DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 215–222

ON THE SPECIAL CLASS OF CURVES ON SOME FOUR-DIMENSIONAL SEMIPARALLEL SUBMANIFOLDS

KAARIN RIIVES

Abstract. A submanifold M^{m} in the Euclidean space E^{n} is said to be parallel if it satisfies the property ¯∇h = 0, where h is the second fundamental form and ¯∇ denotes the van der Waerden-Bortolotti connection. A submanifold is said to be semiparallel if it satisfies the property R̄ ∘ h = 0, where R̄ denotes the curvature operator of the connection. It has been proved by Lumiste [11] that the semiparallel submanifolds are the second order envelopes of the parallel ones. Using the Cartan moving frame method and the exterior differential calculus the paper describes some special classes of curves on irreducible envelopes of the reducible symmetric submanifolds V^{2}(r_{1}) × S^{1}(r_{2}) × S^{1}(r_{3}) with a Veronese component, which is a Veronese surface in E^{5}.

0. Introduction

Let us examine a submanifold M^{m} in Euclidean space E^{n} by the Cartan moving frame method with the help of exterior differential calculus (see [6], Ch I §1). Then in the formulae of infinitesimal displacement

(0.1)  dx⃗ = ω^{I}e⃗_{I}, de⃗_{I} = ω_{I}^{K}e⃗_{K}

of an orthonormal frame bundle O(E^{n}) = {x, e⃗_{I}(I = 1, …, n)}, x ∈ M^{m}, we have ω_{I}^{K} + ω_{K}^{I} = 0, where I, K = 1, …, n and in addition the structure equations

(0.2)  dω^{I} = ω^{K} ∧ ω_{K}^{I}, dω_{I}^{K} = ω_{I}^{L} ∧ ω_{L}^{K} .

If the bundle O(E^{n}) is reduced to

O(M^{m}, E^{n}) = {x, e⃗_{i} (i = 1, …, m), e⃗_{α} (α = m + 1, …, n)}

at every point x ∈ M^{m}, then in (0.1)

ω^{α} = 0

1991 Mathematics Subject Classification. 53B25, 53C40. Key words and phrases. Curves on semiparallel submanifolds, parallel submanifolds, semi-

parallel submanifolds, Veronese surfaces.

This paper is in final form and no version of it will be submitted for publication elsewhere.

and the exterior differentiation of this system leads by means of (0.2) and the Cartan lemma to

ω_{i}^{α} = h_{ij}^{α}ω^{j}, h_{ij}^{α} = h_{ji}^{α} .

These determine the second fundamental form with values in the normal vector bundle T^{⊥}M^{m} of the manifold

h : (e⃗_{i}X^{i}, e⃗_{j}Y^{j}) ↦ e⃗_{α}h_{ij}^{α}X^{i}Y^{j} .

The next step of exterior differentiation leads to the third fundamental form

¯∇h : (e⃗_{i}X^{i}, e⃗_{j}Y^{j}, e⃗_{k}Z^{k}) ↦ e⃗_{α}h_{ijk}^{α}X^{i}Y^{j}Z^{k} ,

whose components are given by

¯∇h_{ij}^{α} = ¯∇_{k}h_{ij}^{α}ω^{k} = h_{ijk}^{α}ω^{k} .

Here the coefficients h_{ijk}^{α} are symmetric with respect to the tangent indices due to the formulae of Peterson-Mainardi-Codazzi ¯∇_{k}h_{ij}^{α} = ¯∇_{j}h_{ik}^{α}. Continuing by exterior differentiation we get the fourth fundamental form

¯∇( ¯∇h) ≡ ¯∇^{2}h : (e⃗_{i}X^{i}, e⃗_{j}Y^{j}, e⃗_{k}Z^{k}, e⃗_{l}W^{l}) ↦ e⃗_{α}h_{ijkl}^{α}X^{i}Y^{j}Z^{k}W^{l} ,

whose components are given by

¯∇h_{ijk}^{α} = ¯∇( ¯∇_{k}h_{ij}^{α}) = ¯∇_{l}( ¯∇_{k}h_{ij}^{α})ω^{l} = h_{ijkl}^{α}ω^{l} .

