A regular curve C is parametrized by the arc-length parameter s, that is, c : L → R n (L 3 s 7→ c(s) ∈ R n ). Then the tangent vector field dcds along C has unit length, that is, k dc(s)ds k = 1 for all s ∈ L.

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2. Preliminaries

Let R ⁿ be an n-dimentional Euclidean space with Cartesian coordinates (x ₁ , · · · , x _n ).

By a parametrized curve C of class C ^∞ , we mean a mapping c of a certain interval I into R ⁿ given by c(t) = (x ₁ (t), · · · , x _n (t)), for all t ∈ I. If k ^dc(t) _dt k = h ^dc(t) _dt , ^dc(t) _dt i ^1/2 6= 0 for all t ∈ I, then C is called a regular curve in R ⁿ . Here h·, ·i denotes the Euclidean inner product on R ⁿ .

A regular curve C is parametrized by the arc-length parameter s, that is, c : L → R ⁿ (L 3 s 7→ c(s) ∈ R ⁿ ). Then the tangent vector field ^dc _ds along C has unit length, that is, k ^dc(s) _ds k = 1 for all s ∈ L.

Hereafter, curves considered are regular C ^∞ -curves in R ⁿ parametrized by the arc- length parameter. Let C be a curve in R ⁿ , that is, c(s) ∈ R ⁿ for all s ∈ L. Let t(s) = ^dc _ds for all s ∈ L. The vector field t is called a unit tangent vector field along C, and we assume that the curve C satisfies the following conditions (C ₁ ) ∼ (C _n−1 ):

(C ₁ ) : κ ₁ (s) = k dt(s)

ds k = k d ² c(s)

ds ² k > 0 for all s ∈ L.

Then we obtain a well-defined vector field n ₁ along C, that is, for all s ∈ L, n ₁ (s) = 1

κ ₁ (s) · dt(s) ds , and we obtain,

ht(s), n 1 (s)i = 0, hn 1 (s), n 1 (s)i = 1.

(C ₂ ) : κ ₂ (s) = k dn ₁ (s)

ds + κ ₁ (s)t(s)k > 0 for all s ∈ L.

Then we obtain a well-defined vector field n ₂ along C, that is, for all s ∈ L, n ₂ (s) = 1

κ ₂ (s)

 dn ₁ (s)

ds + κ ₁ (s)t(s)  , and we obtain, for i, j = 1, 2,

ht(s), n _i (s)i = 0, hn _i (s), n _j (s)i = δ _j ⁱ ,

where δ _j ⁱ denotes the Kronecker delta. By an inductive procedure, for l = 3, · · · , n − 2, (C l ) : κ l (s) = k dn _l−1 (s)

ds + κ l−1 (s)n l−2 (s)k > 0 for all s ∈ L.

Then we obtain, for l = 3, · · · , n − 2, a well-defined vector field n _l along C, that is, for all s ∈ L,

n _l (s) = 1 κ _l (s)

 dn _l−1 (s)

ds + κ _l−1 (s)n _l−2 (s) 

,

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and for i, j = 1, 2, · · · , n − 2,

ht(s), n _i (s)i = 0, hn _i (s), n _j (s)i = δ _j ⁱ . And

(C _n−1 ) : κ _n−1 (s) = h dn _l−2 (s)

ds , n _n−1 (s)i 6= 0 for all s ∈ L,

where the unit vector field n _n−1 along C is determined by the fact that the frame {t, n ₁ , · · · , n _n−1 } is of orthonormal and of positive orientation. We remark that the functions κ ₁ , · · · , κ _n−2 are of positive and the function κ _n−1 is of non-zero. Such a curve C is called a special Frenet curve in R ⁿ . The term ”special” means that the vector field n _i+1 is inductively defined by the vector fields n _i and n _i−1 and the positive functions κ _i and κ _i−1 . Each function κ _i is called the i-curvature function of C(i = 1, 2, · · · , n − 1).

The orthonormal frame {t, n 1 , · · · , n n−1 } along C is called the special Frenet frame along C.

Thus we obtain the Frenet equations:

dt(s)

ds = κ ₁ (s)n ₁ (s) dn ₁ (s)

ds = −κ ₁ (s)t(s) + κ ₂ (s)n ₂ (s)

· · · dn _l (s)

ds = −κ _l (s)n _l−1 (s) + κ _l+1 (s)n _l+1 (s)

· · · dn _n−2 (s)

ds = −κ _n−2 (s)n _n−3 (s) + κ _n−1 (s)n _n−1 (s) dn _n−1 (s)

ds = −κ n−1 (s)n n−2 (s)

for all s ∈ L. And, for j = 1, 2, · · · , n − 1, the unit vector field n j along C is called the Frenet j-normal vector field along C.

Definition 2.1. The Frenet (i 1 , · · · , i m )-normal plane of C at c(s) is a plane spanned by n i

(s), · · · , n i

(s) through c(s), where 1 ≤ i 1 < i 2 < · · · < i m ≤ n − 1.

Note that for each point c(s) of C, the Frenet i-normal plane is also called the Frenet i-normal line, where i ∈ {1, · · · , n − 1}.

Definition 2.2. A C ^∞ -special Frenet curve C in R ⁿ (c : L → R ⁿ ) is called a (i ₁ , · · · , i _m )-

Bertrand curve if there exist another C ^∞ -special Frenet curve ¯ C (¯ c : ¯ L → R ⁿ ), distinct

from C, and a regular C ^∞ -mapping ϕ : L → ¯ L (¯ s = ϕ(s), ^dϕ(s) _ds 6= 0 for all s ∈ L)

such that curves C and ¯ C have the same Frenet (i ₁ , · · · , i _m )-normal plane at each pair

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of corresponding points c(s) and ¯ c(¯ s) = ¯ c(ϕ(¯ s)) under ϕ. Here s and ¯ s are arc-length parameters of C and ¯ C respectively. In this case, ¯ C is called a (i ₁ , · · · , i _m )-Bertrand mate of C.

In this thesis, the notation K _j,i (s) stands for −κ _i (s)δ _j ⁱ⁻¹ + κ _i+1 (s)δ _j ⁱ⁺¹ . Then for all s ∈ L, we have the following properties:

1. K _j,i (s) =



 

 

κ _j (s) if j − i = 1

−κ i (s) if i − j = 1 0 otherwise, 2. K _j,i (s) = −K _i,j (s).

Using the above notation, we rewrite the Frenet equations as n ⁰ _i (s) =

n−1

X

j=0

K j,i (s)n j (s), i = 0, · · · , n − 1,

where n ₀ (s) = t(s).