ū 1 ) and Hh S 2,v 2 )ū 2 ) are equal b) There is an isomorphism γ : R 2 h S 1,v 1 ) R 2 h S 2,v 2 ) such that γh S 1,v 1 )=h S 2,v 2 The proof follows by direct analogy with the proof for Theorem 53 so we omit it 6 Generic properties In this section we consider generic properties of hypersurfaces in H n + 1) The main tool is a kind of transversality theorem We consider the space of embeddings EmbU, H n + 1)) with Whitney C -topology We define two functions: a) H T : H n + 1) H n + 1) R; H T u, v) = u, v ; and b) H S : H n + 1) S n 1 R; H S u, v) = u, v We claim that h T v respectively, H S v) is a submersion for any v H+ 1) n respectively, v S1 n ), where h T v u) = H T u, v) respectively, h S vu) = H S u, v)) For any x EmbU, H+ 1)), n we have H T = H T x id H n + 1)) and H S = H S x id S n 1 ) We also have the l-jet extensions j l 1H T : U H n + 1) J l U, R) respectively, j l 1H S : U S n 1 J l U, R)) defined by j1h l T u, v) =j l h T v u) respectively, j1h l S u, v) =j l h S vu)) We consider the trivialization J l U, R) U R J l n 1, 1) For any submanifold Q J l n 1, 1), we define Q = U R Q Then we have the following proposition as a corollary of Lemma 6 in Wassermann [16] see also [14] and [10]) Proposition 61 Let Q be a submanifold of J l n 1, 1) Then the set T X Q = {x EmbU, H n + 1)) j l 1H X is transversal to Q} is a residual subset of EmbU, H n + 1)) IfQ is a closed subset, then T Q is open Here, X is T or S In the case when n 6, we have finitely many R-orbits in J l n, 1) consisting of the jet z = j l f0) with dim R 1 f) n Let Σ l 0n, 1) be the union of R-orbits consisting of the jet z = j l f0) with dim R 1 f) >n It is known that Σ l 0n, 1) is a semi-algebraic subset of J l n, 1) with codim Σ l 0n, 1) > 2n + 1 for sufficiently large l Therefore, we have a stratification of J l n, 1) \ Σ l 0n, 1)) Σ l 0n, 1) by finitely many R-orbits in J l n, 1) \ Σ l 0n, 1)) and semi-algebraic stratum of Σ l 0n, 1) with codimension greater than 2n +1 By the above proposition, the appendix and the characterization of R + -versal unfolding cf [1]), we have the following theorem

18 Singularities of evolutes of hypersurfaces in hyperbolic space 147 Theorem 62 Suppose that n 6, then there is an open dense subset O EmbU, H n + 1)) such that for any x O, the germ of the Lagrangian lift of the hyperbolic respectively, de Sitter) evolute of x at each point is Lagrangian stable 7 Surfaces in hyperbolic 3-space In this section we stick to the case when n = 3 Let x : U H+ 1) 3 be a surface, H T : U H+ 1) 3 R be a hyperbolic timelike height function and H S : U S1 3 R be a hyperbolic spacelike height function We consider the hyperbolic evolutes HE ± M respectively, de Sitter evolutes SE ± M ) In the case n =3,ifp 0 = xu 0,v 0 ) is not an umbilic point, then h T a 0 respectively, h S a 0 ) has the A k 2 -type singularity at p 0 by Proposition 32, where a 0 =HE ± M u 0,v 0 ) respectively, a 0 =SE ± M u 0,v 0 )) Since HE ± M respectively, SE± M )is the bifurcation set of H T respectively, H S ), it is non-singular if and only if h T a 0 respectively, h S a 0 ) has the A 2 -type singularity ie the fold-type singularity) at p 0 Therefore, we have the following proposition Proposition 71 1) p 0 = xu 0,v 0 ) is a timelike ridge point if and only if p 0 is a non-umbilic point and the corresponding point a 0 =HE ± M u 0,v 0 ) is a singular point of HE ± M u, v) 2) p 0 = xu 0,v 0 ) is a spacelike ridge point if and only if p 0 is a non-umbilic point and the corresponding point a 0 =SE ± M u 0,v 0 ) is a singular point of SE ± M u, v) A line of de Sitter principal curvature on M is a curve which is everywhere tangent to these de Sitter principal directions Let xt) = xut),vt)) be a regular curve on xu) =M and consider the corresponding curve on HE ± M respectively, SE± M ), which is given by κt) at) =± xut),vt)) + 1 ) κ2 t) 1 κt) eut),vt)), where κu, v) is a corresponding de Sitter principal curvature with κ 2 u, v) > 1 respectively, κ 2 u, v) < 1) on U and κt) =κut),vt)) We have the following characterization of the ridge point Corollary 72 Let p 0 = xu 0,v 0 ) be a non-umbilic point of M Then we have the following assertions 1) We assume that κ 2 t) > 1, in which case p 0 = xut 0 ),vt 0 )) is a timelike ridge point if and only if there exists a line of de Sitter principal curvature xt) = xut),vt)) with p 0 = xut 0 ),vt 0 )) and κt 0 )=0 2) We assume that κ 2 t) < 1, in which case p 0 = xut 0 ),vt 0 )) is a spacelike ridge point if and only if there exists a line of de Sitter principal curvature xt) = xut),vt)) with p 0 = xut 0 ),vt 0 )) and κt 0 )=0

