Existence and Uniqueness Theorem for Frenet Frame Supercurves

CRISTEA VALENTIN GABRIEL

Abstract. In the first part of this paper, using the Banach Grassmann algebra B_{L} given by Rogers in her paper [10], a new scalar product and a new definition of the orthogonality are introduced on the (m,n)-dimensional total supereuclidean space B_{L}^{m+n} .Using the GH^{∞} functions given by Rogers in [10], the new definitions of the supercurve, of the supersmooth supercurve, of the supersmooth supercurve in general position and of the Frenet frame associated to a supersmooth supercurve in general position are given.In the second part of this paper, using the classical results described in [9] , the new existence and uniqueness theorem for some supercurves which admit Frenet frame is proved.

Keywords: the (m,n)-dimensional total supereuclidean space B_{L}^{m+n} , the (m,n)-dimensional supereuclidean space B_{L}^{m,n} , the GH^{∞} functions, supersmooth supercurve, supersmooth supercurve in general position, Frenet frame associated to a supersmooth supercurve, Frenet formulas for the supersmooth supercurve. Mathematics Subject Classification (1991): 58A50

1. Supersmooth supercurve in general position and Frenet frame associated to a supersmooth supercurve in general position

Let us consider only algebras over the reals. For each positive integer L, B_{L} [10] will denote the Grassmann algebra over the reals with generators 1^{(L)} ,β_{1}^{(L)},..., β_{L}^{(L)} and relations

(1) 1^{(L)}β_{i}^{(L)} = β_{i}^{(L)}1^{(L)} = β_{i}^{(L)} i = 1, ..., L,

(2) β_{i}^{(L)}β_{j}^{(L)} = −β_{j}^{(L)}β_{i}^{(L)} i, j = 1, ..., L.

B_{L} is a graded algebra [12] and can be written as a direct sum

B_{L} = (B_{L})_{0} ⊕ (B_{L})_{1}

where (B_{L})_{0} and (B_{L})_{1} are the even and odd part of (B_{L}) respectively. Let M_{L} denote (due to Kostant [7]) the set of finite sequences of positive integers μ = (μ_{1}, ..., μ_{k}) with 1 ≤ μ_{1} < ... < μ_{k} ≤ L. M_{L} includes the sequence with no elements, denoted ϕ. As it follows in [10] for each μ in M_{L},

(3) β_{μ}^{(L)} := β_{μ_{1}}^{(L)} ⋯ β_{μ_{k}}^{(L)}, and

(4) β_{ϕ}^{(L)} := 1^{(L)}

a typical element b of B_{L} may be expressed as

(5) b = ∑_{μ∈M_{L}} b^{μ}β_{μ}^{(L)},

where the coefficient b^{μ} are real numbers. We consider the body map (in DeWitt’s terminology [4] )

ε^{(L)} : B_{L} → R

given by

(6) ε^{(L)}(b) = b^{ϕ}.

With the norm on B_{L} defined by

(7) ||b|| := ∑_{μ∈M_{L}} |b^{μ}|,

B_{L} is a Banach algebra [11] . Considering L′ also a positive integer, with L ≥ L′ , then there is a natural injection i_{L′,L} : B_{L′} → B_{L} [10], which is the unique algebra homomorphism satisfying

(8) i_{L′,L}(β_{i}^{(L′)}) = β_{i}^{(L)} i = 1, ..., L i_{L′,L}(1^{(L′)}) = 1^{(L)}.

B_{L} naturally has a B_{L′} module structure [10] with, given a ∈ B_{L′} and b ∈ B_{L},

(9) ab := i_{L′,L}(a)b.

We define the (m, n)−dimensional total supereuclidean space B_{L}^{m+n} [1] as the space which is the cartesian product of m + n copies of B_{L} and has the graduation

B_{L}^{m+n} = (B_{L}^{m+n})_{0} ⊕ (B_{L}^{m+n})_{1}

A typical element of B_{L}^{m+n} is written (x^{1}, ..., x^{m}, θ^{1}, ..., θ^{n}) or simply (x, θ), an element of (B_{L}^{m+n})_{0} is called c−type or even element and is written in the form (x^{′1}, ..., x^{′m}, θ^{′1}, ..., θ^{′n}) with x^{′1}, ..., x^{′m} ∈ (B_{L})_{0} and θ^{′1}, ..., θ^{′n} ∈ (B_{L})_{1} and an element of (B_{L}^{m+n})_{1} is called a−type or odd element and is written in the form (x^{′′1}, ..., x^{′′m}, θ^{′′1}, ..., θ^{′′n}) with x^{′′1}, ..., x^{′′m} ∈ (B_{L})_{1} and θ^{′′1}, ..., θ^{′′n} ∈ (B_{L})_{0}. An even element has the parity 0 and an odd element has the parity 1.

Definition 1. [4] V is called the supervector space over the space B_{L} if and only if V is the supervector space given by the 1)−5) conditions from the DeWitt’s definition (see DeWitt’s book pages 14-15) where instead of Λ_{∞} we put B_{L} .

Example 1. The (m, n)−dimensional total supereuclidean space B_{L}^{m+n} with the above graduation

B_{L}^{m+n} = (B_{L}^{m+n})_{0} ⊕ (B_{L}^{m+n})_{1}

is a supervector space of dimension (m, n).

Remark 1. Let us consider the (m, n)−dimensional supereuclidean space B_{L}^{m,n} given by Rogers in [10]. We note that the space B_{L}^{m,n} is not a supervector space over B_{L} as in definition 1.

A useful map is [10]

ε_{m,n}^{(L)} : (B_{L}^{m+n})_{0} → R^{m}

with

(10) ε_{m,n}^{(L)}(x^{′1}, ..., x^{′m}, θ^{′1}, ..., θ^{′n}) := (ε^{(L)}(x^{′1}), ..., ε^{(L)}(x^{′m}))

and another useful map is

ε_{m,n}^{′(L)} : (B_{L}^{m+n})_{1} → R^{n}

with

(11) ε_{m,n}^{′(L)}(x^{′′1}, ..., x^{′′m}, θ^{′′1}, ..., θ^{′′n}) := (ε^{(L)}(θ^{′′1}), ..., ε^{(L)}(θ^{′′n}).

Remark 2. Let us consider the (1, 1)−dimensional total supereuclidean space B_{L}^{2} , (2, 0)−dimensional total supereuclidean space B_{L}^{2} and (0, 2)−dimensional total supereuclidean space B_{L}^{2} and the element (1, 0) which belongs to these three spaces.We note that the element (1, 0) is c-type for the first two spaces and is a-type for the last space. We may write the supervector (1, 0) in a standard basis [4] in the form:

(1, 0) = 1 · (1, 0) + 0 · (0, β^{1})

for the (1, 1)−dimensional total supereuclidean space B_{L}^{2} and (0, 2)−dimensional total supereuclidean space B_{L}^{2} where a standard basis [4] in these spaces is {(1, 0), (0, β^{1})} with (1, 0) c−type supervector and (0, β^{1}) a−type supervector and the supervector (1, 0) can be written in a standard basis [4] in the form:

(1, 0) = 1 · (1, 0) + 0 · (0, 1)

for the (2, 0)−dimensional total supereuclidean space B_{L}^{2}, where a standard basis [4] in this space is {(1, 0), (0, 1)} with (1, 0) c−type supervector and (0, 1) a−type supervector

Definition 2. [10] Suppose U ⊂ R^{m} is open and L′ is a positive integer with L′ ≤ L. Let C^{∞}(U, B_{L′}) denote the B_{L′} module of C^{∞} functions of U into B_{L′} ;(recall that B_{L′} is a Banach algebra, and hence a fotiori a Banach space).Then the map

Z_{L′,L} : C^{∞}(U, B_{L′}) → [ε_{m,0}^{(L)}(U)]^{B_{L}}

is defined by [10]

(12) Z_{L′,L}(f)(x^{1}, ..., x^{m}) = ∑_{i_{1}=0...i_{m}=0}^{L} \frac{1}{i_{1}!...i_{m}!}·

