On Normal Sections of Stiefel Submanifold

K. Arslan and C. Özgür

Abstract

In this study we consider Stiefel submanifold V_{n,k} which is a projective mspace P^{m} (n = mk) isometrically imbedded in R^{m+ \frac{1}{2} m(m+1)} by its first standard imbedding. We also consider the normal sections of V_{n,k} and show that V_{n,k} has P 2 − P NS property.

Mathematics Subject Classification: 53C40, 53C42 Key words: Normal section, Stiefel manifold

1 Introduction

Let M be an n-dimensional submanifold in (m + d)- dimensional Euclidean space R^{n+d}. Let ∇, \overline{∇}, 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 + \overline{∇}_{X}υ, where h is the second fundamental form and A_{υ} is the shape operator of M [6]. For tangent vector fields X, Y, Z over M we define \overline{∇}h as usual by

\overline{∇}_{X}h(Y, Z) = (\overline{∇}_{X}h)(Y, Z) + h(∇_{X}Y, Z) + h(Y, ∇_{X}Z).

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}. 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 choise 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) [8]. 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.

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 7-14 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Definition 1.1. 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. \overline{∇}h = 0) or hypersurfaces [1], see also[10] and [2]. Theorem 1.1. [1] Let M be a submanifold of R^{n+d} then M has pointwise 2-planar normal sections (P 2 − P NS) if and only if

(2)  ∥h(X, X)∥^{2} (\overline{∇}_{X}h)(X, X) = ⟨(\overline{∇}_{X}h)(X, X), h(X, X)⟩h(X, X).

2 Stiefel Submanifolds V_{n,k}

For each n, k (k ≤ n) the Stiefel manifold V_{n,k} has as its points all orthonormal frames x = (e_{1},...,e_{k}) of k vectors in Euclidean n−space (i.e. ordered sequences of k orthonormal vectors in R^{n} ). Any orthogonal matrix A of degree n sends any such orthonormal frame x to another, namely Ax = (Ae_{1}, ..., Ae_{k}), this defines an action of O(n) on V_{n,k} which is transitive (see [9]).

Each Stiefel manifold V_{n,k} can be realized as a non-singular surface in the Euclidean space R^{nk} in the following way. Fix on an orthonormal basis for R^{n} (e.g. the standard basis), and introduce the following notation for the components with respect to this basis of any k-frame (e_{1},...,e_{k}) (i.e. point of V_{n,k} ):

e_{i} = (x_{i1}, ..., x_{in}) , i = 1, ..., k.

The nk quantities x_{ij} , i = 1, ..., k ; j = 1, ..., n, (in lexicographic order, say) are now to be regarded as the co-ordinates of a point in nk- dimensional Euclidean space R^{nk}, related by the following k(k + 1)/2 equations:

⟨e_{i}, e_{j}⟩ = δ_{ij} ⟺ ∑_{s=1}^{n} x_{is}x_{js} = δ_{ij} , i, j = 1, ..., k, i ≤ j

We now investigate the isotropy group of this homogenous space. Take any orthonormal k-frame e_{1},...,e_{k} and enlarge it to an orthonormal basis e_{1},...,e_{n} for the whole of Euclidean n-space. Any orthogonal transformation fixing the vectors e_{1},...,e_{k} must (relative to the above basis for R^{n}) have the form

[ \begin{matrix}1 & & 0 \\ & ... & & 0 \\ & & 1 \\ & 0 & & A \end{matrix} ] A ∈ O(n − k)

whence the isotropy group is isomorphic to O(n, k), and V_{n,k} can be identified with O(n)/O(n − k). In fact V_{n,k} ≅ O(n)/O(n − k).

The Stiefel manifolds V_{n,k} for k < n are also homogeneous spaces for the group SO(n) (see [9]). From this point of view the isotropy group is clearly (isomorphic to) SO(n − k), and therefore also

V_{n,k} ≅ SO(n)/SO(n − k).