19 148 S Izumiya, D Pei and M Takahashi Proof 1) Suppose that p 0 = xut 0 ),vt 0 )) is a timelike ridge point By Proposition 71, there exists a regular curve xt) =xut),vt)) with p 0 = xut 0 ),vt 0 )) and ȧt 0 )=0 We can calculate that κt) ȧt) =± κ 2 t) 1) 3/2 ± κt) κ2 t) 1 xut),vt)) + 1 ) κt) eut),vt)) dx dt ut),vt)) + 1 κt) de κt) ut),vt)) dt κt) 2 eut),vt)) Since dx/dt +1/κ) de/dt is a tangent vector and x, e are linearly independent normal vectors, ȧt) =0 if and only if κt) = 0 and dx/dt +1/κ)de/dt = 0 Therefore, we have κt 0 ) = 0 and dx/dtt 0 )+1/κ)de/dtt 0 )=0 This means that the tangent vector dx/dtt 0 ) gives a de Sitter principal direction at p 0 We can choose the curve xut),vt)) as a line of de Sitter principal curvature with xut 0 ),vt 0 )) = p 0 The converse assertion follows by straightforward calculations The proof of 2) is also given by similar arguments to those for the proof of 1) By Theorems 62 and A 2 and the classification of function germs under R + - codimension less than or equal to 3, we have the following classification theorem Theorem 73 There exists an open dense subset O EmbU, H 3 + 1)) such that for any x O, the corresponding Lagrangian immersion germ LH T ) at any point u 0,v 0 ), a 0 ) U H 3 + 1) \ M) is Lagrangian equivalent to a Lagrangian immersion germ LF ):CF ), 0) T R 3 whose generating family F u, v, q) q =q 1,q 2,q 3 ) R 3 ) is one of the germs in the following list 1) u 3 + v 2 + q 1 u fold) 2) ±u 4 + v 2 + q 1 u + q 2 u 2 ±cusp) 3) u 5 + v 2 + q 1 u + q 2 u 2 + q 3 u 3 swallowtail) 4) u 3 uv 2 + q 1 u + q 2 v + q 3 u 2 + v 2 )pyramid) 5) u 3 + v 3 + q 1 u + q 2 v + q 3 uv purse) We also have exactly the same result for de Sitter evolutes However, we only change the notation in the above theorem; we omit the detailed statement for de Sitter evolutes here We now apply Theorem 53 to the above classification theorem Let F u, v, q) beone of the germs in the above list We write fu, v) =F u, v, 0), then we define Ff) asthe singular foliation germ in R 2, 0) defined by f ie {f 1 c)} c R,0) ) As a corollary of the above classification theorem and Theorem 53, we have the following Corollary 74 There exists an open dense subset O EmbU, H 3 + 1)) such that for any x Oand any point u 0,v 0 ), a 0 ) U H 3 + 1) \ M), the osculating spherical foliation germ x 1 F h T a0 ), u 0,v 0 )) is diffeomorphic to a foliation germ Ff), 0), where F u, v, q) is one of the germs in the list of Theorem 73 )

