On Normal Sections of Veronese Submanifold

Kadri Arslan & Cihan Özgür

Abstract In this study we consider Veronese Submanifold V^{m} which is a projective m-space P^{m} isometrically imbedded in R^{m+ \frac{1}{2} m(m+1)} by its first standard imbedding. We also consider the normal sections of V^{m}. Finally we show that V^{m} is of AW (3)-type.

Mathematics Subject Classifications: 53C40 Key words: Normal section, Veronese submanifold, submanifold of AW (3) type.

1 Introduction

Let M be an n-dimensional submanifold in (m + d)− dimensional Euclidean space

R^{n+d}. Let ∇, ∇^{−}, and ∇^{∼} denote the covariant derivatives in T (M), N(M) and R^{n+d} respectively. Thus ∇^{∼}_{X} is just the directional derivative in the direction X in R^{n+d}. Then for tangent vector fields X, Y and Z and normal vector field υ over M we

have ∇^{∼}_{X} Y = ∇_{X}Y + h(X, Y) and ∇^{∼}_{X} υ = −A_{υ}X+ ∇^{−}_{X} υ where h is the second fundamental form and A_{υ} is the shape operator of M [3].

For tangent vector fields X, Y, Z, W over M we define ∇^{−} h and ∇^{−}∇^{−} h as usual by

∇^{−}_{X} h(Y, Z) = (∇^{−}_{X} h)(Y, Z) + h(∇_{X}Y, Z) + h(Y, ∇_{X}Z),

∇^{−}_{W} ((∇^{−}_{X} h)(Y, Z)) = (∇^{−}_{W} ∇^{−}_{X} h)(Y, Z) + (∇^{−}_{X} h)(∇_{W} Y, Z)−

−(∇^{−}_{X} h)(Y, ∇_{W} Z) + (∇^{−}_{Y} h)(∇_{W} X, Z).

2 Normal sections

Let M be a smooth n-dimensional submanifold in (n+d)-dimensional Euclidean space R^{n+d}. For a point x in M and a non-zero tangent vector X ∈ T_{x}M, we define the (d + 1)−dimensional affine subspace E(x, X) of R^{n+d} by

E(x, X) = x + span{X, T_{x}^{⊥}M}.

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 1-8 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

In a neighborhood of x the intersection M ∩E(x, X) is a regular curve γ : (−ε, ε) → M. We suppose the parameter t ∈ (−ε, ε) is a multiple of the arc-length such that γ(0) = x and γ′(0) = X. Each choose of X ∈ T (M) yields a different curve which is called the normal section of M at x in the direction of X, where X ∈ T_{x}(M) [5]. For such a normal section we can write

(1)  γ(t) = x + λ(t)X + N(t),

where N(t) ∈ T_{x}^{⊥}M and λ(t) ∈ R. Definition 1. The submanifold M is said to have pointwise k-planar normal sections (P k − P NS) if for each normal section γ, the higher order derivatives γ′(0), γ′′(0),..., γ^{(k+1)}(0) are linearly dependent as vectors in R^{n+d}.

If k = 1, then M is totally geodesic. Taking k = 2, we note that submanifolds with pointwise 2-planar normal sections have been classified. They have parallel second

fundamental form (i. e. ∇^{−} h = 0) or hypersurfaces [6], [1]. Taking k = 3, we note that submanifolds with pointwise 3-planar normal sections have been studied by various mathematicians (see [7],[2]). Proposition 1. Let γ(t) be a normal section of an n−dimensional submanifold M in R^{n+d}. Then M has P k − P NS if and only if for every normal section, the normal vectors N′′(0), N′′′(0),..., N^{(k+1)}(0) are linearly dependent. Proposition 2. Let γ(t) be a normal section of M at point γ(0) = x in the direction of γ′(0) = X. Then

N′′(0) = h(X, X),

N′′′(0) = (∇^{−}_{X} h)(X, X),

N^{(iv)}(0) = 4λ^{(iv)}(0)h(X, X) + 3h(A_{h(X,X)}X, X) + (∇^{−}_{X} ∇^{−}_{X} h)(X, X).