·i_{L′,L}(∂_{1}^{i_{1}}...∂_{m}^{i_{m}}f(ε^{(L)}(x^{1}), ..., ε^{(L)}(x^{m}))) × s(x^{1})^{i_{1}} ⋯ s(x^{m})^{i_{m}}

where [10]

(13) s(x^{i}) = x^{i} − ε^{(L)}(x^{i})1, i = 1, ..., m. Definition 3. [10] Suppose V is open in B_{L}^{m,n} (with respect to its usual finitedimensional vector space topology) and U = ε_{m,n}^{(L)}(V), suppose L > 2n and L′ = [\frac{1}{2}L], the last integer not less than \frac{1}{2}L. GH^{∞}(V) denotes the set of functions f : V → B_{L} for which there exist [10] f_{μ} ∈ C^{∞}(U, B_{L′}) such that

(14) f(x, θ) = ∑_{μ∈M_{n}} Z_{L′,L}(f_{μ})(x)θ^{μ} where

(15) θ^{μ} = θ^{μ_{1}} ⋯ θ^{μ_{k}} and

(16) θ^{ϕ} = 1^{(L)}. Definition 4. [10] With the notation of Definition 2., let f be an element of GH^{∞}(V), with expansion (12) . Then, for i = 1, ..., m

G_{i}f : V → B_{L}

is defined by [10]

(17) G_{i}f(x; θ) = ∑_{μ∈M_{n}} Z_{L′,L}(∂_{i}f_{μ})(x)θ^{μ}. Also, for j = 1, ..., n,

G_{j+m}f : V → B_{L}

is defined by [10]

(18) G_{j+m}f(x; θ) = ∑_{μ∈M_{n}} Z_{L′,L}(f_{μ})(x)θ^{μ/j} × (−1)^{|f_{μ}(x)|}, where |f_{μ}(x)| is the Grassmann parity of f_{μ}(x), and

θ^{μ/j} = θ^{μ_{1}} ⋯ θ^{μ_{i−1}}θ^{μ_{i+1}} ⋯ θ^{μ_{k}}(−1)^{i−1},

if j = μ_{i} for some i, 1 ≤ i ≤ k, θ^{μ/j} otherwise.

For the first time, I have introduced on B_{L}^{m+n} with n = 2r , the scalar product ⟨v, w⟩ = x^{1}y^{1} + ⋯ + x^{m}y^{m} + θ^{1}θ_{1}^{r+1} + ⋯ + θ^{r}θ_{1}^{n} − θ^{r+1}θ_{1}^{1} − ⋯ − θ^{n}θ_{1}^{r}

(∀) v = (x^{1}, …, x^{m}, θ^{1}, …, θ^{n}), w = (y^{1}, …, y^{m}, θ_{1}^{1}, …, θ_{1}^{n}) ∈ B_{L}^{m+n} which has these properties: a) ⟨v, w⟩ = (−1)^{|v|·|w|}⟨w, v⟩ (supersymmetry)

b) ⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩  (∀) u, v, w ∈ B_{L}^{m+n}  (linearity)
c) ⟨v, ·⟩ = 0  if and only if  v = 0.

Proposition 1. (Number of scalar products on B_{L}^{m+n} ) On B_{L}^{m+n} with n = 2r , we may give r! different scalar products.

Proof. There are r! one-to-one functions f : {1, …, r} → {r+1, …, 2r}. For each function f we have the following scalar product between v = (x^{1}, …, x^{m}, θ^{1}, …, θ^{n}) and w = (y^{1}, …, y^{m}, θ_{1}^{1}, …, θ_{1}^{n})

⟨v, w⟩_{f} = ∑_{k=1}^{m} x^{k}y^{k} + ∑_{j_{1}=1}^{r}(θ^{j_{1}}θ_{1}^{f(j_{1})} − θ^{f(j_{1})}θ_{1}^{j_{1}}).

One can easily verify the a), b) and c) relations of a scalar product. Let us prove a) in the case when v and w are odd elements of B_{L}^{m+n} , that is, v and w belong to (B_{L}^{m+n})_{1}. Then, we shall have v = (x^{′′1}, …, x^{′′m}, θ^{′′1}, …, θ^{′′n}) and w = (y^{′′1}, …, y^{′′m}, θ′′_{1}^{1}, …, θ′′_{1}^{n}) .When we compute the scalar product between v and w, we shall get

⟨v, w⟩_{f} = ∑_{k=1}^{m} x^{′′k} · y^{′′k} + ∑_{j_{1}=1}^{r}(θ_{1}^{′′j}θ′′_{1}^{f(j_{1})} − θ^{′′f(j_{1})}θ′′_{1}^{j_{1}}) =

= − ∑_{k=1}^{n} y^{′′k} · x^{′′k} + ∑_{j_{1}=1}^{r}(θ′′_{1}^{f(j_{1})}θ′′^{j_{1}} − θ′′_{1}^{j_{1}}θ′′^{f(j_{1})})

Now, the scalar product between v and w becomes

⟨v, w⟩_{f} = −(∑_{k=1}^{n} y^{′′k} · x^{′′k} + ∑_{j_{1}=1}^{r}(θ′′_{1}^{j_{1}}θ′′^{f(j_{1})} − θ′′_{1}^{f(j_{1})}θ′′^{j_{1}}))

Thus, we get

⟨v, w⟩_{f} = − ⟨w, v⟩_{f}.

This ends the proof.

Definition 5. (Orthogonality on B_{L}^{m+n} ) We say that the supervector v of B_{L}^{m+n} is orthogonal to the supervector w of B_{L}^{m+n} if and only if ε^{(L)}(⟨v, w⟩) = 0 .

The column supervectors E_{1} = (1, …, 0), …, E_{m} = (0, …, 1, 0, …, 0), where 1 is written on the m^{th} place, E_{m+1} = (0, …, 0, −1, 0, …, 0), where -1 is written on the (m + r + 1)^{th} place, …, E_{m+r} = (0, …, 0, −1), where -1 is written on the (m + n)^{th} place, E_{m+r+1} = (0, …, 0, 1, 0, …, 0), where 1 is written on the (m + 1)^{th} place, …, E_{m+n} = (0, …, 0, 1, 0, …, 0), where 1 is written on the (m + r)^{th} place, form the standard basis on B_{L}^{m+n} [4] where the first m supervectors are c−type and the last n supervectors are a−type.

Definition 6. (Supersmooth supercurve) [10] Let suppose L > 2n and let B_{L}^{m+n} be an (m, n)−dimensional total supereuclidean space, let V be an open set in B_{L}^{1,1} , let c : V ⊂ B_{L}^{1,1} → B_{L}^{m+n} be a function, and for every θ ∈ V ∩ (B_{L})_{1} let define

c_{θ,0} : V ∩ (B_{L})_{0} → (B_{L}^{m+n})_{0}

given by

c_{θ,0}(t) = (c(t, θ))_{0}

where (c(t, θ))_{0} is the even part of the supervector c(t, θ) and

c_{θ,B} : V ∩ (B_{L})_{0} → R^{m}

given by

c_{θ,B}(t) = ε_{m,n}^{(L)} ∘ c_{θ,0}(t)

for all t ∈ V ∩ (B_{L})_{0} . The function c will be said to be supercurve if and only if c_{θ,B}|_{V ∩R} will be a curve. The function c is called supersmooth if and only if

c^{i} ∈ GH^{∞}(V) (∀) i ∈ {1, …, m} and c^{j+m} ∈ GH^{∞}(V) (∀) j ∈ {1, …, n} where c^{i} = x^{i} ∘ c (∀) i ∈ {1, …, m} and c^{j+m} = θ^{j} ∘ c (∀) j ∈ {1, …, n} .