Here the components h_{ijkl}^{α} need not to be symmetric with respect to the lower indices. Continuing analogously with exterior differentiation we obtain the set of fundamental forms ¯∇^{s}h (s ≥ 0). The submanifold M^{m} in Euclidean space (or even more generally, in space form N^{n}(c)) is said to be s-parallel if ¯∇^{s}h = 0, ¯∇^{s−1}h ≠ 0 [13]. In other words, an s-parallel submanifold has a parallel fundamental form of order s + 1 and a nonparallel fundamental form of order s. Thus 0-parallel submanifolds are totally geodesic, 1-parallel submanifolds are the wellknown parallel ones, which have been completely described by Ferus [5] in terms of Lie algebras. The parallel submanifolds are analytically determined by the system ¯∇h = 0 of differential equations whose integrability condition is R̄ ∘ h = 0. Here the components of R̄ ∘ h are given by

¯∇( ¯∇_{k}h_{ij}^{α}) ∧ ω^{k} = −h_{kj}^{α}Ω_{i}^{k} − h_{ik}^{α}Ω_{j}^{k} + h_{ij}^{β}Ω_{β}^{α} ,

ON THE SPECIAL CLASS OF CURVES … 217

where Ω_{i}^{k} = ω_{i}^{α} ∧ ω_{α}^{k} and Ω_{β}^{α} = ω_{β}^{k} ∧ ω_{k}^{α} are the curvature forms of Levi-Civita and the normal connections, respectively. If R̄ ∘ h = 0, the submanifold M^{m} is said to be semiparallel [1], [2]. This condition leads to the symmetry of coefficients h_{ijkl}^{α} with respect to the lower indices. It was proved by Lumiste [11] that the semiparallel submanifolds are the second order envelopes of the parallel ones. It is proved in [7] that the system ¯∇^{2}h = 0 of differential equations determining 2parallel submanifolds implies R̄ ∘ h = 0. Thus a submanifold with a parallel third fundamental form is always a semiparallel one. This means that the classification of semiparallel submanifolds describes the connections between different classes of submanifolds. The classification has not yet been completed even for M^{m} in Euclidean spaces E^{n}. Surveys of results have been given by Deprez [3] up to 1989 and by Lumiste [13] up to 1991. The classification problem has been solved for m = 2 [1], 3 [10,15] (partly together with the present author), n−2 [9], n−1 [2]. All normally flat semiparallel M^{m} have been described by Lumiste [12] in E^{n} and by Dillen and Nölker [4] in the space forms N^{n}(C). Normally non-flat reducible cases for m = 4 have been described in [14,15] except the case in which M^{4} is a second order envelope of the reducible symmetric submanifold V^{2}(r_{1}) × S^{1}(r_{2}) × S^{1}(r_{3}) with a Veronese component, which is a Veronese surface in E^{5}, and its special case V^{2}(r_{1}) × S^{1}(r_{2}) × E^{1} (r_{3} = ∞). For the most general case r_{2} < ∞, r_{3} < ∞ of the submanifolds M^{4} mentioned above, the following results have been proved [16].

Proposition 1. The tangent distributions T V^{2}(r_{1}) and T S^{1}(r_{2}) × T S^{1}(r_{3}) on M^{4} determine foliations.

The integral submanifolds of the foliations are Veronese leaves M_{V er}^{2} and Clifford leaves M_{Cliff}^{2}, respectively. The Veronese leaves M_{V er}^{2} are the most general semiparallel surfaces with non-flat Levi-Civita and non-flat normal connections. The Clifford leaves M_{Cliff}^{2} are surfaces with flat van der Waerden-Bortolotti connection. In this paper the investigations will be continued by a description of special classes of curves on the submanifolds M^{4} under consideration.