In particular, we have

V_{n,n} ≅ O(n) , V_{n,n−1} ≅ SO(n) , V_{n,1} ≅ S^{n−1}.

In [11] Jimenez shown that; i) V_{n,k} belongs to the class of reduced Riemannian ∑-spaces, ii) if n = 2k > 30 and k = n − 2, V_{n,k} belongs to the class of all pointwise Riemannian k -symmetric spaces (k ≥ 2) and furthermore, it is not diffeomorphic to the underlying manifold of any symmetric in the class of all regular Riemannian k-symmetric spaces (k ≥ 2), and iii) if n and k are appropriately chosen V_{n,k} is not diffeomorphic to the underlying manifold of a space in class of all pointwise Riemannian r-symmetric spaces.

3 The First Standard Imbedding Of The Real Projective Space P^{m}

In this section we will consider the Stiefel submanifold as the first standard imbedding of the real projective space P^{m} (m = nk) in R^{m+ \frac{1}{2} m(m+1)}. The notation here is essentially the same as in [7].

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^{n+1} : ζ^{T} ζ = I_{k}}; m = nk,

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

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

(3)  φ((π(ζ)) = φ̃(ζ) = ζζ^{T} .

where π : S^{m} → P^{m} is a Riemannian submersion [7]. 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 trace A = 1}, m = nk, .

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

(4)  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

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

Proposition 3.1 [7]. Let Y be a vector field tangent to P^{m} and X ∈ T_{A}(P^{n}). 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 3.2. Let A(t) = ζ(t)ζ(t)^{T}; ζ(t) ∈ R^{m+1} be a curve in φ(P^{m}) then

(5)  A(0) = X = ξa^{T} + aξ^{T}

where

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

A′(t) = ζ′ζ^{T} + ζζ′^{T} .

At point t = 0 since ζ(0) = ξ, ζ′(0) = a we get the result.

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 (5) we obtain

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

Theorem 3.3 [7]. The isometric imbedding φ : S^{m} → H(m + 1; R), determined by (3), is the first standard imbedding of P^{n} into H(n + 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}(n + 1)}.

Definition 3.1. A real projective m-space P^{m} (m = nk) isometrically imbedded in R^{m+ \frac{1}{2} m(m+1)} by its first standard imbedding φ determined by (3) is called the Stiefel submanifold V_{n,k}.

4 Normal Sections Of V_{n,k}

An element A ∈ V_{n,k} can be written as A = ζζ^{T} where ζ^{T} ζ = I_{k×k} and ζ = ξm + η, ξ^{T} η = 0, m ∈ M_{k×k}(R). Since ζ is not unique we can replace ζ by ζu where u ∈ O(k). Put m = ξ^{T} ζ then we can find u ∈ O(k) so that mu is lower triangular and the diagonal element of mu are ≥ 0 by using the Gram-Schmidt orthonormalization process. Further mu is uniquely defined by m and ξ^{T} (ζ − ξm) = ξ^{T} ζ − m = 0. Thus we can write A = ζζ^{T} where ζ = ξm + η where m is lower triangular k × k with diagonal term ≥ 0 and ξ^{T} η = 0, m ∈ M_{k×k}(R) where m and η are unique. So A = ξmm^{T} ξ^{T} +ξmη^{T} +ηm^{T} ξ^{T} +ηη^{T} . Interchanging A with the curve A(t) in φ(P^{m}) we get

(6) A(t) = ζ(t)ζ(t)^{T} = ξm(t)m(t)^{T} ξ^{T} + ξm(t)η(t)^{T} + η(t)m(t)^{T} ξ^{T} + η(t)η(t)^{T} .