Definition 7. (Supercurve in general position) Let suppose L > 2n and let B_{L}^{m+n} be an (m, n)−dimensional total supereuclidean space, let V be an open set in B_{L}^{1,1} and let c : V ⊂ B_{L}^{1,1} → B_{L}^{m+n} be a supersmooth supercurve . We say that the supercurve c is in general position if and only if

{G_{1}c(t, θ), …, G_{1}^{(m−1)}c(t, θ), G_{2}c(t, θ), G_{1}G_{2}c(t, θ), …
…, G_{1}^{(n−1)}G_{2}c(t, θ)}

are linear independent (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and where by G_{1}c(t, θ) we understand the supervector

(G_{1}c^{1}(t, θ), …, G_{1}c^{m}(t, θ), G_{1}c^{m+1}(t, θ), …, G_{1}c^{m+n}(t, θ)) (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}

by G_{2}c(t, θ) we understand the supervector

(G_{2}c^{1}(t, θ), …, G_{2}c^{m}(t, θ), G_{2}c^{m+1}(t, θ), …, G_{2}c^{m+n}(t, θ)) (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1},

G_{1}^{(s)}c(t, θ) = G_{1} … G_{1}c(t, θ), where G_{1}^{(0)}c(t, θ) = c(t, θ) and G_{1}^{(1)}c(t, θ) = G_{1}c(t, θ) and “ … “ means that G_{1} is applied by s times.

Definition 8. (Frenet Frame associated to a supersmooth supercurve) Let suppose L > 2n and let B_{L}^{m+n} be an (m, n)−dimensional total supererclidean space, let V be an open set in B_{L}^{1,1} and let c : V ⊂ B_{L}^{1,1} → B_{L}^{m+n} be a supersmooth supercurve. By a Frenet frame associated to a supersmooth supercurve c : V ⊂ B_{L}^{1,1} → B_{L}^{m+n} we shall mean a system of m+n supervector fields {e_{1}, …, e_{m+n}} along to the supersmooth supercurve c such that (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} we have the following properties:

(19) ⟨e_{k}(t, θ), e_{h}(t, θ)⟩ = δ_{kh} (∀) k, h ∈ {1, …, m} (19′) ⟨e_{m+r+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ = −δ_{j_{1}j_{2}} (∀) j_{1}, j_{2} ∈ {1, …, r} (19′′) ⟨e_{m+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ = δ_{j_{1}j_{2}} (∀) j_{1}, j_{2} ∈ {1, …, r} (19′′′) ⟨e_{m+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ = 0, (∀) j_{1}, j_{2} ∈ {1, …, r} (19^{IV}) ⟨e_{m+r+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ = 0, (∀) j_{1}, j_{2} ∈ {1, …, r} (19^{V}) ⟨e_{i}(t, θ), e_{m+j}(t, θ)⟩ = ⟨e_{m+j}(t, θ), e_{i}(t, θ)⟩ = 0,

(∀) i ∈ {1, …, m} and j ∈ {1, …, n}

(20) span(G_{1}c(t, θ), …, G_{1}^{(k)}c(t, θ)) = span(e_{1}(t, θ), …, e_{k}(t, θ)) (∀) k ∈ {1, …, m − 1} where G_{1}^{(1)}c(t, θ) = G_{1}c(t, θ) and (20′) span(G_{2}c(t, θ), G_{1}G_{2}c(t, θ), …, G_{1}^{(j−1)}G_{2}c(t, θ)) =

= span(e_{m+1}(t, θ), …, e_{m+j}(t, θ)) (∀) j ∈ {1, …, n}

where G_{1}^{(0)}G_{2}c(t, θ) = G_{2}c(t, θ) and G_{1}^{(1)}G_{2}c(t, θ) = G_{1}G_{2}c(t, θ) and we shall mean by span(e_{1}(t, θ), …, e_{k}(t, θ)) the supervector space [4], spanned by e_{1}(t, θ), …, e_{k}(t, θ) (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} .The systems of supervectors

(21) {G_{1}c(t, θ), …, G_{1}^{(k)}c(t, θ)} and

{e_{1}(t, θ), …, e_{k}(t, θ)}

are directed in the same way (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and (∀) k ∈ {1, …, m − 1}, that is, ε^{(L)}(detM_{1}) > 0 where M_{1} is the matrix when we change from the frame {G_{1}c(t, θ), …, G_{1}^{(k)}c(t, θ)} to the frame {e_{1}(t, θ), …, e_{k}(t, θ)} and the systems of supervectors (21′) {G_{2}c(t, θ), G_{1}G_{2}c(t, θ), …, G_{1}^{(j−1)}G_{2}c(t, θ)} and

{e_{m+1}(t, θ), …, e_{m+j}(t, θ)}

are directed in the same way (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} (∀) j ∈ {1, …, n},that is, ε^{(L)}(detM_{2}) > 0 where M_{2} is the matrix when we change from the frame

{G_{2}c(t, θ), G_{1}G_{2}c(t, θ), …, G_{1}^{(j−1)}G_{2}c(t, θ)} to the frame

{e_{m+1}(t, θ), …, e_{m+j}(t, θ)}.

The system of supervectors

(22) {e_{1}(t, θ), …, e_{m+n}(t, θ)}

is positive directed (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}, that is, ε^{(L)}(sdet(e_{q}^{s}(t, θ))) > 0 where by sdet(e_{q}^{s}(t, θ)) we understand det(A − C · B^{−1} · D) · (detB)^{−1} [2], [8], [4], [1], [5] where

(e_{q}^{s}(t, θ))_{1≤s,q≤m+n} = ( \begin{matrix}A & C \\ D & B \end{matrix} ) ,

where A, B, C and D are m × m, n × n, m × n and n × m matrices with elements in B_{L}, respectively.

2. Existence and Uniqueness Theorem for Frenet Frame Supercurves

Theorem 1. (Existence and Uniqueness Theorem for Frenet Frame Supercurves) Let suppose L > 2n and let B_{L}^{m+n} be an (m, n)−dimensional total supereuclidean space, let V be an open set of B_{L}^{1,1} and let c : V ⊂ B_{L}^{1,1} → B_{L}^{m+n} be a supersmooth supercurve in general position which satisfy the following relation:

(1) ε^{(L)}(⟨G_{2}c(t, θ), G_{1}^{(r)}G_{2}c(t, θ)⟩) > 0

(2) ε^{(L)}(⟨G_{1}^{(j_{1})}G_{2}c(t, θ)), G_{1}^{(r+j_{1})}G_{2}c(t, θ)⟩) > 0

(∀) j_{1} ∈ {1, …, r − 1}, and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1},

(3) ε^{(L)}(⟨G_{2}c(t, θ), G_{1}^{(j)}G_{2}c(t, θ)⟩) = 0,

(∀) j ∈ {0, …, n − 1}, with j ≠ r and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1},

(4) ε^{(L)}(⟨G_{1}^{(j′)}G_{2}c(t, θ)), G_{1}^{(j)}G_{2}c(t, θ)⟩) = 0

(∀) j′ ∈ {1, …, n − 1}, (∀) j ∈ {1, …, n − 1} with j ≠ j′ + r and j′ < j and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Then there exists a unique Frenet frame {e_{1}, …, e_{m+n}} associated to the supercurve c and we have the following formulas (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}:

(5)  G_{1}e_{k}(t, θ) = ∑_{h=1}^{m} a_{kh}(t, θ) · e_{h}(t, θ)  (∀) k ∈ {1, …, m},

where

(6)  a_{kh}(t, θ) + a_{hk}(t, θ) = 0  (∀) k, h ∈ {1, …, m}

and

(7)  a_{kh}(t, θ) = 0  if  h > k + 1,  (∀) k, h ∈ {1, …, m}

(8)  G_{1}e_{m+j}(t, θ) = ∑_{l=1}^{n} a_{m+j m+l}(t, θ) · e_{m+l}(t, θ)  (∀) j ∈ {1, …, n},

where

(9)  a_{m+j_{1} m+j_{2}}(t, θ) + a_{m+r+j_{2} m+r+j_{1}}(t, θ) = 0  (∀) j_{1}, j_{2} ∈ {1, …, r},
(10)  a_{m+r+j_{1} m+j_{2}}(t, θ) − a_{m+r+j_{2} m+j_{1}}(t, θ) = 0  (∀) j_{1}, j_{2} ∈ {1, …, r},
(11)  a_{m+j_{1} m+r+j_{2}}(t, θ) − a_{m+j_{2} m+r+j_{1}}(t, θ) = 0  (∀) j_{1}, j_{2} ∈ {1, …, r},

and

(12) a_{i m+j}(t, θ) = 0, a_{m+j i}(t, θ) = 0 (∀) i ∈ {1, …, m}, (∀) j ∈ {1, …, n},