1. Preliminaries and apparatus

Let a semiparallel submanifold M^{4} in Euclidean space E^{n} be a second order envelope of the normally non-flat reducible symmetric submanifold V^{2}(r_{1})×S^{1}(r_{2})× S^{1}(r_{3}) with a Veronese component V^{2}(r_{1}), which is a Veronese surface in E^{5}. Let the reduced orthonormal frame bundle O(M^{4}, E^{n}) at every point x ∈ M^{4} be specialized so that e⃗_{1}, e⃗_{2} belong to T_{x}V^{2}(r_{1}), e⃗_{3} to T_{x}S^{1}(r_{2}) and e⃗_{4} to T_{x}S^{1}(r_{3}). Further, let e⃗_{5}, e⃗_{6}, e⃗_{7} determine the first normal space N_{x}^{1}V^{2}(r_{1}) and e⃗_{8}, e⃗_{9} the normal spaces N_{x}^{1}S^{1}(r_{2}) and N_{x}^{1}S^{1}(r_{3}), respectively. Let the rest frame vectors be denoted by e⃗_{ξ}, ξ = 10, …, n. In the formulae of the infinitesimal displacement of such a subbundle O(M^{4}, E^{n}),

(1)  dx⃗ = ω^{I}e⃗_{I} , de⃗_{I} = ω_{I}^{K}e⃗_{K} ,

with ω_{I}^{K} + ω_{K}^{I} = 0, where I, K, ⋯ = 1, …, n, and also in the structure equations,

(2)  dω^{I} = ω^{K} ∧ ω_{K}^{I} , dω_{K}^{I} = ω_{K}^{L} ∧ ω_{L}^{I} ,

we have

(3)  ω^{5} = ω^{6} = ω^{7} = ω^{8} = ω^{9} = ω^{ξ} = 0 ,
(4)  ω_{1}^{ξ} = ω_{2}^{ξ} = ω_{3}^{ξ} = ω_{4}^{ξ} = 0, ξ = 10, …, n .

Due to the product structure of the symmetric submanifold V^{2}(r_{1}) × S^{1}(r_{2}) × S^{1}(r_{3}), which by assumption has second order tangency with M^{4} at every point x ∈ M^{4}, the following equations are satisfied:

ω_{1}^{8} = ω_{2}^{8} = ω_{1}^{9} = ω_{2}^{9} = 0 , ω_{3}^{5} = ω_{3}^{6} = ω_{3}^{7} = ω_{3}^{9} = 0 ,

(5)  ω_{4}^{5} = ω_{4}^{6} = ω_{4}^{7} = ω_{4}^{8} = 0 .

A further adaption of the frame part {e⃗_{1}, e⃗_{2}, e⃗_{5}, e⃗_{6}, e⃗_{7}} to V^{2}(r_{1}), as has been done in [15], leads to

ω_{1}^{5} = ρ_{1}\sqrt{3}ω^{1} , ω_{1}^{6} = ρ_{1}ω^{1} , ω_{1}^{7} = ρ_{1}ω^{2} ,
(6)  ω_{2}^{5} = ρ_{1}\sqrt{3}ω^{2} , ω_{2}^{6} = −ρ_{1}ω^{2} , ω_{2}^{7} = ρ_{1}ω^{1} ,
ω_{3}^{8} = ρ_{2}ω^{3} , ω_{4}^{9} = ρ_{3}ω^{4} ,

where ρ_{1} = r_{1}^{−1}, ρ_{2} = r_{2}^{−1}, ρ_{3} = r_{3}^{−1}. Let us suppose that ρ_{2} and ρ_{3} do not vanish, i.e. the circles S^{1}(r_{2}), S^{1}(r_{3}) do not become straight lines. The semiparallel submanifold M^{4} under consideration is determined by the system (3)-(6) of Pfaff equations.