Proof. Let M be an n-dimensional submanifold of R^{n+d}. Let X be a unit vector tangent to M at x ∈ M and let γ(s) be a normal section of M at p in the direction of X with its arc-length and γ(0) = x. We denote T = γ′(s) the unit vector tangent to the normal section. Then we have

γ′′(s) =∇^{∼}_{T} T = ∇_{T} T + h(T, T)

γ′′′(s) =∇^{∼}_{T} ∇^{∼}_{T} T = ∇_{T} ∇_{T} T + h(T, ∇_{T} T) − A_{h(T,T)}T + ∇^{∼}_{T} (h(T, T))

γ^{(iv)}(s) =∇^{∼}_{T} ∇^{∼}_{T} ∇^{∼}_{T} T = ∇_{T} ∇_{T} ∇_{T} T + 2h(∇_{T} ∇_{T} T, T) − h(A_{h(T,T)}T, T)

−∇_{T} (A_{h(T,T)}T) − 3A_{h(∇_{T} T,T)}T + 6(∇^{−}_{T} h)(∇_{T} T, T)

−A_{(∇^{−}_{T} h)(T,T)}T + h(∇_{T} T, ∇_{T} T) + (∇^{−}_{T} ∇^{−}_{T} h)(T, T),

at s = 0. After some calculation we obtain

γ′(0) = X , N′(0) = 0
γ′′(0) = h(X, X) = N′′(0)

γ′′′(0) = ∇_{X}∇_{X}X − A_{h(X,X)}X + (∇^{−}_{X} h)(X, X) = λ′′′(0)X + N′′′(0)

γ^{(iv)}(0) = (terms in T_{x}M)+4λ^{(iv)}(0)h(X, X)+3h(A_{h(X,X)}X, X)+(∇^{−}_{X} ∇^{−}_{X} h)(X, X) =

= λ^{(iv)}(0)X + N^{(iv)}(0).

For the normal parts we obtain the result. Definition 2. Submanifolds are of AW (3) type if

(2)  ∥N′′(0)∥^{2} N^{(iv)}(0) =< N′′(0), N^{(iv)}(0) > N′′(0).

(see [2]).

3 Veronese submanifolds

In this section we will consider the Veronese submanifold which is the first standard imbedding of the real projective space P^{m} in R^{m+ \frac{1}{2} m(m+1)}. The notation here is essentially the same as in [4].

Let M(m+1; R) be the space of (m+1)×(m+1) matrices over R. It is considered as an (m + 1)^{2}-dimensional Euclidean space with the inner product < A, B >=

\frac{1}{2}traceAB^{T}, where B^{T} is the transpose of the matrix B. Someone consider P^{m} as the quotient space of the hypersphere

S^{m} = {ζ ∈ R^{m+1} : ζ^{T} ζ = 1}

obtained by identifying ζ with ζλ, where ζ is a column vector and λ ∈ R such that |λ| = 1.

Define a mapping

φ̃: S^{m} → H(m + 1; R) = {A ∈ M(m + 1; R) : A^{T} = A}

as follows

(3)  φ̃(ζ) = ζζ^{T} = ( \begin{matrix}|ζ_{0}|^{2} & ζ_{0}ζ_{1} & − & ζ_{0}ζ_{m} \\ − & − & − & − \\ ζ_{m}ζ_{0} & ζ_{m}ζ_{1} & − & |ζ_{m}|^{2} \end{matrix} )

for

ζ = (ζ_{i}) ∈ S^{m} ⊂ R^{m+1}, 0 ≤ i ≤ m.

Then it is easy to verify that φ̃ induces a mapping φ of P^{m} into H(m + 1; R):

(4)  φ(π(ζ)) = φ̃(ζ) = ζζ^{T} ,

where π : S^{m} → P^{m} is a Riemannian submersion [4]. We simply denote φ(π(ζ)) by φ(ζ).

From (3) the image of P^{m} under φ is given by

φ(P^{m}) = {A ∈ H(m + 1; R) : A^{2} = A and traceA = 1}.

Let A = ζζ^{T} be a point in φ(P^{m}). Consider the curve

(5)  A(t) = ζζ^{T}; ζ ∈ R^{m+1}

in φ(P^{m}) with A(0) = A and A′(0) = X ∈ T_{A}(P)^{m}). From A^{2}(t) = A(t) one gets XA^{T} + AX^{T} = X. So we have

(6)  T_{A}(P^{m}) = {X ∈ H(m + 1; R) : XA^{T} + AX^{T} = X}.

Proposition 3 [4]. Let Y be a vector field tangent to P^{m} and X ∈ T_{A}(P^{m}). Consider a curve A(t) in φ(P^{m}) so that A(0) = A and A′(0) = X. Denote by Y (t) the restriction of Y to A(t). Then

A(t)Y (t) + Y (t)A(t) = Y (t).