20 Singularities of evolutes of hypersurfaces in hyperbolic space fold + cusp cusp swallowtail pyramid purse Figure 1 Generic osculating spherical foliation germs n = 3) We can draw the corresponding pictures of the foliation germs as in Figure 1 8 Examples In the last part of the paper, we give some examples and draw their pictures Since we only consider the local situation, we use the notion of the hyperbolic Monge H-Monge) form of a surface, which was introduced in [6] Let fu, v) be a function with f0)=0 and f u 0) = f v 0) = 0 Then we have a surface in H 3 + 1) defined by x f u, v) = f 2 u, v)+u 2 + v 2 +1,fu, v),u,v) We can easily calculate that e0)=0, 1, 0, 0) We call x f a hyperbolic Monge form briefly, H-Monge form) If the point x f 0, 0) = 1, 0, 0, 0) is umbilic with principal curvature κ0) = κ, the function f can be written in the form fu, v) =κ/2)u 2 + v 2 )+ gu, v), where Hf0) = 0 Under the assumption that 0 <κ 2 1, the centre of the osculating sphere is given by a 0 =κ/ κ 2 1 )1, 1/κ, 0, 0) Therefore, the osculating spherical foliation is given by { u, v) { } κ 1 κ2 1 κ fu, v)+ f 2 u, v)+u 2 + v 2 +1 = } κ κ2 1 + c c R,0)

21 150 S Izumiya, D Pei and M Takahashi fold + cusp cusp swallowtail pyramid purse Figure 2 Osculating spherical foliation germs for examples 1) 5) For example, we consider the case where κ = 1 2 and the gu, v) are given as follows 1) gu, v) =3u 3 v 2 ) fold) 2) gu, v) =3±u 4 + v 2 )±cusp) 3) gu, v) =3u 5 v 2 ) swallowtail) 4) gu, v) =3u 3 uv 2 ) pyramid) 5) gu, v) =3u 3 + v 3 ) purse) Since κ = 1 2, the centre of the osculating sphere is located on S3 1 and the total evolute is the de Sitter evolute It might be very hard to draw the picture of the de Sitter evolute for each surface However, we can easily draw the picture of the osculating spherical foliation for each surface by using the package ImplicitPlot of Mathematica as in Figure 2 Appendix A Lagrangian singularities and unfoldings of functions In this section we give a brief review of the theory of Lagrangian singularities given in [1] We consider the cotangent bundle π : T R r R r over R r Let u, p) =

22 Singularities of evolutes of hypersurfaces in hyperbolic space 151 u 1,,u r,p 1,,p r ) be the canonical coordinate on T R r Then the canonical symplectic structure on T R r is given by the canonical 2-form ω = r i=1 dp i du i Let i : L T R r be an immersion We say that i is a Lagrangian immersion if dim L = r and i ω = 0 In this case the critical value of π i is called the caustic of i : L T R r and it is denoted by C L The main result in the theory of Lagrangian singularities is to describe Lagrangian immersion germs by using families of function germs Let F :R n R r, 0, 0)) R, 0) be an r-parameter unfolding of function germs We call CF )= { x, u) R n R r, 0, 0)) F x, u) = = F x, u) =0 x 1 x n the catastrophe set of F and { B F = u R r, 0) 2 there exist x, u) CF ) such that rank F }, ) x, u) x i x j } <n the bifurcation set of F Let π r :R n R r, 0) R r, 0) be the canonical projection, then we can easily show that the bifurcation set of F is the critical value set of π r CF ) We say that F is a Morse family if the map germ F F =,, F ) :R n R r, 0) R r, 0) u 1 u r is non-singular, where x, u) =x 1,,x n,u 1,,u r ) R n R r, 0) In this case we have a smooth submanifold germ CF ) R n R r, 0) and a map germ LF ):CF), 0) T R r defined by LF )x, u) = u, F,, F ) u 1 u r We can show that LF ) is a Lagrangian immersion Then we have the following fundamental theorem [1, p 300] Proposition A 1 All Lagrangian submanifold germs in T R r are constructed by the above method Using the above notation, we call F a generating family of LF ) We define an equivalence relation among Lagrangian immersion germs Let i :L, x) T R r,p) and i : L,x ) T R r,p ) be Lagrangian immersion germs Then we say that i and i are Lagrangian equivalent if there exist a diffeomorphism germ σ :L, x) L,x ), a symplectic diffeomorphism germ τ :T R r,p) T R r,p ), and a diffeomorphism germ τ :R r,πp)) R r,πp )) such that τ i = i σ and π τ = τ π, where π :T R r,p) R r,πp)) is the canonical projection and a symplectic diffeomorphism germ is a diffeomorphism germ which preserves symplectic structure on T R r In this case, the caustic C L is diffeomorphic to the caustic C L by the diffeomorphism germ τ A Lagrangian immersion germ into T R r at a point is said to be Lagrangian stable if for every map with the given germ there is a neighbourhood in the space of Lagrangian