Due to the structure equations (2) and the Cartan lemma, exterior differentiation of the equations of the system (3)-(6) leads to the system

ω_{1}^{3} = ρ_{1}Cω^{1} , ω_{2}^{3} = ρ_{1}Cω^{2} , ω_{5}^{8} = −(\sqrt{3})^{−1}ρ_{1}Cω^{3} , ω_{6}^{8} = ω_{7}^{8} = 0 ,
(7) ω_{1}^{4} = ρ_{2}Dω^{1} , ω_{2}^{4} = ρ_{2}Dω^{2} , ω_{5}^{9} = −(\sqrt{3})^{−1}ρ_{2}Dω^{3} , ω_{6}^{9} = ω_{7}^{9} = 0 ,

and the following consequences:

ω_{5}^{ξ} = A_{1}^{ξ}ω^{1} + A_{2}^{ξ}ω^{2} , ω_{6}^{ξ} = A_{3}^{ξ}ω^{1} + A_{4}^{ξ}ω^{2} ,
ω_{7}^{ξ} = (\sqrt{3}A_{2}^{ξ} + A_{4}^{ξ})ω^{1} + (\sqrt{3}A_{1}^{ξ} − A_{3}^{ξ})ω^{2} ,
ω_{8}^{ξ} = B^{ξ}ω^{3} , ω_{9}^{ξ} = C^{ξ}ω^{4} ,
ω_{3}^{4} = ρ_{2}E_{1}ω^{3} − ρ_{3}E_{2}ω^{4} , ω_{8}^{9} = ρ_{2}E_{2}ω^{3} − ρ_{3}E_{1}ω^{4} ,
(8)  ω_{5}^{6} = \sqrt{3}λ_{1}ω^{1} + \sqrt{3}λ_{2}ω^{2} , ω_{5}^{7} = −\sqrt{3}λ_{2}ω^{1} + \sqrt{3}λ_{1}ω^{2} ,
ω_{6}^{7} = 2ω_{1}^{2} − 5λ_{2}ω^{1} − 5λ_{1}ω^{2} ,
dlnρ_{1} = −2λ_{1}ω^{1} + 2λ_{2}ω^{2} + ρ_{1}Cω^{3} + ρ_{1}Dω^{4} ,
dρ_{2} = γ_{3}ω^{3} + ρ_{2}^{2}E_{1}ω^{4} , dρ_{3} = ρ_{3}^{2}E_{2}ω^{3} + γ_{4}ω^{4} .

After the exterior differentiation of (7) there arise, in particular, the expressions

dC = 2λ_{1}Cω^{1} − 2λ_{2}Cω^{2} + ρ_{2}DE_{1}ω^{3} − ρ_{3}DE_{2}ω^{4} ,
(9)  dD = 2λ_{1}Dω^{1} − 2λ_{2}Dω^{2} − ρ_{2}CE_{1}ω^{3} + ρ_{3}CE_{2}ω^{4} .

If we denote

(10) ω̂^{3} = Cω^{3} + Dω^{4} , ω̂^{12} = λ_{1}ω^{1} − λ_{2}ω^{2} , ω̂^{34} = ρ_{2}E_{1}ω^{3} − ρ_{3}E_{2}ω^{4} ,

then it follows from (9) that

(11)  dC = 2Cω̂^{12} + Dω̂^{34} , dD = 2Dω̂^{12} − Cω̂^{34} ,

and the exterior differentiation of (10), in conjunction with the structure equations (2), leads to

dω̂^{3} = 2ω̂^{12} ∧ ω̂^{3} , dω̂^{12} = 0 , dω̂^{34} = 0 .

2. Results

As by the Proposition 1 the tangent distributions T V^{2}(r_{1}) and T S^{1}(r_{2}) × T S^{1}(r_{3}) on M^{4} determine foliations, then let us denote the corresponding integral submanifolds by M_{V er}^{2} = M^{4}(mod ω^{3}, ω^{4}) and M_{Cliff}^{2} = M^{4}(mod ω^{1}, ω^{2}). Here M_{V er}^{2} is the 2nd order envelope of Veronese surfaces (Veronese leaf) and M_{Cliff}^{2} is the 2nd order envelope of the Clifford tori (Clifford leaf).