Corollary 4. Let A(t) = ζζ^{T}; ζ ∈ R^{m+1} be a curve in φ(P^{m}). Then

(7)  A(0) = X = ξa^{T} + aξ^{T} ,

where

(8)  ζ(0) = ξ, ζ′(0) = a.

Proof. Let A(t) = ζζ^{T} be a curve in φ(P^{m}) with A(0) = A and A′(0) = X ∈ T_{A}(P^{m}). Differentiating (5) we get

(9)  A′(t) = ζ′ζ^{T} + ζ(ζ′)^{T} .

Substituting (8) into (9) we obtain (7).

A vector ν in H(m + 1; R) is normal to P^{m} at A if and only if < X, ν >= 0 for all X in T_{A}(P^{m}). Thus, ν is in T_{A}^{⊥}(P^{m}) if and only if trace(Xν) = 0 for all X in T_{A}(P^{m}). Therefore by (6) we obtain

(10)  T_{A}^{⊥}(P^{m}) = {ν ∈ H(m + 1; R) : Aν = νA}.

Proposition 5 [4]. If Y is a vector field as in the previous proposition and X ∈ T_{A}(P^{m}), then

∇^{∼}_{X} Y = Y′(0) = A(∇^{∼}_{X} Y) + (∇^{∼}_{X} Y)A + XY + Y X,

(11)  h^{∼} (X, Y) = (XY + Y X)(I − 2A),

and

∇_{X}Y = 2(XY + Y X)A + A(∇^{∼}_{X} Y) + (∇^{∼}_{X} Y)A,

where h^{∼} is the second fundamental form of P^{m} in H(m + 1; R) at A, ∇ and ∇^{∼} denote the induced connection on P^{m} and the Riemannian connection of the Euclidean space H(m + 1; R) at A respectively. Corollary 6. Let A(t) = ζζ^{T} be a curve in φ(P^{m}) with A(0) = A and A′(0) = X ∈ T_{A}(P^{m}). Then

(12)  h^{∼} (X, X) = 2aa^{T} − 2ξa^{T} aξ^{T} .

Proof. Differentiating ξ^{T} ξ = 1 and using (9) we obtain

(13)  a^{T} ξ = 0.

Substituting (13) and (7) into (11) we get the result. Theorem 7 [4]. The isometric imbedding φ : S^{m} → H(m + 1; R), determined by (4), is the first standard imbedding of P^{m} into H(m + 1, R) and P^{m} lies in a hypersphere

S(r) of H(m + 1; R) centered at \frac{1}{m + 1}I and with radius r = \sqrt{\frac{m}{2}(m + 1)}.

Definition 3. A real projective m-space P^{m} isometrically imbedded in R^{m+ \frac{1}{2} m(m+1)} by its first standard imbedding φ determined by (4) is called the Veronese submanifold V^{m} [4].

4 Normal section of V^{m}

Let A ∈ H(m + 1; R) be a symmetric matrix. Then we can decompose A into

\begin{aligned}A & = & ξξ^{T} Aξξ^{T} + [(1 − ξξ^{T})Aξξ^{T} + \\ & + & ξξ^{T} A(1 − ξξ^{T})] + (1 − ξξ^{T})A(1 − ξξ^{T}), \end{aligned}

where ξ ∈ R^{m+1} is the constant vector.

Interchanging A with the curve A(t) in φ(P^{m}) we get

\begin{aligned}A(t) & = & ξξ^{T} A(t)ξξ^{T} + [(1 − ξξ^{T})A(t)ξξ^{T} + \\ & + & ξξ^{T} A(t)(1 − ξξ^{T})] + (1 − ξξ^{T})A(t)(1 − ξξ^{T}). \end{aligned}

We consider A(t) as a normal section of V^{m} at point A(0) in the direction of A′(0). Then by (1) we get

(14)  λ(t)X = (1 − ξξ^{T})A(t)ξξ^{T} + ξξ^{T} A(t)(1 − ξξ^{T})

and

(15)  N(t) = ξξ^{T} A(t)ξξ^{T} + (1 − ξξ^{T})A(t)(1 − ξξ^{T}).

Combining (14) with (15) we have

(16)  λ(t)a = (1 − ξξ^{T})A(t)ξ,

where ξ = ζ(0) and a = ζ′(0). Theorem 8. Veronese submanifold V^{m} is of AW (3) type. Proof. The normal section A(t) of V^{m} at point A(0) ∈ V^{m} in the direction of

A′(0) = X = ξa^{T} + aξ^{T}

is given by

A(t) = ζ(t)ζ^{T} (t),

where ζ(t) ∈ V^{m} ⊂ R^{m+1} and

(17)  ζ^{T} (t)ζ(t) = 1.