(13) a_{m+j m+l}(t, θ) = 0 if l ≠ j + 1 (∀) j, l ∈ {1, …, n}, where

(14) a_{m+j_{1} m+j_{2}}(t, θ) = ⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩,

(15) a_{m+j_{1} m+r+j_{2}}(t, θ) = −⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩,

(16) a_{m+r+j_{1} m+j_{2}}(t, θ) = ⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩,

(17) a_{m+r+j_{1} m+r+j_{2}}(t, θ) = −⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩, (∀) j_{1}, j_{2} ∈ {1, …, r},

(18)  a_{kh}(t, θ) = ⟨G_{1}e_{k}(t, θ), e_{h}(t, θ)⟩,  (∀) k, h ∈ {1, …, m},
(19)  a_{k m+j}(t, θ) = ⟨G_{1}e_{k}(t, θ), e_{m+j}(t, θ)⟩,  (∀) k ∈ {1, …, m}

and (∀) j ∈ {1, …, n},

(20) a_{m+j k}(t, θ) = ⟨G_{1}e_{m+j}(t, θ), e_{k}(t, θ)⟩, (∀) k ∈ {1, …, m} and (∀) j ∈ {1, …, n}. Proof. We use the proof of the existence and uniqueness theorem for Frenet frame curves from [8] in our proof. From the (1) and (2) relations we have:

ε^{(L)}(⟨G_{2}c(t, θ), G_{1}^{(r)}G_{2}c(t, θ)⟩) ≠ 0 and

ε^{(L)}(⟨G_{1}^{(j_{1})}G_{2}c(t, θ)), G_{1}^{(r+j_{1})}G_{2}c(t, θ)⟩) ≠ 0

∀ j_{1} ∈ {1, …, r − 1}. Let us consider

λ_{1}(t, θ) = ⟨G_{2}c(t, θ), G_{1}^{(r)}G_{2}c(t, θ)⟩

and

λ_{j_{1}}(t, θ) = ⟨G_{1}^{(j_{1}−1)}G_{2}c(t, θ), G_{1}^{(r+j_{1}−1)}G_{2}c(t, θ)⟩

(∀) j_{1} ∈ {2, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Because of ε^{(L)}(λ_{j_{1}}(t, θ)) ≠ 0 (∀) j_{1} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} it results that there exist (λ_{j_{1}}(t, θ))^{−1}. Let us consider

e_{m+1}(t, θ) = (λ_{1}(t))^{−1} · G_{2}c(t, θ)

and

e_{m+j_{1}}(t, θ) = (λ_{j_{1}}(t))^{−1} · G_{1}^{(j_{1}−1)}G_{2}c(t, θ)

(∀) j_{1} ∈ {2, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and let us consider

e_{m+r+j_{2}}(t, θ) = G_{1}^{(r+j_{2}−1)}G_{2}c(t, θ)

(∀) j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

The supervectors {e_{m+1}(t, θ), …, e_{m+n}(t, θ)} are linear independent because the supersmooth supercurve c is general position. From the definition of these supervectors, we have:

span(G_{1}c(t, θ), …, G_{1}^{(k)}c(t, θ)) = span(e_{1}(t, θ), …, e_{k}(t, θ)) (∀) k ∈ {1, …, m − 1}

span(G_{2}c(t, θ), G_{1}G_{2}c(t, θ), …, G_{1}^{(j−1)}G_{2}c(t, θ)) =

= span(e_{m+1}(t, θ), …, e_{m+j}(t, θ)) (∀) j ∈ {1, …, n}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

From the (1) and (2) relation we have ε^{L}((λ_{j_{1}}(t, θ))) > 0 (∀) j_{1} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. The ” body ” of the determinants of the matrices when we change from the system {G_{2}c(t, θ), G_{1}G_{2}c(t, θ), …, G_{1}^{(j−1)}G_{2}c(t, θ)} to the system {e_{m+1}(t, θ), …, e_{m+j}(t, θ)} (∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} is

ε^{(L)}(| \begin{matrix}(λ_{1}(t, θ))^{−1} & ⋯ & 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ... & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & ⋯ & (λ_{j}(t, θ))^{−1} & 0 & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & 1 & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ... & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & ⋯ & 0 & 0 & 0 & ⋯ & 1 \end{matrix} |) =

= ε^{(L)}((λ_{1}(t, θ))^{−1} · . . . · (λ_{j}(t, θ))^{−1}) =

= ε^{(L)}((λ_{1}(t, θ))^{−1}) · . . . · ε^{(L)}((λ_{j}(t, θ))^{−1}) > 0.
We have (∀) j_{1} ∈ {1, …, r} and (∀) t ∈ I that
⟨e_{m+j_{1}}(t), e_{m+r+j_{1}}(t)⟩ = (λ_{j_{1}}(t, θ))^{−1} · ⟨G_{1}^{(j_{1}−1)}G_{2}c(t, θ)), G_{1}^{(r+j_{1}−1)}G_{2}c(t, θ)⟩ =
= (λ_{j_{1}}(t))^{−1} · λ_{j_{1}}(t) = 1.

Thus, we get the (19′′) formula from the first part of this paper. The (19′) formula, from the first part of this paper, results from the supersymmetry property of the scalar product ⟨, ⟩ and the (19^{IV}) and (19′′′) formulas , from the first part of this paper, result from the (3) and (4) relations.We shall specify if a relation belongs to the first part of this paper and we shall not specify if a relation belongs to the second part of this paper.

Because the supermooth supercurve c is in general position, it results that the supervector f_{1}(t, θ) = G_{1}c(t, θ) is nonzero (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Let us consider ||v||′ = \sqrt{ε^{(L)}(⟨v, v⟩)} (21) (∀) v ∈ B_{L}^{m+n} .

We put e_{1}(t, θ) = f_{1}(t, θ) · (||f_{1}(t, θ)||′)^{−1} (22) (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Let the supervector f_{2}(t, θ) be G_{1}^{(2)}c(t, θ)+(ε^{(L)}(A(t, θ))+s(A(t, θ)))·e_{1}(t, θ) (23). We shall get the functions ε^{(L)}(A(t, θ)) and s(A(t, θ)) such that the supervectors f_{2}(t, θ) and e_{1}(t, θ) to be orthogonal, that is,

(24) ε^{(L)}(⟨f_{2}(t, θ), e_{1}(t, θ)⟩) = 0

(t, θ) ∈ V ⊂ B_{L}^{1,1}. From (23) and (24), we have

0 = ⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩ + (ε^{(L)}(A(t, θ)) + s(A(t, θ))) · ⟨e_{1}(t, θ), e_{1}(t, θ)⟩

from which we obtain

0 = ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) + s(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩)+
+ε^{(L)}(A(t, θ)) · ε^{(L)}(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) + ε^{(L)}(A(t, θ)) · s(⟨e_{1}(t), e_{1}(t)⟩)+
+s(A(t, θ)) · s(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) + s(A(t, θ)) · ε^{(L)}(⟨e_{1}(t, θ), e_{1}(t, θ)⟩).

Thus we have

ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) + ε^{(L)}(A(t, θ)) · ε^{(L)}(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) = 0

and

s(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) + ε^{(L)}(A_{B}(t, θ)) · s(⟨e_{1}(t, θ), e_{1}(t, θ)⟩)+
+s(A(t, θ)) · s(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) + s(A(t, θ)) · ε^{(L)}(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) = 0.

