Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 37 (1996), No. 1, 75-86.

Immersions into Spheres with Parallel Higher Fundamental Forms

Antonella Carfagna D’Andrea

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma “La Sapienza”, Via A. Scarpa, 16, 00161 Roma

Abstract. In this note we prove helical minimal immersions of a compact rank one symmetric space into spheres are 2-symmetric submanifolds.

We give conditions under which a nicely-curved 2-symmetric submanifold of a space of constant curvature c ≥ 0 is a helical immersion and vice versa.

MSC 1991: 53C42, 53A07

Introduction

The notion of k-symmetric submanifold of the Euclidean space was given by Kowalski and Külich [5]. In the case k = 2, 2-symmetric submanifolds are a generalization of the symmetric submanifolds of the Euclidean space introduced and classified by Ferus [2].

In [1] a local extension of the notion of 2-symmetric submanifolds is given considering immersions into a space of constant curvature. Moreover it is proved that a nicely curved submanifold [see definition 1.2] of a space of constant curvature c ≥ 0 is a 2-symmetric

submanifold if and only if all higher fundamental forms S^{k} are parallel, i.e., ∇^{k+1} S^{k} = 0 where

∇^{k+1} is the derivative of S^{k} as a multilinear form with values in the (k +1)-th normal bundle.

In this note we prove that the minimal immersions of a compact rank one symmetric space into a unit sphere S_{1}^{n} with higher fundamental forms isotropic [10] have all higher fundamental forms parallel, so they are 2-symmetric submanifolds of S_{1}^{n}. As a consequence helical minimal geodesic immersions into spheres (see for the definition [7]) and standard minimal immersions into spheres (see [3]) are examples of 2-symmetric submanifolds.

In general, when the immersions are not minimal, we give conditions under which a nicely curved 2-symmetric submanifold of a space of constant curvature c ≥ 0 is a helical geodesic immersion and vice versa, applying the Frenet theory of immersed curves in Riemannian manifolds.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1996 Heldermann Verlag, Berlin

76 A. Carfagna D’Andrea: Immersions into Spheres with Parallel …

In section 1 the notions of the higher fundamental forms, and some results about the isotropic fundamental forms are recalled.

Section 2 deals with the minimal immersions of a compact rank one symmetric space into a unit sphere and helical geodesic immersions.

Sections 3 and 4 are devoted to the study of the relations between the isotropic minimal immersions, helical geodesic immersions and 2-symmetric submanifolds.

1. Riemannian submanifolds and higher fundamental forms

Let: φ : M → M̃ be an isometric immersion of a m-dimensional Riemannian manifold M into a n-dimensional Riemannian manifold M̃ of constant curvature.

For any p ∈ M denote by T_{p}M the tangent space of M at p and by N_{p}M the normal space of the immersion φ at p.

In a small neighbourhood U of any p ∈ M, φ is an imbedding, thus locally we can identify a vector field Y on M with its image dφ(Y) on φ(M) ⊂ M̃.

Denote by ˜∇ the Riemannian Levi-Civita connection of M̃ and by ∇ the induced connection on M via the projection on the tangent space T M of M, the Gauss formula is

˜∇_{X_{p}}Y = ∇_{X_{p}}Y + s^{0}(X_{p}, Y_{p})

where X_{p}, Y_{p} ∈ T_{p}M, Y is an extension of Y_{p} and s^{0}(X_{p}, Y_{p}) = ( ˜∇_{X_{p}}Y)^{⊥} is the second fundamental form of φ at p (⊥ means taking the orthogonal complement in T_{φ(p)}M̃).

We define the first normal space N^{1}_{p}M of M at p to be the linear span of the image

of s^{0} in the normal space N_{p}M.

For convenience, we put N^{0}_{p}M = T_{p}M.

We can define the third fundamental form S^{1}_{p} of φ at p by

(1)  S^{1}_{p}(X_{p}, Y_{p}, Z_{p}) = ( ˜∇_{X_{p}} s^{0}(Y, Z))^{⊥_{2}}

for Y, Z vector fields arbitrarily extensions of Y_{p}, Z_{p} ∈ T_{p}M, where (,)^{⊥_{2}} is the nomal

projection in N_{p}M on the space orthogonal to ⨁_{i=0}^{1} N^{i}_{p}M = T_{p} ⊕ N^{1}_{p}M.

