Singularities of pedal curves produced by singular dual curve germs in Sn

Abstract For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are singular, singularity types of pedal curves depend on singularity types of the n-th curvature function germs and the locations of pedal points. In this paper, we investigate sigularity types of pedal curves in such cases.


Introduction
Let I be an open interval such that 0 ∈ I and S n be the n-dimensional unit sphere in R n+1 (n ≥ 2). A C ∞ non-singular map r : I → S n is said to be a spherical unit speed curve if each of the following u i (s) (1 ≤ i ≤ n − 1) is inductively well-defined for any s ∈ I, where initial information are u −1 (s) ≡ 0, u 0 (s) = r(s), u ′ 0 (s) ≡ 1 and κ 0 (s) ≡ 0.
The function κ i : I → R is called the i-th curvature function of r. For a spherical unit speed curve two vectors u i (s) and u j (s) (0 ≤ i, j ≤ n − 1, i = j) are perpendicular ( [17]). Thus we can define one more vector u n (s) uniquely so that {u 0 (s), u 1 (s), . . . , u n (s)} is an orthogonal moving frame and det(u 0 (s), . . . , u n (s)) = 1 for any s ∈ I. The map u n : I → S n , which is called the dual curve of r ( [1], [21]), defines the n-th curvature function in the following way, where the dot in the center is the scalar product.
κ n (s) = u ′ n−1 (s) · u n (s). We see that the dual curve u n is non-singular at s if and only if κ n (s) = 0 (see §2).

T. Nishimura
For any i (−1 ≤ i ≤ n), we put S i u i (s) = (S n − {±u n (s)}) ∩ u −1 (s), . . . , u i (s) R , where u −1 (s), . . . , u i (s) R means the vector subspace spanned by the vectors u −1 (s), . . . , u i (s). Given a spherical unit speed curve r : I → S n , choosing a point P of S n − {±u n (s) | s ∈ I} gives the map which maps s ∈ I to the unique nearest point in S n−1 u n−1 (s) from P . Such a map, which is called the pedal curve relative to the pedal point P for an n-dimensional unit speed curve r, is denoted by ped r,P . Note that since all points in S n−1 u n−1 (s) are the nearest points from ±u n (s) the pedal point P for the map-germ ped r,P at s must be outside {±u n (s)}.
In [17] we have shown the following.
1. The pedal point P is inside S n u n (0) − S n−2 u n−2 (0) if and only if the map-germ ped r,P : (I, 0) → S n is C ∞ left equivalent to the map-germ given by s → (s, 0, . . . , 0).

For any
if and only if the map-germ ped r,P : (I, 0) → S n is C ∞ left equivalent to the map-germ given by the following: Here, two map-germs f, g : (R, 0) → (R n , 0) are said to be C ∞ left equivalent if there exists a germ of C ∞ diffeomorphism h t : (R n , 0) → (R n , 0) such that the identity g = h t • f is satisfied.
The purpose of this paper is to investigate singularities of pedal curves when κ n (0) = 0. We say that the n-th curvature function κ n has an A k -type Theorem 2. Let r : I → S n be an n-dimensional spherical unit speed curve. Suppose that P ∈ S n u n (0) − S n−1 u n−1 (0) . Then the following holds. 1. If κ n has an A k -type singularity at 0 (0 ≤ k ≤ n − 2), then the map-germ ped r,P : (I, 0) → S n is C ∞ left equivalent to the map-germ given by s → (s k+2 , s k+3 , . . . , s 2k+3 (k+2) elements , 0, . . . , 0 (n−k−2) elements ).
Singularities of pedal curves produced by singular dual curve germs in S n 449 2. If κ n has an A n−1 -type singularity at 0, then the map-germ ped r,P : (I, 0) → S n is C ∞ right-left equivalent to the map-germ given by s → (s n+1 , s n+2 , . . . , s 2n ).
Here, two map-germs f, g : (R, 0) → (R n , 0) are said to be C ∞ right-left equivalent if there exist germs of C ∞ diffeomorphisms h s : (R, 0) → (R, 0) and h t : (R n , 0) → (R n , 0) such that the identity g = h t • f • h −1 s is satisfied. In the case that n = 2 Theorem 2 has been announced in [20]. In the case that n ≥ 3 it seems to be almost impossible to obtain similar results when κ n has an A n -type singularity at 0. We may observe its reason in the following way. It is possible to show that ped r,P is C ∞ right-left equivalent to ϕ(s) = (s n+2 , s n+3 + ϕ 2 (s), . . . , s 2n+1 + ϕ n (s)) where ϕ j (s) = o(s 2n+1 ). However, ϕ is not A-simple since in the case that n = 3 fencing curves due to Arnol'd ( [2]) have the form of ϕ and for n ≥ 3 the local multiplicity of ϕ is more than n 2 (n−1) which is an upper bound for the local multiplicity of an A-simple map-germ; and the codimension of T A(ϕ) in T K(ϕ) is positive (for the restriction on the local multiplicity of an A-simple map-germ, see [18], [19]). Thus, there must exist strong restrictions on higher terms ϕ j which can be truncated.
Next, we investigate singularity types of pedal curves when P ∈ S n−1 u n−1 (0) . We concentrate on the case that κ n has an A 0 -type singularity at 0. Note that κ n has an A 0 -type singularity at 0 if and only if the function-germ κ n : (I, 0) → (R, 0) is non-singular, and the dual curve germ is an ordinary cusp in this case.
Theorem 3. Let r : I → S n be an n-dimensional spherical unit speed curve. Suppose that κ n has an A 0 -type singularity at 0. Then the following hold.