Therefore

(25) ε^{(L)}(A(t, θ)) = −ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) and

(26) s(A(t, θ)) = (−s(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) − ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩)·

·s(⟨e_{1}(t), e_{1}(t)⟩)) · (1 + s(⟨e_{1}(t), e_{1}(t)⟩))^{−1}.

From (23), (25) and (26), we get

(27) f_{2}(t, θ) = G_{1}^{(2)}c(t, θ) + (−ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) + s(A(t, θ))) · e_{1}(t, θ).

Because the supervectors G_{1}c(t, θ) and G_{1}^{(2)}c(t, θ) are linear independent (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} from (27) we have ε^{(L)}(⟨f_{2}(t, θ), f_{2}(t, θ⟩) ≠ 0 (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Therefore ||f_{2}(t, θ)||′ ≠ 0 (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

We set e_{2}(t, θ) = f_{2}(t, θ) · (||f_{2}(t, θ)||′)^{−1} (28).

We note that ||e_{1}(t, θ)||′ = ||e_{2}(t, θ)||′ = 1 but ⟨e_{1}(t, θ), e_{1}(t, θ)⟩ = 1+

+s(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) where s(⟨e_{1}(t, θ), e_{1}(t, θ)⟩) has not importance and we have the same for ⟨e_{2}(t, θ), e_{2}(t, θ)⟩ . From the formulas (22), (27) and (28) formulas we have:

(29) G_{1}c(t, θ) = ||G_{1}c(t, θ)||′ · e_{1}(t, θ)

and

(30) G_{1}^{(2)}c(t, θ) = ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) − s(A(t, θ)) · e_{1}(t, θ) + ||f_{2}(t, θ)||′ · e_{2}(t, θ).

The relations (29) and (30) show us that

span(G_{1}c(t, θ), G_{1}^{(2)}c(t, θ)) = span(e_{1}(t, θ), e_{2}(t, θ)).

Because of

ε^{(L)}(| \begin{matrix}||G_{1}c(t, θ)||′ & 0 \\ ε^{(L)}(⟨G_{1}^{(2)}c(t, θ), e_{1}(t, θ)⟩) − s(A(t, θ)) & ||f_{2}(t, θ)||′ \end{matrix} |) > 0,

(t, θ) ∈ V ⊂ B_{L}^{1,1} it results that the systems of supervectors {G_{1}c(t, θ), G_{1}^{(2)}c(t, θ)} and {e_{1}(t, θ), e_{2}(t, θ)} are directed in the same way (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

We assume that we have constructed the unit, orthogonal two by two supervectors e_{1}(t, θ), …, e_{h−1}(t, θ) (h < m) with the properties that:

span(G_{1}c(t, θ), G_{1}^{(2)}c(t, θ), …, G_{1}^{(h−1)}c(t, θ)) = span(e_{1}(t, θ), …, e_{h−1}(t, θ))
and the systems of supervectors  {G_{1}c(t, θ), G_{1}^{(2)}c(t, θ), …, G_{1}^{(h−1)}c(t, θ)}  and

{e_{1}(t, θ), …, e_{h−1}(t, θ)} are directed in the same way (∀)(t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus we construct the supervector f_{h}(t, θ) :

(31) f_{h}(t, θ) = G_{1}^{(h)}c(t, θ) + ∑_{k=1}^{h−1}(ε^{(L)}(A_{k}(t, θ)) + s(A_{k}(t, θ))) · e_{k}(t, θ), h < m where (t, θ) → A_{k}(t, θ) are supersmooth functions (∀) k ∈ {1, …, h − 1} which will be determined by the conditions:

(32) ⟨f_{h}(t, θ), e_{i}(t, θ)⟩ = 0, h < m, i ∈ {1, …, h − 1}.

By (31) and (32) we get

⟨G_{1}^{(h)}c(t, θ), e_{i}(t, θ)⟩ + ∑_{k=1}^{h−1}(ε^{(L)}(A_{k}(t, θ)) + s(A_{k}(t, θ))) · ⟨e_{k}(t, θ), e_{i}(t, θ)⟩ = 0

(∀) i ∈ {1, …, h − 1} or equivalent to

(33) ⟨G_{1}^{(h)}c(t, θ), e_{i}(t, θ)⟩ + (ε^{(L)}(A_{i}(t, θ)) + s(A_{i}(t, θ))) · (1 + s(⟨e_{i}(t, θ), e_{i}(t, θ)⟩)) = 0

(∀) i ∈ {1, …, h − 1} . From (33) we have:

(34) ε^{(L)}(⟨G_{1}^{(h)}c(t, θ), e_{i}(t, θ)⟩) + ε^{(L)}(A_{i}(t, θ)) = 0

and

(35) s(⟨G_{1}^{(h)}c(t, θ), e_{i}(t, θ)⟩) + ε^{(L)}(A_{i}(t, θ)) · s(⟨e_{i}(t, θ), e_{i}(t, θ)⟩)+

+s(A_{i}(t, θ)) · (1 + s(⟨e_{i}(t, θ), e_{i}(t, θ)⟩)) = 0 (∀) h < m, i ∈ {1, …, h − 1}.

Because the supervectors G_{1}c(t, θ), G_{1}^{(2)}c(t, θ), …, G_{1}^{(h)}c(t, θ) (h < m) are linear independent it results that f_{h}(t, θ) ≠ 0.

We put (36) e_{h}(t, θ) = f_{h}(t, θ) · (||f_{h}(t, θ)||′)^{−1}, h < m.

Thus, we have constructed the unit and orthogonal two by two supervectors e_{1}(t, θ), …, e_{m−1}(t, θ). On the other hand, from (31) and (36) we get the fol-

lowing relations for all h < m :

(37) G_{1}^{(h)}c(t, θ) = (ε^{(L)}(⟨G_{1}^{(h)}c(t, θ), e_{1}(t, θ)⟩) − s(A(t, θ))) · e_{1}(t, θ) + ⋯

⋯ + (ε^{(L)}(⟨G_{1}^{(h)}c(t, θ), e_{h−1}(t)⟩) − s(A_{h−1}(t, θ))) · e_{h−1}(t, θ) + ||f_{h}(t, θ)||′ · e_{h}(t, θ).

From (29), (30) and (37) we obtain:

span(G_{1}c(t, θ), G_{1}^{(2)}c(t, θ), …, G_{1}^{(h)}c(t, θ)}) = span(e_{1}(t, θ), …, e_{h}(t, θ)).

Taking account of (29), (30), and (37) we get that the ” body ” of the determinant of the matrix of the linear transformation when we change from the basis {e_{1}(t, θ), …, e_{h}(t, θ)} to the basis {G_{1}c(t, θ), G_{1}^{(2)}c(t, θ), …, G_{1}^{(h)}c(t, θ)} (h < m) ε^{(L)}(Δ(t, θ)) is given by:

ε^{(L)}(Δ(t, θ)) = ||f_{1}(t, θ)||′ · · · · · ||f_{h}(t, θ)||′ > 0 (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Therefore ε^{(L)}(Δ(t, θ)) > 0 (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and the systems of supervectors {e_{1}(t, θ), …, e_{h}(t, θ)} and {G_{1}c(t, θ), G_{1}^{(2)}c(t, θ), …, G_{1}^{(h)}c(t, θ)} (h < m) are directed in the same way (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

By our construction, the functions (t, θ) → e_{k}(t, θ) (∀) k ∈ {1, …, m − 1} and (t, θ) → e_{m+j}(t, θ) (∀) j ∈ {1, …, n} are supersmooth. We shall get e_{m}(t, θ) from the relations

(38) ⟨e_{m}(t, θ), e_{k}(t, θ)⟩ = 0 (∀) k ∈ {1, …, m − 1}

and

(38) ⟨e_{m}(t, θ), e_{m+j}(t, θ)⟩ = 0 (∀) j ∈ {1, …, n}

Thus, we have:

e_{m}^{1}(t, θ) · e_{1}^{1}(t, θ) + ⋯ + e_{m}^{m}(t, θ) · e_{1}^{m}(t, θ) + e_{m}^{m+1}(t, θ) · e_{1}^{m+r+1}(t, θ) + ⋯