Denote by N^{2}_{p}M the linear span of the image of S^{1}_{p}, and call it the second normal space of M at p.

From the fact M̃ has constant curvature, S^{1} is symmetric and moreover induces a

linear map from T_{p}M × T_{p}M × T_{p}M into N^{2}_{p}M.

If k is any integer, the fundamental form S^{k−1}_{p} and the normal space at p of order k are defined by recursion:

(2)  S^{k−1}_{p}(X_{p}, X_{1}, …, X_{k}) = ( ˜∇_{X_{p}} S^{k−2}(X̃_{1}, …, X̃_{k}))^{⊥_{k}}

for X_{p}, X_{1}, …, X_{k} ∈ T_{p}M arbitrarily extended to vector fields X̃_{1}, …, X̃_{k} on M and (,)^{⊥_{k}}

is the orthogonal projection on the space normal to ⨁_{i=0}^{k−1} N^{i}_{p}M.

Clearly the process must stop since dim(T_{p}M ⊕ N^{1}_{p}M ⊕ … ⊕ N^{k}_{p}M) ≤ dim T_{φ(p)}M̃.

For each k, S^{k}_{p} is a symmetric k +2-linear form on T_{p}M with values in N^{k+1}_{p}M, because M̃ is of constant curvature. For more details on higher fundamental forms see [8], [3].

Definition 1.1 Let m be the first integer, m ≥ 1 such that S^{m}_{p} ≠ 0 but S^{m+1}_{p} ≡ 0. We call m the degree of the immersion φ at p.

Definition 1.2 The submanifold φ(M) ⊂ M̃ is called nicely curved if the dimension of

each normal space N^{k}_{p}M does not depend on p ∈ M.

Definition 1.3 We say that an immersion φ : M → M̃ is full if φ(M) is not contained in a totally geodesic submanifold of M̃.

It is known that if M̃ is a space of constant curvature and φ(M) is a connected, nicely curved submanifold of M̃, φ is full in a totally geodesic submanifold of M̃ of dimension

equal to dimension of ⨁_{i=0}^{m} N^{i}_{p}M. In particular N_{p}M = ⨁_{i=1}^{m} N^{i}_{p}M [8].

Clearly, for a nicely curved submanifold we have a vector bundle N^{k}M whose fiber at

p is N^{k}_{p}M, for each k.

The connection ˜∇ induces a connection ∇^{k} on the vector bundle N^{k}M by the formula

∇^{k}_{X_{p}} ξ^{k} =⊥_{k} ( ˜∇_{X_{p}} ξ^{k} ) X_{p} ∈ T_{p}M, ξ^{k} ∈ Γ(N^{k}M)

where Γ(N^{k}M) are the sections of N^{k}M, and ⊥_{k} is the orthogonal projection on N^{k}_{p}M. Setting

\begin{aligned}⊥_{k−1} ( ˜∇_{X_{p}} ξ^{k} ) & = − A^{k} (X_{p}, ξ^{k}_{p}) \\ ⊥_{k+1} ( ˜∇_{X_{p}} ξ^{k} ) & = s^{k} (X_{p}, ξ^{k}_{p}) \end{aligned}

we have the Frenet formulas [8]

(3)  ˜∇_{X_{p}} ξ^{k} = − A^{k}(X_{p}, ξ^{k}_{p}) + ∇^{k}_{X_{p}} ξ^{k} + s^{k}(X_{p}, ξ^{k}_{p})

It can easily be seen that A^{1} is the restriction of classical Weingarten map to N^{1}M and

this justifies the notion Weingarten map of order k for A^{k}.

By (1) and (2) we have

\begin{aligned}s^{0}(X_{1}, X_{2}) & = S^{0}(X_{1}, X_{2}) \\ s^{1}(X_{1}, s^{0}(X_{2}, X_{3})) & = S^{1}(X_{1}, X_{2}, X_{3}) \end{aligned}

and in general

S^{k}(X_{1}, …, X_{k+2}) = s^{k}(X_{1}, s^{k−1}(X_{2}, …, s^{0}(X_{k+1}, X_{k+2}) .

If we denote by ⟨ , ⟩ the inner product on the normal bundles, then

⟨A^{k}(X_{p}, ξ^{k}), η^{k−1}⟩ = ⟨ξ^{k}, s^{k−1}(X_{p}, η^{k−1})⟩

where ξ^{k} ∈ N^{k}M, η^{k−1} ∈ N^{k−1} M, see [8].