Let us introduce the function γ^{2} = C^{2} + D^{2}, which shall be called the function of reducibility as its values describe the reducibility properties of the submanifolds M^{4} under consideration [16,17]. From the decomposition theorems [7,8] and (7) there follows

Proposition 2. The second order envelope M^{4} of the reducible symmetric submanifold V^{2}(r_{1}) × S^{1}(r_{2}) × S^{1}(r_{3}) is irreducible iff neither of the parameters C and D vanish.

The orthogonal vectors ε⃗_{3} = Ce⃗_{3} +De⃗_{4} and ε⃗_{4} = −De⃗_{3} +Ce⃗_{4} with |ε⃗_{3}| = |ε⃗_{4}| = γ^{2} ≠ 0 determine an orthogonal net of curves on the Clifford leaf M_{Cliff}^{2}. Let us denote the curves with tangent vectors ε⃗_{3} by s(I) and the curves with tangent vectors ε⃗_{4} by s(II), respectively. By (1) and (11) we have

dε⃗_{3} = 2ω̂^{12}ε⃗_{3} − ρ_{1}γ^{2}(ω^{1}e⃗_{1} + ω^{2}e⃗_{2}) + Cρ_{2}ω^{3}e⃗_{8} + Dρ_{3}ω^{4}e⃗_{9},
(12)  dε⃗_{4} = 2ω̂^{12}ε⃗_{4} − ρ_{2}Dω^{3}e⃗_{8} + ρ_{3}Cω^{4}e⃗_{9},

Proposition 3. The first curvatures of the curves s(I) and s(II) are k(I) = γ^{−1}\sqrt{ρ_{2}^{2}C^{4} + ρ_{3}^{2}D^{4}} and k(II) = γ^{−1}\sqrt{ρ_{2}^{2}D^{4} + ρ_{3}^{2}C^{4}}, respectively. Their vectors of normal curvature have a nonzero component in the direction of one of the normal vectors of M_{V er}^{2} and the components in the direction of e⃗_{ξ} can be non-zero if at least one of the vectors b⃗ = B^{ξ}e⃗_{ξ}, c⃗ = C^{ξ}e⃗_{ξ} does not vanish.

Proof. Let us transform the part e⃗_{3}, e⃗_{4} of the reduced orthonormal frame bundle O(M^{4}, E^{n}) by the rotation

ε⃗_{30} = e⃗_{3} cos α + e⃗_{4} sin α , ε⃗_{40} = −e⃗_{3} sin α + e⃗_{4} cos α ,

or

e⃗_{3} = ε⃗_{30} cos α − ε⃗_{40} sin α , e⃗_{4} = ε⃗_{30} sin α + ε⃗_{40} cos α ,

where cos α = γ^{−1}C, sin α = γ^{−1}D, α ≠ 0, α ≠ \frac{π}{2} on the Clifford leaves M_{Cliff}^{2} under consideration. Then the first formula of an infinitesimal displacement of the described subbundle O(M^{4}, E^{n}) along the curve s(I) is as follows:

dx⃗ = ω^{30}ε⃗_{30} = ω^{30}γ^{−1}Ce⃗_{3} + ω^{30}γ^{−1}De⃗_{4} .

From here it is obvious that

ω^{3} = γ^{−1}Cω^{30} = ω^{30} cos α , ω^{4} = γ^{−1}Dω^{30} = ω^{30} sin α .

Then from (12) we get

dε⃗_{30} = ω^{30}(ρ_{2} cos^{2} αe⃗_{8} + ρ_{3} sin^{2} αe⃗_{9})

and denoting

ε⃗_{31} = ρ_{2} cos^{2} αe⃗_{8} + ρ_{3} sin^{2} αe⃗_{9}

we have

|ε⃗_{31}|^{2} = ρ_{2}^{2} cos^{4} α + ρ_{3}^{2} sin^{4} α .