⋯ + e_{m}^{m+r}(t, θ) · e_{1}^{m+n}(t, θ) − e_{m}^{m+r+1}(t, θ) · e_{1}^{m+1}(t, θ) − … − e_{m}^{m+n}(t, θ) · e_{1}^{m+r}(t, θ) = 0

⋮ (39)

⋮

e_{m}^{1}(t, θ) · e_{m+n}^{1}(t, θ) + ⋯ + e_{m}^{m}(t, θ) · e_{m+n}^{m}(t, θ) + e_{m}^{m+1}(t, θ) · e_{m+n}^{m+r+1}(t, θ) + ⋯

⋯ + e_{m}^{m+r}(t, θ) · e_{m+n}^{m+n}(t, θ) − e_{m}^{m+r+1}(t, θ) · e_{m+n}^{m+1}(t, θ) − … − e_{m}^{m+n}(t, θ) · e_{m+n}^{m+r}(t, θ) = 0

where e_{k}^{1}(t, θ), …, e_{k}^{m+n}(t, θ) are the components of the supervector e_{k}(t, θ) (∀) k ∈ {1, …, m} and e_{m+j}^{1}(t, θ), …, e_{m+j}^{m+n}(t, θ) are the components of the supervector e_{m+j}(t, θ) (∀) j ∈ {1, …, n}.

Let us consider (39) as a linear and homogeneous system of m + n − 1 equations with the m + n unknowns e_{m}^{1}(t, θ), …, e_{m}^{m+n}(t, θ). Because the supervectors e_{1}(t, θ), …, e_{m−1}(t, θ), e_{m+1}(t, θ), …, e_{m+n}(t, θ) (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} are linear independent, it follows that the rank of the matrix M(t, θ) :

( \begin{matrix}e_{1}^{1} & ⋯ & e_{1}^{m} & e_{1}^{m+r+1} & ⋯ & e_{1}^{m+n} & −e_{1}^{m+1} & ⋯ & −e_{1}^{m+r} \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ e_{m−1}^{1} & ⋯ & e_{m−1}^{m} & e_{m−1}^{m+r+1} & ⋯ & e_{m−1}^{m+n} & −e_{m−1}^{m+1} & ⋯ & −e_{m−1}^{m+r} \\ e_{m+1}^{1} & ⋯ & e_{m+1}^{m} & e_{m+1}^{m+r+1} & ⋯ & e_{m+1}^{m+n} & −e_{m+1}^{m+1} & ⋯ & −e_{m+1}^{m+r} \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ e_{m+n}^{1} & ⋯ & e_{m+n}^{m} & e_{m+n}^{m+r+1} & ⋯ & e_{m+n}^{m+n} & −e_{m+n}^{m+1} & ⋯ & −e_{m+n}^{m+r} \end{matrix} )

is m + n − 1, where e_{q}^{s} means e_{q}^{s}(t, θ) with s ∈ {1, …, m + n} and q ∈ {1, …, m−1, m+1, …, m+n}. Let Δ_{q}(t, θ) be the minor of order m+n−1 obtained by omitting the q colunm from the matrix M(t, θ) (∀) q ∈ {1, …, m+n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Then from (39), we get:

(40)  e_{m}^{q}(t, θ) = (−1)^{q−1} · ν(t, θ) · Δ_{q}(t, θ),  (∀) q ∈ {1, …, m + n},

where ν(t, θ) has to fulfil the condition:

(41) ||e_{m}(t, θ)||′ = 1.

Because of rank(M(t, θ)) = m + n − 1 we have:

(42) ε^{(L)}(Δ(t, θ)) = (ε^{(L)}(Δ_{1}(t, θ)))^{2} + ⋯ + (ε^{(L)}(Δ_{m}(t, θ)))^{2} > 0.

From (40) and (41) we get:

(43) ν(t, θ) = η · (\sqrt{(Δ(t, θ))})^{−1}

where η = 1 or η = −1; this resulting from the condition that the frame {e_{1}(t, θ), …, e_{m+n}(t, θ)} to be positiv directed. By (40) and (43) we get:

(44) e_{m}^{q}(t, θ) = (−1)^{q−1} · η · Δ_{q}(t, θ) · (\sqrt{(Δ(t, θ))})^{−1} (∀) q ∈ \{1, …, m + n\},

where η verify the conditions |η| = 1 and

ε^{(L)}(sdet(e_{q}^{s}(t, θ))_{1≤s,q≤m+n}) > 0.

By (44) and (41) it follows that the functions (t, θ) → e_{m}^{s}(t, θ), 1 ≤ s ≤ m+n are supersmooth and the function (t, θ) → e_{m}(t, θ) is supersmooth. From our construction results (19) and (19^{V}) from the first part of this paper. The uniqueness of the Frenet frame results by our construction. We fix an index k ∈ {1, …, m}. We express G_{1}e_{k}(t, θ) in the frame {e_{1}(t, θ), …, e_{m+n}(t, θ)} and we have:

(45) G_{1}e_{k}(t, θ) = ∑_{h=1}^{m} a_{kh}(t, θ) · e_{h}(t, θ) + ∑_{j=1}^{n} a_{kj}(t, θ) · e_{m+j}(t, θ).

Computing the scalar product between the (45) relations and e_{i}(t, θ) (∀) i ∈ {1, …, m} we get:

(46) a_{ki}(t, θ) = ⟨G_{1}e_{k}(t, θ), e_{i}(t, θ)⟩

because ⟨e_{m+j}(t, θ), e_{i}(t, θ)⟩ = 0 (∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus we proved the (18) relation (∀) k, i ∈ {1, …, m} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Computing the scalar product between the (45) relations and e_{m+l}(t, θ) (∀) l ∈ {1, …, n} we get:

(47) a_{k m+l}(t, θ) = ⟨G_{1}e_{k}(t, θ), e_{m+l}(t, θ)⟩

because ⟨e_{h}(t, θ), e_{m+l}(t, θ)⟩ = 0 (∀) h ∈ {1, …, m} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus we proved the (19) relation (∀) h ∈ {1, …, m}, l ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Derivating the following relation by G_{1}

⟨e_{k}(t, θ), e_{h}(t, θ)⟩ = δ_{kh} (∀) k, h ∈ {1, …, m}, (∀)(t, θ) ∈ V ⊂ B_{L}^{1,1}.

we get:

⟨G_{1}e_{k}(t, θ), e_{h}(t, θ)⟩ + ⟨e_{k}(t, θ), G_{1}e_{h}(t)⟩ = 0 (∀) k, h ∈ {1, …, m},

(∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. By the (46) relation we have:

a_{kh}(t, θ) + a_{hk}(t, θ) = 0

that means we proved the (6) relation (∀) k, h ∈ {1, …, m} and

(∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Because {e_{1}(t, θ), …, e_{m+n}(t, θ)} is a Frenet frame we have:

(48) G_{1}^{(k)}c(t, θ) ∈ span(e_{1}(t, θ), …, e_{k}(t, θ)), (∀) k ∈ {1, …, m − 1}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and

(49) e_{k}(t, θ) ∈ span(G_{1}c(t, θ), …, G_{1}^{(k)}c(t, θ)) (∀) k ∈ {1, …, m − 1}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

By (48) and (49) we get:

(50) G_{1}e_{k}(t, θ) ∈ span(G_{1}c(t, θ), …, G_{1}^{(k)}c(t, θ), G_{1}^{(k+1)}c(t, θ))

(∀) k ∈ {1, …, m − 1} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. By (48) and (50) we have:

(51) G_{1}e_{k}(t) ∈ span(e_{1}(t, θ), …, e_{k+1}(t, θ)) (∀) k ∈ {1, …, m − 1}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