For any N^{k}-valued symmetric k-linear form Q on T_{p}M, we can define the derivative by the formula

(4) (∇^{k} Q)(X, X_{1}, …, X_{k}) = ∇^{k}_{X}Q(X̃_{1}, …, X̃_{k}) − ∑_{i=1}^{k} Q(X_{1}, …, ∇_{X} X̃_{i}, …, X_{k})

for X, X_{1}, …, X_{k} ∈ T_{p}M arbitrarily extended to vector fields X̃_{1}, …, X̃_{k} on M.

O’Neill [6] has given the following

Definition 1.4 The immersion φ is isotropic at a point p if the norm ∥ s^{0}(X, X)∥ of the

vector s^{0}(X, X) is equal to a constant for any unit vector X ∈ T_{p}M.

These three conditions are equivalent :

1) φ is isotropic at p;

2) ∑ A^{1}(X, s^{0}(Y, Z)) = λ^{2} ∑⟨X, Y ⟩Z

for every X, Y, Z ∈ T_{p}M where ∑ denotes the cyclic sum with respect to X, Y, Z, [4];

3) ⟨s^{0}(X, X), s^{0}(X, Y)⟩ = 0 for any two orthonormal vectors X and Y in T_{p}M, [6].

In general we have

Definition 1.5 The higher fundamental form S^{k} is λ-isotropic at p if ∥ S^{k}(X^{k+2})∥ = con-

stant = λ, for any unit vector X ∈ T_{p}M [11], where S^{k}(X^{k+2}) = S^{k}(X, …, X).

These three conditions are equivalent:

1) S^{k} is λ-isotropic;

2) ∑_{2k+4}⟨S^{k}(X_{1}, X_{2}, …, X_{k+2}), S^{k}(X_{k+3}, …, X_{2k+4}) =

= ∑_{2k+4}⟨X_{1}, X_{2}⟩ … ⟨X_{2k+3}, X_{2k+4}⟩ where ∑_{2k+4} denotes the symmetrized of order 2k + 4, [11];

3) ⟨S^{k}(X^{k+2}), S^{k}(X^{k+1}, Y)⟩ = 0

for any unit vector Y which is orthogonal to X, [4].

2. Minimal immersion of a compact rank one symmetric space

into a unit sphere

Let M = G/K be a n-dimensional compact homogeneous Riemannian manifold with an irreducible linear isotropy group and V^{s} the s-th eigenspace of the Laplacian Δ_{M} corresponding to the s-th eigenvalue λ_{s}. We define an inner product ⟨ , ⟩ in V^{s} by

⟨f, h⟩ = ∫_{M} fh dv f, h ∈ V^{s} .

where dv is the volume element of (M, g).

For simplicity, we normalize the canonical measure dv of (M, g) in such a way that ∫_{M} dv = dim V^{s} = m_{s} + 1. Let \{\underline{φ^{i}, i = 0, …, m_{s}}\} be an orthonormal basis for V^{s} and define a map φ : M → R^{m_{s}+1} by φ(p) = {φ^{0}(p), …, φ^{m_{s}}(p)}. The transitive action of G on M induces a natural action on V^{s} by (g ∘ f)(p) = f ∘ (g^{−1}p), g ∈ G, p ∈ M.

It is easily seen that φ(M) is contained into a sphere S_{1}^{m_{s}} of curvature 1 and the metric on M induced by φ coincides with g up to a constant. Moreover φ is an immersion of (M, g) into S_{1}^{m_{s}}.

By a theorem of Takahashi [9], φ is then a minimal immersion of (M, cg) into S_{1}^{m_{s}}

with c = \frac{λ_{s}}{n}, called the s-th standard minimal immersion of M into a sphere associated to the eigenvalue λ_{s}.

All standard minimal immersions are full and equivariant, i.e. for each g ∈ G there exists an element ρ(g) of the isometry group of S_{1}^{m_{s}} such that φ(gp) = ρ(g)φ(p), p ∈ M. As a consequence of the invariance, we have that φ(M) is nicely curved. Moreover the

higher fundamental forms S^{k} of a standard minimal immersion are isotropic [10].

Do Carmo and Wallace showed that the degree of s-th standard minimal immersion is s when M is a sphere S_{k}^{m_{s}} of curvature k.