Differentiating (5) and (17) (surpressing the dependence of ζ on t to simplify the notation) we get

A′(t) = ζ′ζ^{T} + ζ(ζ′)^{T} = λ′(t)X + N′(t),
(ζ′)^{T} ζ + ζ^{T} ζ′ = 0.

Differentiating the previous equations and (10) respectively we obtain

(18)  (ζ′′)^{T} ζ + 2(ζ′)^{T} ζ′ + ζ^{T} ζ′′ = 0,

(19)  A′′(t) = ζ′′ζ^{T} + 2ζ′(ζ′)^{T} + ζ(ζ′′)^{T} ,

and

(20)  λ′′(t)a = (1 − ξξ^{T})A′′(t)ξ.

At the point t = 0, substituting (8) into (18)-(20) we have

(21)  ζ′′ = −ξ ∥ a ∥^{2},

(22)  λ′′(0) = 0,

(23)  A′′(0) = −2 ∥ a ∥^{2} ξξ^{T} + 2aa^{T} = N′′(0).

Differentiating (19)-(21) we get

(24)  (ζ′′′)^{T} ζ + 3(ζ′′)^{T} ζ′ + 3(ζ′)^{T} ζ′′ + ζ^{T} ζ′′′ = 0

(25)  A′′′(t) = ζ′′′ζ^{T} + 3ζ′′(ζ′)^{T} + 3ζ′(ζ′′)^{T} + ζ(ζ′′′)^{T} ,

(26)  λ′′′(t)a = (1 − ξξ^{T})A′′′(t)ξ.

At the point t = 0, substituting (8), and (21) into (24)-(26) we have

(27)  (ζ′′′)^{T} (0)ξ = 0,

\begin{aligned}A′′′(0) & = & ζ′′′(0)ξ^{T} − 3 ∥ a ∥^{2} ξa^{T} − 3 ∥ a ∥^{2} aξ^{T} + ξ(ζ′′′)^{T} (0) \\ & = & λ′′′(0)X + N′′′(0), \end{aligned}

(28)  λ′′′(0)a = ζ′′′(0) − 3a ∥ a ∥^{2} .

So by the use of (27) we also get

λ′′′(0)X = ζ′′′(0)ξ^{T} + ξ(ζ′′′)^{T} (0) − 3a ∥ a ∥^{2} ξ^{T} − 3ξa^{T} ∥ a ∥^{2},

which implies

(29)  N′′′(0) = 0.

By definition V^{m} has parallel second fundamental form.

Now we want to show that V^{m} is of AW (3) type. First, differentiating (24)-(27) we get

(30)  (ζ^{(iv)})^{T} ζ + 4(ζ′′′)^{T} ζ′ + 6(ζ′′)^{T} ζ′′ + 4(ζ′)^{T} ζ′′′ + ζ^{T} ζ^{(iv)} = 0,

(31)  A^{(iv)}(t) = ζ^{(iv)}ζ^{T} + 4ζ′′′(ζ′)^{T} + 6ζ′′(ζ′′)^{T} + 4ζ′(ζ′′′)^{T} + ζ(ζ^{(iv)})^{T} ,

(32)  λ^{(iv)}(t)a = (1 − ξξ^{T})A^{(iv)}(t)ξ.

At the point t = 0, substituting (8), (21) and (27) into (30)-(32) we have

ζ^{(iv)}(0) =∥ a ∥^{4} ξ, λ^{(iv)}(0) = 0,

(33)  A^{(iv)}(0) = 8 ∥ a ∥^{4} ξξ^{T} − 8 ∥ a ∥^{4}= N^{(iv)}(0).

Comparing (33) with (23) we get N^{(iv)}(0) = 4 ∥ a ∥^{2} N′′(0). So by Definition 2, V^{m} is of AW (3) type.

References

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[3] B.Y. Chen, Geometry of Submanifolds, Dekker 1973.

[4] B.Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, 1984.

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[6] D. Ferus, Immersions with Parallel Second Fundamental Form., Math Z. 140(1974), 87-93.

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Uludag University Faculty of Art and Sciences Department of Mathematics Campus of Görükle, 16059 BURSA-TURKEY e-mail: arslan@uu20.bim.edu.tr e-mail: cozgur@mail.balikesir.edu.tr