By (51), we note that in the writing

G_{1}e_{k}(t, θ) = ∑_{h=1}^{m} a_{kh}(t, θ) · e_{h}(t, θ) + ∑_{j=1}^{n} a_{k m+j}(t, θ) · e_{m+j}(t, θ) (∀) k ∈ {1, …, m}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} the coeficients a_{kh}(t, θ) are zero if h > k + 1 and a_{k m+j}(t, θ) = 0 (∀) k ∈ {1, …, m}, j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Thus, we get the (5) formula (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} :

G_{1}e_{k}(t, θ) = ∑_{h=1}^{m} a_{kh}(t, θ) · e_{h}(t, θ) (∀) k ∈ {1, …, m}

and the (7) formula a_{kh}(t, θ) = 0 if h > k + 1, (∀) k, h ∈ {1, …, m} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Because of our construction of the frame {e_{1}(t, θ), …, e_{m+n}(t, θ)}, we have:

(52) G_{1}^{(j−1)}G_{2}c(t, θ) ∈ span(e_{m+j}(t, θ)),

and

(53) e_{m+j}(t, θ) ∈ span(G_{1}^{(j−1)}G_{2}c(t, θ))

(∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. From now, we have:

(54) G_{1}e_{m+j}(t, θ) ∈ span(G_{1}^{(j)}G_{2}c(t, θ))

(∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. By the (52) and (54) relations we get

(55) G_{1}e_{m+j}(t, θ) ∈ span(e_{m+j+1}(t, θ))

(∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. By (55) we note that in writing

G_{1}e_{m+j}(t, θ) = ∑_{k=1}^{m} a_{m+j k}(t, θ) · e_{k}(t, θ) + ∑_{l=1}^{n} a_{m+j m+l}(t, θ) · e_{m+j}(t, θ)

(∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}, the coeficients a_{m+j m+l}(t, θ) are zero if l ≠ j + 1 and a_{m+j k}(t, θ) = 0 (∀) k ∈ {1, …, m}, j ∈ {1, …, n} (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Thus we proved the (8), (12) and (13) relations.

We fix an index j_{1} ∈ {1, …, r}. We express G_{1}e_{m+j_{1}}(t, θ) in the frame {e_{1}(t, θ), …, e_{m+n}(t, θ)}. We have:

(56) G_{1}e_{m+j_{1}}(t, θ) = ∑_{h=1}^{m} a_{m+j_{1} h}(t, θ) · e_{h}(t, θ) + ∑_{j=1}^{n} a_{m+j_{1} m+j}(t) · e_{m+j}(t, θ)

(∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Computing the scalar product between the (56) relation and e_{k}(t, θ) (∀) k ∈ {1, …, m} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} we get:

(57) a_{m+j_{1} k}(t) = ⟨G_{1}e_{m+j_{1}}(t, θ), e_{k}(t, θ)⟩ (∀) k ∈ {1, …, m}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} because ⟨e_{m+j}(t, θ), e_{k}(t, θ)⟩ = 0 (∀) k ∈ {1, …, m}, (∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus, we proved the (20) relation (∀) j_{1} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Computing the scalar product between the (56) relation and e_{m+j_{2}}(t, θ) (∀) j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. we get:

(58) a_{m+j_{1} m+r+j_{2}}(t, θ) = −⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ (∀) j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus we proved the (15) relation (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Computing the scalar product between the (56) relation and e_{m+r+j_{2}}(t, θ) (∀) j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} we get:

(59) a_{m+j_{1} m+j_{2}}(t, θ) = ⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ (∀) j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus, we proved the (14) relation (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

We fix an index j_{1} ∈ {1, …, r}. We express G_{1}e_{m+r+j_{1}}(t, θ) in the frame {e_{1}(t, θ), …, e_{m+n}(t, θ)}. We have:

(60) G_{1}e_{m+r+j_{1}}(t, θ) = ∑_{h=1}^{m} a_{m+r+j_{1} h}(t) · e_{h}(t, θ)+

+ ∑_{j=1}^{n} a_{m+r+j_{1} m+j}(t, θ) · e_{m+j}(t, θ) (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}.

Computing the scalar product between the (60) relation and e_{k}(t, θ) (∀) k ∈ {1, …, m} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} we get:

(61) a_{m+r+j_{1} k}(t, θ) = ⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{k}(t, θ)⟩ (∀) k ∈ {1, …, m}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} because ⟨e_{m+j}(t, θ), e_{k}(t, θ)⟩ = 0 (∀) k ∈ {1, …, m}, (∀) j ∈ {1, …, n} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus, by (57) and (61), we proved the (20) relation (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Computing the scalar product between the (60) relation and e_{m+j_{2}}(t, θ) (∀) j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} we get:

(62) a_{m+r+j_{1} m+r+j_{2}}(t, θ) = −⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ (∀) j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus we proved the (17) relation (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Computing the scalar product between the (60) relation and e_{m+r+j_{2}}(t, θ) (∀) j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} we get:

(63) a_{m+r+j_{1} m+j_{2}}(t, θ) = ⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ (∀) j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Thus, we proved the (16) relation (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1}. Derivating the following relation by G_{1}

⟨e_{m+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ = δ_{j_{1}j_{2}} (∀) j_{1}, j_{2} ∈ {1, …, r}

we get

⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ + ⟨e_{m+j_{1}}(t, θ), G_{1}e_{m+r+j_{2}}(t, θ)⟩ = 0

(∀) j_{1}, j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} which is equivalent to

⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ − ⟨G_{1}e_{m+r+j_{2}}(t, θ), e_{m+j_{1}}(t, θ)⟩ = 0

(∀) j_{1}, j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and using the (14) and (17) relations we get:

a_{m+j_{1} m+j_{2}}(t, θ) + a_{m+r+j_{2} m+r+j_{1}}(t, θ) = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} which is the (9) relation.

Derivating the following relation by G_{1}

⟨e_{m+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

we get

⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ + ⟨e_{m+j_{1}}(t, θ), G_{1}e_{m+j_{2}}(t, θ)⟩ = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} which is equivalent to

⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+j_{2}}(t, θ)⟩ − ⟨G_{1}e_{m+j_{1}}(t, θ), e_{m+j_{1}}(t, θ)⟩ = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and using the (15) relation we get:

−a_{m+j_{1} m+r+j_{2}}(t, θ) + a_{m+j_{2} m+r+j_{1}}(t, θ) = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} which is the (11) relation.

Derivating the following relation by G_{1}

⟨e_{m+r+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

we get

⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ + ⟨e_{m+r+j_{1}}(t, θ), G_{1}e_{m+r+j_{2}}(t, θ)⟩ = 0

(∀) j_{1}, j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} which is equivalent to

⟨G_{1}e_{m+r+j_{1}}(t, θ), e_{m+r+j_{2}}(t, θ)⟩ − ⟨G_{1}e_{m+r+j_{2}}(t, θ), e_{m+r+j_{1}}(t, θ)⟩ = 0

(∀) j_{1}, j_{2} ∈ {1, …, r} and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} and using the (16) relation we get:

a_{m+r+j_{1} m+j_{2}}(t, θ) − a_{m+r+j_{2} m+j_{1}}(t, θ) = 0 (∀) j_{1}, j_{2} ∈ {1, …, r}

and (∀) (t, θ) ∈ V ⊂ B_{L}^{1,1} which is the (10) relation. ♢

Corollary 1. The (5) and (8) relations extend the Frenet formulas for the curves.

Remark 1. By (6), (7), (12), (13), (9), (10) and (11) we get:

A = (a_{sq})_{1≤s,q≤m+n} = ( \begin{matrix}A_{1} & A_{3} \\ A_{4} & A_{2} \end{matrix} ) ,

where

A_{1} = ( \begin{matrix}0 & a_{12} & ⋯ & 0 & 0 \\ −a_{12} & 0 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & a_{m−1 m} \\ 0 & 0 & ⋯ & −a_{m−1 m} & 0 \end{matrix} ) ,

A_{2} = ( \begin{matrix}a_{m+1 m+1} & ⋯ & 0 & 0 & ⋯ & 0 \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & ⋯ & a_{m+r m+r} & 0 & ⋯ & 0 \\ 0 & ⋯ & 0 & −a_{m+1 m+1} & ⋯ & 0 \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & ⋯ & 0 & 0 & ⋯ & −a_{m+r m+r} \end{matrix} )

and

A_{3} = A_{4} = ( \begin{matrix}0 & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & 0 \end{matrix} ) .