We recall that two such immersions φ_{1} and φ_{2} are equivalent if there exists an isometry σ of S_{1}^{m_{s}} such that φ_{2} = σ ∘ φ_{1}.

Tsukada [10] proved the following

Theorem 2.1 Let φ : M → S_{1}^{m} be a minimal immersion of a n-dimensional compact rank one symmetric space into a unit sphere corresponding to the s-th eigenvalue. If we assume that

1) φ is full, 2) in the case M = S^{n}, the degree of φ is ≥ [\frac{s}{2}], where [\frac{s}{2}] is the largest integer less

than or equal to \frac{s}{2}, in the other cases the degree of φ is ≥ s,

3) the higher fundamental forms are isotropic for all p ∈ M, then φ is equivalent to the standard minimal immersion corresponding to the same eigenvalue s.

It is known that the standard minimal immersions of compact rank one symmetric space are helical geodesic. We recall the definition of a helical geodesic immersion given by Sakamoto [7]. Let γ : I → M̃ be a curve parametrized by the arc length s. Let γ_{1}(s) = X(s) the unit tangent vector and put k_{1} = ∥ ˜∇_{X}X∥. If k_{1} is not identically zero, then we define γ_{2} by k_{1}γ_{2} = ˜∇_{X}X on the set I_{1} = {s ∈ I | k_{1}(s) ≠ 0}. Set k_{2} = ∥ ˜∇_{X}γ_{2} + k_{1}X∥. If k_{2} vanishes on I_{1} then γ is said of order 2. If k_{2} is not identically zero on I_{1}, then we can define γ_{3} by ˜∇_{X}γ_{2} = −k_{1}X + k_{2}γ_{3}.

Inductively, we can define γ_{d} and k_{d} = ∥ ˜∇_{X}γ_{d} + k_{d−1}γ_{d−1}∥ and if k_{d} = 0 identically on I_{d−1} = {s ∈ I | k_{d−1}(s) ≠ 0} then γ is said of order d. γ_{1}, γ_{2}, …, γ_{d} are called Frenet vectors and k_{1}, k_{2}, …, k_{d−1} Frenet curvatures.

Definition 2.2 Let φ : M → M̃ be an isometric immersion of a connected Riemannian manifold M into a Riemannian manifold M̃ and σ : I → M be an arbitrary geodesic in M parametrized by the arc length if the curve γ = φ ∘ σ in M̃ is of order d and has constant curvatures k_{1}, k_{2}, …, k_{d−1} which do not depend on σ, then φ is called a helical immersion of order d.

It is known [11]

Theorem 2.3 Let φ : M → S_{1}^{m} be an immersion of a compact rank one symmetric space M into a unit sphere. φ is a helical geodesic immersion if and only if the i-th fundamental form is isotropic for 0 ≤ i ≤ degree of φ.

3. Isotropic minimal immersions and 2-symmetric submanifolds

In the paper [1] the notion of 2-symmetric submanifold of a space M̃ of constant curvature is given.

Definition 3.1 f : M → M̃ is a 2-symmetric submanifold (or immersion) if, for every point p ∈ M, there exist neighbourhoods U of p in M and Ũ of f(p) in M̃ and an isometry σ : Ũ → Ũ such that

i) σ^{2} = IdŨ ii) f(U) ⊆ Ũ, and σ(f(U)) ⊆ f(U) iii) f(p) is an isolated fixed point of σ.

It is proved [1] that Theorem 3.2 f : M → M̃ is a nicely curved 2-symmetric submanifold of a space M̃ of

constant curvature c ≥ 0 if and only if ∇^{h+1} s^{h} = 0 or equivalently ∇^{h+1} S^{h} = 0 for any h.

Theorem 3.3 Let φ : M → S_{1}^{l} be a minimal immersion of a compact rank one symmetric space into a unit sphere S_{1}^{l} of R^{l+1}. We assume φ is full, and the higher fundamental forms

S^{k} are isotropic for each k. Then we see that φ(M) is a minimal 2-symmetric submanifold of S_{1}^{l}.