Now due to

dε⃗_{30} = ω^{30}k(I)ε⃗_{310} = ω^{30}|ε⃗_{31}|ε⃗_{310}

it follows that k(I) = |ε⃗_{31}| > 0. In particular,

k(I) = \sqrt{ρ_{2}^{2} cos^{4} α + ρ_{3}^{2} sin^{4} α} = γ^{−1}\sqrt{ρ_{2}^{2}C^{4} + ρ_{3}^{2}D^{4}} .

Analogously the formulae of infinitesimal displacement of the bundle O(M^{4}, E^{n}) along the curve s(II) on the Clifford leaf are, due to (12), as follows:

dx⃗ = ω̂^{4}ε⃗_{40} = −ω̂^{4}γ^{−1}De⃗_{3} + ω̂^{4}γ^{−1}Ce⃗_{4} = ω̂^{4}(− sin αe⃗_{3} + cos αe⃗_{4}) ,
dε⃗_{40} = ω̂^{4}(ρ_{2} sin^{2} αe⃗_{8} + ρ_{3} cos^{2} αe⃗_{9}) .

Recall that here ε⃗_{4} = −De⃗_{3} + Ce⃗_{4} = γε⃗_{40}. Denoting

ε⃗_{41} = ρ_{2} sin^{2} αe⃗_{8} + ρ_{3} cos^{2} αe⃗_{9} = |ε⃗_{41}|ε⃗_{410}

we have

|ε⃗_{41}|^{2} = ρ_{2}^{2} sin^{4} α + ρ_{3}^{2} cos^{4} α .

Now from

dε⃗_{40} = ω̂^{4}k(II)ε⃗_{410} = ω̂^{4}|ε_{41}|ε⃗_{410}

it follows that k(II) = |ε_{41}| > 0, and in particular,

k(II) = \sqrt{ρ_{2}^{2} sin^{4} α + ρ_{3}^{2} cos^{4} α} =
= γ^{−1}\sqrt{ρ_{2}^{2}D^{2} + ρ_{3}^{2}C^{2}} .

Description of the second normal spaces of considered curves are given by the differentials dε⃗_{310} and dε⃗_{410}, respectively:

dε⃗_{310} = −k(I)ω^{30}ε⃗_{30} + ω^{30}(κ_{I}^{40}ε⃗_{40} + κ_{I}^{5}e⃗_{5} + κ_{I}^{8}e⃗_{8} + κ_{I}^{9}e⃗_{9} + κ_{I}^{ξ}e⃗_{ξ}) ,
dε⃗_{410} = −k(II)ω̂^{4}ε⃗_{40} + ω̂^{4}(κ_{II}^{30}ε⃗_{30} + κ_{II}^{5}e⃗_{5} + κ_{II}^{8}e⃗_{8} + κ_{II}^{9}e⃗_{9} + κ_{II}^{ξ}e⃗_{ξ}) .

Calculations, which have been done with the help of (1), (8)-(11), lead to

κ_{I}^{40} = \frac{cos α · sin α(ρ_{2}^{2} cos^{2} α − ρ_{3}^{2} sin^{2} α)}{k(I)} ,

κ_{I}^{5} = \frac{γ}{\sqrt{3}}k(I) ≠ 0 ,

κ_{I}^{ξ}e⃗_{ξ} = \frac{1}{k(I)}(ρ_{2} cos^{3} αb⃗ + ρ_{3} sin^{3} αc⃗) = \frac{1}{k(I)}(B^{ξ}ρ_{2} cos^{3} α + C^{ξ}ρ_{3} cos^{3} α)e⃗_{ξ} ,

κ_{II}^{30} = \frac{cos α · sin α(ρ_{2}^{2} sin^{2} α − ρ_{3}^{2} cos^{2} α)}{k(II)} ,

κ_{II}^{5} = −\frac{γ}{\sqrt{3}k(II)} cos α · sin α(ρ_{2}^{2} sin^{2} α − ρ_{3}^{2} cos^{2} α) = −\frac{γ}{\sqrt{3}}k_{II}^{30} ,

κ_{II}^{ξ}e⃗_{ξ} = \frac{1}{k(II)}(−ρ_{2} sin^{3} αb⃗+ρ_{3} cos^{3} αc⃗) = −\frac{1}{k(II)}(−B^{ξ}ρ_{2} sin^{3} α +C^{ξ}ρ_{3} cos^{3} α)e⃗_{ξ} .