23 152 S Izumiya, D Pei and M Takahashi immersions in the Whitney C -topology) and a neighbourhood of the original point such that each Lagrangian immersion belonging to the first neighbourhood has in the second neighbourhood a point at which its germ is Lagrangian equivalent to the original germ We can interpret the Lagrangian equivalence by using the notion of generating families We denote by E m the local ring of function germs R m, 0) R with the unique maximal ideal M m = {h E m h0)=0} Let F, G :R n R r, 0) R, 0) be function germs We say that F and G are P R + -equivalent if there exists a diffeomorphism germ Φ : R n R r, 0) R n R r, 0) of the form Φx, u) =Φ 1 x, u),φu)) and a function germ h :R r, 0) R such that Gx, u) =F Φx, u)) + hu) For any F 1 M n+r and F 2 M n +r, F 1, F 2 are said to be stably P R + -equivalent if they become P R + -equivalent after the addition to the arguments to x i of new arguments y i and to the functions F i of non-degenerate quadratic forms Q i in the new arguments ie F 1 + Q 1 and F 2 + Q 2 are P R + -equivalent) Let F :R n R r, 0) R, 0) be a function germ We say that F is an R + -versal deformation of f = F Rn {0} if F E n = J f + u 1 Rn {0},, F u r Rn {0} + 1 R, where f J f =,, f x 1 x n E n Theorem A 2 Let F 1 M n+r and F 2 M n +r be Morse families Then we have the following 1) LF 1 ) and LF 2 ) are Lagrangian equivalent if and only if F 1, F 2 are stably P R + - equivalent 2) LF ) is Lagrangian stable if and only if F is a R + -versal deformation of F R n {0} For the proof of the above theorem, see pp 304 and 325 of [1] The following proposition describes the well-known relationship between bifurcation sets and equivalence among unfoldings of function germs Proposition A 3 Let F, G :R n R r, 0) R, 0) be function germs If F and G are P R + -equivalent then there exists a diffeomorphism germ φ :R r, 0) R r, 0) such that φb F )=B G R References 1 V I Arnol d, S M Gusein-Zade and A N Varchenko, Singularities of differentiable maps, vol I Birkhäuser, 1986) 2 T Banchoff, T Gaffney and C McCrory, Cusps of Gauss mappings, Research Notes in Mathematics, vol 55 Pitman, 1982) 3 D Bleeker and L Wilson, Stability of Gauss maps, Illinois J Math ),

24 Singularities of evolutes of hypersurfaces in hyperbolic space M Golubitsky and V Guillemin, Contact equivalence for Lagrangian manifold, Adv Math ), V V Goryunov, Projections of generic surfaces with boundaries, Adv Soviet Math ), S Izumiya, D-H Pei and T Sano, Singularities of hyperbolic Gauss maps, Proc Lond Math Soc ), S Izumiya, D-H Pei, T Sano and E Torii, Evolutes of hyperbolic plane curves, Acta Math Sinica, in press 8 S Izumiya, D-H Pei and T Sano, Horospherical surface of curve in Hyperbolic space, Publ Math Debrecen, in press 9 E E Landis, Tangential singularities, Funct Analysis Applic ), E J N Looijenga, Structural stability of smooth families of C -functions, PhD thesis, University of Amsterdam 1974) 11 J Martinet, Singularities of smooth functions and maps, London Mathematical Society Lecture Notes Series, vol 58 Cambridge University Press, 1982) 12 D K H Mochida, R C Romero-Fuster and M A Ruas, The geometry of surfaces in 4-space from a contact viewpoint, Geom Dedicata ), J A Montaldi, On contact between submanifolds, Michigan Math J ), J A Montaldi, On generic composites of maps, Bull Lond Math Soc ), I Porteous, The normal singularities of submanifold, J Diff Geom ), G Wassermann, Stability of caustics, Math Ann ), 43 50

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