Example 1. Let B_{L}^{2+2} be the (2, 2)−dimensional total supereuclidean space, V be an open subset of B_{L}^{1,1} and c : V ⊂ B_{L}^{1,1} → B_{L}^{2+2} be a supersmooth supercurve given by

c(t, θ) = (t^{2}, θ · β^{2}, θ + 2 · β^{1} · t, θ · t^{2})

where V ∩ R be the open set (0, 1) .

We note that the supercurve c is supersmooth because the functions

c^{1}(t, θ) = t^{2},  c^{2}(t, θ) = θ · β^{2},  c^{3}(t, θ) = θ + 2 · β^{1} · t,  c^{4}(t, θ) = θ · t^{2}

are supersmooth. Let us compute G_{1}c(t, θ), G_{2}c(t, θ), and G_{1}G_{2}c(t, θ) :

G_{1}c(t, θ) = (2 · t, 0, 2 · β^{1}, 2 · θ · t)
G_{2}c(t, θ) = (0, β^{2}, 1, t^{2})  and  G_{1}G_{2}c(t, θ) = (0, 0, 0, 2 · t).

By

ε^{(L)}(⟨G_{2}c(t, θ), G_{1}G_{2}c(t, θ)⟩) = ε^{(L)}(0 · 0 + β^{2} · 0 + 1 · 2 · t − t^{2} · 0) = 2 · t > 0

(∀) t ∈ (0, 1),

ε^{(L)}(⟨G_{2}c(t, θ), G_{2}c(t, θ)⟩) = ε^{(L)}(0 · 0 + β^{2} · β^{2} + 1 · t^{2} − t^{2} · 1) = 0

(∀) t ∈ (0, 1) and

ε^{(L)}(⟨G_{1}G_{2}c(t, θ), G_{1}G_{2}c(t, θ)⟩) = ε^{(L)}(0 · 0 + 0 · 0 + 0 · 2 · t − 2 · 0 · t) = 0

we conclude that the supercurve c fulfils the (1), (2), (3), (4), relations from the Theorem 1.

Computing G_{1}c(t, θ), G_{2}c(t, θ), and G_{1}G_{2}c(t, θ), we get that the supervectors {G_{1}c(t, θ), G_{2}c(t, θ), G_{1}G_{2}c(t, θ)} are linear independent. Thus we conclude that the supercurve c is in general position. Let us get the Frenet frame of the supercurve c, {e_{1}(t, θ), e_{2}(t, θ), e_{3}(t, θ), e_{4}(t, θ)} and the matrix A = (a_{sq})_{1≤s,q≤4}. Let f_{1}(t, θ) be G_{1}c(t, θ) = (2 · t, 0, 2 · β^{1}, 2 · θ · t) and

⟨G_{1}c(t, θ), G_{1}c(t, θ)⟩ = 2·t·2·t+0·0+2·β^{1}·2·θ·t−2·θ·2·β^{1}·t = 4·t^{2}+4·β^{1}·θ·t−4·θ·β^{1}·t

We have ⟨G_{1}c(t, θ), G_{1}c(t, θ)⟩ = 4·t^{2} +8·β^{1} ·θ·t and ε^{(L)}(⟨G_{1}c(t, θ), G_{1}c(t, θ)⟩) = = 4 · t^{2}. Thus we get ||G_{1}c(t, θ)||′ = \sqrt{ε^{(L)}(⟨G_{1}c(t, θ), G_{1}c(t, θ)⟩)} = \sqrt{4 · t^{2}} = 2 · t.

By the above theorem 1 we compute the Frenet frame of c, {e_{1}(t, θ), e_{2}(t, θ), e_{3}(t, θ), e_{4}(t, θ)} and we get

e_{1}(t, θ) = (1, 0, 2 · β^{1} · (2 · t)^{−1}, θ),
e_{3}(t, θ) = (0, β^{2} · (2 · t)^{−1}, (2 · t)^{−1}, 2^{−1} · t)

and

e_{4}(t, θ) = (0, 0, 0, 2 · t).

Let the matrix M(t, θ) be

( \begin{matrix}1 & 0 & θ & −2 · β^{1} · (2 · t)^{−1} \\ 0 & β^{2} · (2 · t)^{−1} & 2^{−1} · t & −(2 · t)^{−1} \\ 0 & 0 & 2 · t & 0 \end{matrix} )

and computing e_{2}(t, θ) we get

e_{2}(t, θ) = (−2β^{2} · β^{1} · (2 · t)^{−1}, 1, 0, −β^{2}).

Now, we may compute a_{12}(t, θ) and a_{33}(t, θ) and we have

a_{12}(t, θ) = ⟨G_{1}e_{1}(t, θ), e_{2}(t, θ)⟩ = β^{1} · β^{2} · t^{−2}

where

G_{1}e_{1}(t, θ) = (0, 0, −β^{1} · t^{−2}, 0)

and

a_{33}(t, θ) = ⟨G_{1}e_{3}(t, θ), e_{4}(t, θ)⟩ = −t^{−1},

where

G_{1}e_{3}(t, θ) = (0, −β^{2} · 2^{−1} · t^{−2}, −2^{−1} · t^{−2}, 2^{−1}).

We conclude that

A = (a_{sq})_{1≤s,q≤4} =

= ( \begin{matrix}0 & β^{1} · β^{2} · t^{−2} & 0 & 0 \\ −β^{1} · β^{2} · t^{−2} & 0 & 0 & 0 \\ 0 & 0 & −t^{−1} & 0 \\ 0 & 0 & 0 & t^{−1} \end{matrix} ) .

Acknowledgement

The author is very grateful to Max-Planck-Institut für Mathematik for financial support and hospitality.

REFERENCES

1. BARTOCCI, C., BRUZZO, U., HERNANDEZ-RUIPEREZ, D., The Geometry of Supermanifolds, Mathematics and Its Applications, Volume 71, 1991

2. BEREZIN, F.A., Introduction to Superanalysis, D. Reidel Publishing Company, Kluwer Academic Publishers Group Dordrecht, Boston, Lancaster, Tokyo, Volume 7, 1987 3. BEREZIN, F.A., The method of second quantization, Academic Press, New York, 1966 4. DEWITT, B., Supermanifolds, Cambridge, Univ. Press, Cambridge, London, 1984

5. INOUE, A., MAEDA, Y., Foundations of Calculus on Supereuclidean Space based on a Frcehet-Grassmann Algebra, Kodai Math. J. 14(1991)

6. KOBAYASHI, S., NOMIZU, K., Foundations of Differential Geometry, (I) Interscience, New York-London 1963

7. KOSTANT, B., Graded manifolds, Graded Lie Theory and Prequantization, Lect. Notes in Math. no. 570, Springer-Verlag 1977

8. MANIN, Yu.I., Gauge Field Theory and Complex Geometry, A Series of Comprehensive Studies in Mathematics 289, Springer-Verlag 1988

9. NICOLESCU, L., Curs de geometrie, Tipografia Univ. Bucuresti, 1989

10. ROGERS, A., Graded Manifolds, Supermanifolds and Infinite-Dimensional Grassmann Algebras, Commun. Math. Phys. 105, 375-384(1986)

11. ROGERS, A., A global theory of supermanifolds, J. Math. Phys. 21(6), 1352-1365 June 1980

12. SCHEUNERT, M., The theory of Lie superalgebras, Lect. Notes in Math. no. 716, Springer-Verlag 1979

MAX-PLANCK-INSTITUT FÜR MATHEMATIK, PB 7280, D-53072 BONN, GERMANY AND C.P. 101 O.P. 4 TARGOVISTE 0200 ROMANIA E-mail address: cristea@mpim-bonn.mpg.de

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