Proof. We denote by (\overline{∇} S^{k})(X_{1}, X_{2}, …, X_{k+3}) the derivative of S^{k} defined by the formula

(5)  \begin{matrix}(\overline{∇} S^{k}) & (X_{1}, X_{2}, …, X_{k+3}) = \\ = ∇_{X_{1}}^{⊥} S^{k}(X̃_{2}, …, X̃_{k+3}) − ∑_{i=2}^{k+3} S^{k}(X_{2}, …, ∇_{X_{1}} X̃_{i}, …, X_{k+3}) \end{matrix}

for X_{1}, …, X_{k+3} ∈ T_{p}M arbitrarily extended to vector fields X̃_{1}, …, X̃_{k+3} on M, and ∇^{⊥} is the connection on the normal bundle NM.

In the above hypothesis, it is proved that, for any point p ∈ M, the dimension of N^{k}_{p}M does not depend on p, therefore φ(M) is nicely curved [11].

Moreover \overline{∇} S^{k} has component only on the normal spaces N^{k}_{p}M and N^{k+2}_{p}M [11].

By the Frenet formulas (3), we have for X_{1}, X_{2}, …, X_{k+3} ∈ T_{p}M arbitrarily extended to vector fields X̃_{1}, X̃_{2}, …, X̃_{k+3}

(\overline{∇} S^{k})(X_{1}, X_{2}, …, X_{k+3}) =

= − A^{k+1}(X_{1}, S^{k}(X̃_{2}, …, X̃_{k+3})) + ∇^{k+1}_{X_{1}} S^{k}(X̃_{2}, …, X̃_{k+3}) + S^{k+1}(X_{1}, …, X_{k+3})

− ∑_{i=2}^{k+3} S^{k}(X_{2}, …, ∇_{X_{1}} X̃_{i}, …, X_{k+3}) .

Since A^{k+1}(X_{1}, S^{k}) ∈ N^{k}M and S^{k+1} ∈ N^{k+2} M, we have for (4)

( ∇^{k+1} S^{k})(X_{1}, …, X_{k+3}) = 0 .

By the theorems 2.3 and 2.1 we have the following corollaries:

Corollary 3.4 Let φ : M → S_{1}^{l} be a helical minimal immersion of a compact rank one symmetric space into a unit sphere. We assume φ is full. Then φ is a 2-symmetric submanifold of S_{1}^{l}.

Corollary 3.5 Let φ : M → S_{1}^{l} be a standard minimal immersion of compact rank one symmetric space into the unit sphere S_{1}^{l}. Then φ is a 2-symmetric submanifold of S_{1}^{l}.

Observation. For a helical immersion φ : M → S_{1}^{l} of a compact rank one symmetric space into a unit sphere, the Frenet vectors γ_{1}, γ_{2}, …, γ_{d} satisfy the conditions:

γ_{1} ∈ T_{p}M, γ_{2} ∈ N^{1}_{p}M, …, γ_{d} ∈ N^{d−1}_{p}M .

In fact, by theorem 2.3 S^{k} are isotropic, so

γ_{2} = \frac{S^{0}(X, X)}{k_{1}} , γ_{3} = \frac{1}{k_{1}k_{2}} S^{1}(X, X, X) , … γ_{d} = \frac{1}{k_{1}k_{2}, …, k_{d−1}} S^{d−2}(X^{d}) ,

where k_{i} are the normal curvatures, [11].

Moreover, since S^{k} are isotropic, we have A^{1}(X, s^{0}(X, X)) = k_{1}^{2}X and in general

A^{i}(X, S^{i−1}(X^{i+1}) = k_{i}^{2} S^{i−2}(X^{i})

for i = 2, …, d − 1.

4. Helical immersions and 2-symmetric submanifolds

In this section we show the following

Theorem 4.1 Let φ : M → M̃ be a nicely curved, 2-symmetric submanifold of a space M̃ of constant curvature c ≥ 0. Denote by σ : I → M a geodesic of M and by γ = φ∘σ : I → M̃ its image curve in M̃. We assume that the Frenet vectors γ_{1}, …, γ_{d−1} of every curve γ of M̃ satisfy the conditions:

(6)  γ_{h} ∈ N^{h−1}_{p}M , 1 ≤ h ≤ d − 1 .

Then φ is a helical geodesic immersion.

Before the proof, we prepare the following lemmas.

Lemma 4.2 Let φ : M → M̃ be a nicely curved, 2-symmetric submanifold of a space M̃ of constant curvature c ≥ 0. Then all the curvatures of the geodesics of M are constant along the geodesics.

Proof. It is an easy consequence of the symmetry property of the 2-symmetric submanifolds, (see [1]).