Lemma 4.1. Let A(t) be a normal section of V_{n,k} at point A(0) in the direction of A′(0) = X. Then

(7)  λ(t)X = ξm(t)η(t)^{T} + η(t)m(t)^{T} ξ^{T}
(8)  N(t) = ξm(t)m(t)^{T} ξ^{T} + η(t)η(t)^{T}

where aξ^{T} + ξa^{T} = X, ξ^{T} ξ = I_{k} and a^{T} ξ + ξ^{T} a = 0, a = η′(0) ∈ M_{n×k}. Proof. Since ξ^{T} η(t) = 0 then m(t)η(t)^{T} ξ + ξ^{T} η(t)m(t)^{T} = 0 therefore ξm(t)η(t)^{T} + η(t)m(t)^{T} ξ^{T} ∈ T_{x}(V_{n,k}). Also η(t)η(t)^{T} ∈ N_{x}(V_{n,k}) since trace(a^{T} η(t)η(t)^{T} ξ) = 0 (whatever a ∈ M_{n×k}). Also

a^{T} (ξm(t)m(t)^{T} ξ^{T})ξ = trace(a^{T} ξm(t)m(t)^{T}) = trace(m(t)m(t)^{T} ξ^{T} a)
= \frac{1}{2}trace(a^{T} ξ + ξ^{T} a)m(t)m(t)^{T} = 0

if a^{T} ξ + ξ^{T} a = 0. Therefore ξm(t)m(t)^{T} ξ^{T} ∈ N_{x}(V_{n,k}). Hence comparing (1) with (6) we get the result. Lemma 4.2. Let A(t) = ζ(t)ζ(t)^{T} be a normal section of V_{n,k} at point A(0) in the direction of A′(0) = X. Then

(9)  m(t)^{T} m(t) + η(t)^{T} η(t) = I_{k}.

Proof. Since ζ(t) = ξm(t) + η(t),

(10)  ζ(t)^{T} ζ(t) = I_{k},

and

(11)  ξ^{T} η(t) = 0, η ∈ M_{n×k}

we get the result. Theorem 4.3. Stiefel submanifold V_{n,k} (considered as a projective m-space P^{m} (n = mk) isometrically imbedded in R^{m+ \frac{1}{2} m(m+1)} by its first standard imbedding defined by (3)) has P 2 − P NS property. Proof. The normal section A(t) of V_{n,k} at point A(0) = ξξ^{T} in the direction of A′(0) = X is given by A(t) = ζ(t)ζ^{T} (t), where ζ(t) = ξm(t) + η(t), η, m ∈ M_{n×k},

(12)  m(t) = ξ^{T} ζ(t)

and

(13)  η(t)m(t)^{T} = λ(t)a.

At point t = 0 we get

(14)  \begin{matrix}ζ(0) = ξ \\ m(0) = I_{k} \\ η(0) = 0 \\ A(0) = ξξ^{T} \end{matrix} \}

at least for sufficiently small A(t). Thus we now differentiate (9)-(13) and so determine m′, η′, λ′, m′′, η′′, λ′′. Then using N(t) = ξm(t)m(t)^{T} ξ^{T} + η(t)η(t)^{T} we compute N′, N′′, etc.

Differentiating (9), (12) and (11) (surprising the dependence of ζ on t to simplify the notation) we get

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

(16)  ζ′(t)^{T} ζ(t) + ζ(t)^{T} ζ′(t) = 0,

(17)  m′(t) = ξ^{T} ζ′(t)

Putting t = 0 we get

(18)  A′(0) = X = ξa^{T} + aξ^{T}

(19)  ζ(0) = ξ, ζ′(t) = η′(0) = a,

(20)  m′(0) = ξ^{T} a = 0.

Differentiating (13) two times with respect to t we also get

η′′(t)m(t)^{T} + 2η′(t)m′(t)^{T} + η(t)m′′(t)^{T} = λ′′(t)a.

At t = 0 by the use of (14) and (20) the above equation gives

(21)  η′′(0) = 0.