These equalities prove the second assertion of the proposition. The reason being that the normal curvature vector s(I) has a nonzero component k_{I}^{5} in the direction of the normal e⃗_{5} of the Veronese leaf. The non-vanishing or otherwise of the corresponding quantity for s(II) is then of no consequence for this proof. The components of these normal curvature vectors in the direction of e⃗_{ξ} are given by coefficients k_{I}^{ξ} and k_{II}^{ξ}, which depend linearly on the parameters B^{ξ} and C^{ξ} and can vanish only if the vectors b⃗ and c⃗ vanish simultaneously.

Remark. The irreducibilty of the semiparallel submanifold M^{4} under consideration is described by the coefficients k_{I}^{5} and k_{II}^{5} (see the Proposition 2).

References

[1] Deprez, J., Semiparallel surfaces in Euclidean space, J.Geom. 25(1985) 192-200.

[2] Deprez, J., Semiparallel hypersurfaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986) 303-316.

[3] Deprez, J., Semiparallel immersions, Geom. and Topol. of Submanifolds I, Singapore et al., 1989, 73-88.

[4] Dillen, F., Nölker, S., Semi-parallelity, multi-rotation surfaces and the helix-property, J. reine angew. Math. 453 (1993) 33-63.

[5] Ferus, D., Symmetric submanifolds of Euclidean spaces, Math. Ann. 247 (1980) 81-93.

[6] Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, Vol II, Interscience Publishers, New York-London-Sydney, 1969.

[7] Lumiste, Ü., Decomposition and classification theorems for semisymmetric immersions, Proc. Estonian Acad. Sci. Phys. Math. 36 (1987), 414-417.

[8] Lumiste, Ü., Decomposition of semisymmetric submanifolds, Tartu Ülik. Toim. Acta et Comm. Univ. Tartu 803(1988) 69-78.

[9] Lumiste, Ü., Classification of two-codimensional semisymmetric submanifolds, Tartu Ülik. Toim. Acta et Comm. Univ. Tartu 803(1988) 79-94.

[10] Lumiste, Ü., Classification of three-dimensional semisymmetric submanifolds in Euclidean spaces, Tartu Ülik. Toim. Acta et Comm. Univ. Tartu 899(1990) 29-44.

[11] Lumiste, Ü., Semisymmetric submanifold as a second order envelope of symmetric submanifolds, Proc. Estonian Acad. Sci. Phys. Math. 39 (1990) 1-8.

[12] Lumiste, Ü., Semisymmetric envelopes of some symmetric cylindrical submanifolds, Proc. Estonian Acad. Sci. Phys. Math. 40(1991) 245-257.

[13] Lumiste, Ü., Semisymmetric submanifolds, Itogi nauki i tekhn. Probl. geom. 23 (1991) 3-28 (Russian).

[14] Lumiste, Ü., Semiparallel submanifolds of cylindrical or toroidal Segre type, Proc. Estonian Acad. Sci. Phys. Math. 45 (1996), 161-176.

[15] Lumiste, Ü., Riives, K., Semisymmetric envelopes of some four-dimensional reducible symmetric submanifolds, Trans. Tallinn Techn. Univ. Math. Phys. 1(1992) 49-58.

[16] Riives, K., On a class of four-dimensional semiparallel submanifolds in Euclidean spaces, Proceedings of the 4th International Congress of Geometry. Edited by N. K. Stephanidis and N. K. Artemiadis. Thessaloniki (1996), 351-357.

[17] Riives, K., On the function of reducibility of a class of four-dimensional semiparallel submanifolds, Proc. Estonian Acad. Sci. Phys. Math. (to appear).

Department of Pure Mathematics, University of Tartu Vanemuise 46-114, Tartu 51 014, ESTONIA E-mail: riives@math.ut.ee