For any
In the case that n = 2 the "only if" parts of Theorem 3 has been announced in [20]. Note that the first assertion of Theorem 2 yields only the "only if" part of the first assertion of Theorem 3. By obtaining a complete list of locations of pedal points inside S n−1 u n−1 (0) and singularity types of pedal curves (assertions 2 and 3 of Theorem 3) we can obtain "if" part of the first assertion of Theorem 3.
In §2 we give several preparations to prove Theorems 2 and 3. Theorems 2 and 3 are proved in §3 and §4 respectively.
The author would like to express his sincere gratitude to the referee for making valuable suggestions. He also wishes to thank S. Izumiya for sending a useful hand-written note [10].

Preliminaries
We put where u i (s) t means the transposed vector of u i (s). We further put Then, the following Serret Frenet type formula holds.
By Lemma 2.1 we see that the dual curve u n is non-singular at 0 if and only if κ n (0) = 0. By using Lemma 2.1 again and again we obtain the following: Lemma 2.2. Suppose that κ n has an A k type singularity at 0 (k ≤ n − 1).
The pedal curve of r relative to the pedal point P is given by the following expression: Let Ψ P be the C ∞ map from S n − {±P } to S n given by We see that the image Ψ P (S n −{±P }) is inside the open hemisphere centered at P . Let this open hemisphere be denoted by X P and set , the map Ψ P canonically induces the map Ψ P : B P → X P . Then, Lemma 2.3 shows that ped r,P is factored into three maps in the following way: Let p : B → R n be the blow up centered at the origin.
For any positive integer r let E r be the R-algebra of all C ∞ functiongerms at the origin of R r with usual operations, and let m r be the unique maximal ideal of E r .
For any positive integers p, q given a C ∞ map-germ f : For any positive integer r we put θ(r) = θ(id. R r ), where id. R r is the identity map-germ of R r at the origin. An element of m ℓ p θ(f ) is a vector field along f such that the Taylor polynomial up to (ℓ − 1)-th degree of it at the origin is zero. The map f * : E q → E p is defined by f * (u) = u • f . Two homomorphisms tf (tf is an E p -homomorphism) and ωf (ωf is an E q -homomorphism via f * ) are defined in the following way: where df is the differential of f . We put The Taylor polynomial up to r-th degree at the origin of f is called r jet of f at the origin and is denoted by j r f (0).
Two map-germs f, g : , and is said to be finitely A-determined (resp. finitely L-determined) if f is r-A-determined (resp. r-L-determined) by a certain r.
Note that in the case of the assertion 1 of Theorem 2 the following equalities hold: T K(ped r,P ) = T C(ped r,P ) = T A(ped r,P ) = T L(ped r,P ). is 2n-A-determined.
Since n + 1 and n + 2 are relatively prime, we see that gcd(n + 1, . . . , 2n) = 1, where gcd means the greatest common divisor. Thus, the map f C (z) = (z n+1 , . . . , z 2n ) (z ∈ C), which is the complexification of f , is injective. From this and the fact that f C has an isolated singularity at the origin, by the geometric characterization of finite determinacy due to Mather and Gaffney (see §2 of [23]) we see that f is finitely L-determined. Hence, in order to show that f is 2n-A-determined it is sufficient to show that for any C ∞ map-germ h : (I, 0) → R n such that j 2n h(0) = 0 by Mather's lemma (Corollary 3.2 of [14], see also §4 of [23]).
Let h : (I, 0) → R n be a C ∞ map-germ such that j 2n h(0) = 0. Then, we see easily that the following holds.
Since we see easily that where (X 1 , . . . , X n ) ∈ R n , we have that

T. Nishimura
Apply the Malgrange preparation theorem to (f + h) * m n θ(f + h) and f + h. Then, we have the following desired inclusion: Note that in the case of the assertion 2 of Theorem 2 the following equalities hold but the equality for T L(ped r,P ) does not hold: T K(ped r,P ) = T C(ped r,P ) = T A(ped r,P ).

Proof of Theorem 3
Since } gives a stratification of S n − {±u n (0)}, the "if" parts of the assertions 1-3 of Theorem 3 follow from the corresponding "only if" parts. Moreover, since the "only if" part of the first assertion of Theorem 3 is contained in the assertion 1 of Theorem 2, we just need to show the "only if" parts of the assertions 2 and 3 of Theorem 3.

T. Nishimura
Thus, by the Malgrange preparation theorem we have the following desired inclusion: Therefore, in order to finish the proof of the "only if" part of the assertion 2 in Theorem 3, it is enough to show the following lemma: Proof of Lemma 4.2. In the case that i = 1 Lemma 4.2 holds by the Sylvester duality ( [22], see also the comment of the problem 1999-8 in [5]). Thus, in the following we assume that i ≥ 2.
Thus, f is finitely L-determined by the geometric characterization of finite determinacy due to Mather and Gaffney. Therefore, in order to show that f is (2n + 1)-A-determined it is sufficient to show that for any C ∞ map-germ h : (I, 0) → R n such that j 2n+1 h(0) = 0 by Mather's lemma.