Differentiating (9) and (8) we get

(22)  \begin{aligned}0 & = & m′′(t)^{T} m(t) + 2m′(t)^{T} m′(t) + m(t)m′′(t)^{T} + \\ & + & η′′(t)^{T} η(t) + 2η′(t)^{T} η′(t) + η(t)η′′(t)^{T} \end{aligned}

and

(23)  \begin{aligned}N′′(t) & = & ξm′′(t)m(t)^{T} ξ^{T} + 2ξm′(t)m′(t)^{T} ξ^{T} + ξm(t)m′′(t)^{T} ξ^{T} \\ & + & η′′(t)η(t)^{T} + 2η′(t)η′(t)^{T} + η(t)η′′(t)^{T} . \end{aligned}

At t = 0 by (14), (20) and (21) we have

m′′(0)^{T} + m′′(0) = −2a^{T} a

and

(24)  N′′(0) = h(X, X) = −2ξa^{T} aξ^{T} + 2aa^{T} .

Differentiating (22) and (23) with respect to t we get

(25)  \begin{aligned}0 & = & m′′′(t)^{T} m(t) + 3m′′(t)^{T} m′(t) + 3m′(t)^{T} m′′(t) + m^{T} (t)m′′′(t)+ \\ & + & η′′′(t)^{T} η(t) + 3η′′(t)^{T} η′(t) + 3η′(t)^{T} η′′(t) + η^{T} (t)η′′′(t) \end{aligned}

and

(26)  \begin{aligned}N′′′(t) & = & ξm′′′(t)m(t)^{T} ξ^{T} + 3ξm′′(t)m′(t)^{T} ξ^{T} + 3ξm′(t)m′′(t)^{T} ξ^{T} \\ & + & ξm(t)m′′′(t)^{T} ξ^{T} + η′′′(t)η(t)^{T} + 3η′′(t)η′(t)^{T} \\ & + & 3η′(t)η′′(t)^{T} + η(t)η′′′(t)^{T} . \end{aligned}

At t = 0, substituting (14), (20) and (21) into (25) and (26) one can get respectively

m′′′(0)^{T} + m′′′(0) = 0,

and

N′′′(0) = (\overline{∇}_{X}h)(X, X) = ξm′′′(0)ξ^{T} + ξm′′′(0)^{T} ξ^{T} = 0.

This means that V_{n,k} has parallel second fundamental form. Therefore by Theorem 1.1 M has P 2-P NS property. Remark. For k = 1 the Stiefel submanifold V_{n,1} becomes a Veronese submanifold. In [3] we have shown that V_{n,1} has P 2 − P NS property. Acknowledgements. This research is supported by Uludağ University research fund.

References

[1] K. Arslan and A. West, Non-Spherical Submanifolds with Pointwise 2-Planar Normal Sections, Bulletin London Math. Soc.28, 1996, 88-92.

[2] K. Arslan and A. West, Product Submanifolds with Pointwise 3-Planar Normal Sections, Glasgow Math. J. 37 (1995), 73-81.

[3] K. Arslan and C. Özgür, On Normal Sections of Veronese Submanifold, Balkan J. Geo. and its Apl. 4 (1999), 1-8.

[4] K. Arslan, Y. Çelik and C. Özgür, Isoparametric Submanifolds with P2-PNS, Far. East J. Math. Sci. 2 (1996), 269-274.

[5] K. Arslan, Y. Çelik and C. Özgür, Submanifolds of AW(k) type, Banyan Math J. 3(1996), 33-39.

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

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

[8] B.Y. Chen, Submanifolds with Planar Normal Sections, Soochow J.Math.7, 1981, 19-24.

[9] B. A. Dubrovin, A.T. Fomenko and S.P. Nonikov, Modern Geometry Methods and Appliations, Springer-Verlag, 1984.

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

[11] J. Jimenez, A. Stiefel manifolds and the existence of non-trivial generalizations of Riemannian symmetric spaces from a differentiable viewpoint, Diff. Geo. and its Appl. 1(1991), 47-55.

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