A. Carfagna D’Andrea: Immersions into Spheres with Parallel … 83

Lemma 4.3 In the hypothesis of theorem 4.1, we have

(7)  γ_{h} = \frac{1}{k_{1}k_{2}, …, k_{h−1}} S^{h−2}(X^{h}) for any h .

Proof. By Frenet formulas, put

γ_{2} = \frac{s^{0}(X, X)}{k_{1}} , if k_{1} ≠ 0 , γ_{3} is defined by ˜∇_{X} \frac{s^{0}(X, X)}{k_{1}} + k_{1}X = k_{2}γ_{3} ,

so, by the hypothesis (6): γ_{3} ∈ N^{2}_{p}M, we have

(8) A^{1}(X, s^{0}(X, X)) = k_{1}^{2}X ,

for any X and k_{2}γ_{3} = \frac{1}{k_{1}} S^{1}(X^{3}). Since k_{2} = \frac{1}{k_{1}} ∥ S^{1}(X^{3})∥, we have that k_{2} is constant along γ.

If k_{2} ≠ 0 then γ_{3} = \frac{1}{k_{1}k_{2}} S^{1}(X^{3}).

Now we apply the mathematical induction. For this we suppose that k_{1}, k_{2}, …, k_{ρ−1} are constant, different from zero, along γ and γ_{ρ} = \frac{1}{k_{1}k_{2}, …, k_{ρ−1}} S^{ρ−2}(X^{ρ}). By Frenet formulas (3)

\frac{1}{k_{1}k_{2} … k_{ρ−1}} ˜∇_{X} S^{ρ−2}(X^{ρ}) + \frac{k_{ρ−1}}{k_{1}k_{2} … k_{ρ−2}} S^{ρ−3}(X^{ρ−1}) = k_{ρ}γ_{ρ+1}

\frac{1}{k_{1}k_{2} … k_{ρ−1}} [ − A^{ρ−1}(X, S^{ρ−2}(X^{ρ})) + S^{ρ−1}(X^{ρ+1})]+

+\frac{k_{ρ−1}}{k_{1}k_{2} … k_{ρ−2}} S^{ρ−3}(X^{ρ−1}) = k_{ρ}γ_{ρ+1} .
By γ_{ρ+1} ∈ N^{ρ}M, we have
(9) A^{ρ−1}(X, S^{ρ−2}(X^{ρ}) = k_{ρ−1}^{2} S^{ρ−3}(X^{ρ−1})

and k_{ρ}γ_{ρ+1} = \frac{1}{k_{1}k_{2} … k_{ρ−1}} S^{ρ−1}(X^{ρ+1}).

Now we prove Lemma 4.4 In the hypothesis of theorem 4.1, we have that the j-th fundamental forms are isotropic for 0 ≤ j ≤ d.

Proof. The second fundamental form is isotropic if and only if ⟨s^{0}(X, X), s^{0}(X, Y)⟩=0 for any orthonormal vector fields X and Y of M (see section 1).

In our case, by property (8)

⟨s^{0}(X, X), s^{0}(X, Y)⟩ = ⟨A^{1}(X, s^{0}(X, X)), Y ⟩ = k_{1}^{2}⟨X, Y ⟩ = 0 .

In general we have, by property (9)

\begin{aligned}⟨S^{ρ}(X^{ρ+2}), S^{ρ}(X^{ρ+1}, Y)⟩ & = ⟨ A^{ρ+1}(X, S^{ρ}(X^{ρ+2}), S^{ρ−1}(Y, X^{ρ})⟩ = \\ & = k_{ρ+1}^{2}⟨ S^{ρ−1}(X^{ρ+1}), S^{ρ−1}(Y, X^{ρ})⟩ . \end{aligned}

So the statement follows by induction. If S^{ρ−1} is isotropic ⟨ S^{ρ−1}(X^{ρ+1}), S^{ρ−1}(Y, X^{ρ})⟩ = 0 (see

section 1) and then S^{ρ} is isotropic. Now the proof of the theorem 4.1 is an easy consequence of these lemmas. In fact, we proved that all normal curvatures are constant along γ and since all higher fundamental forms are isotropic at p, by (7) all curves emanating from a point p have the same curvatures.

We can also prove the following

Theorem 4.5 Let φ : M → S_{1}^{l} be a helical immersion of order d. We assume that, for any geodesic σ in M, the Frenet vectors of the curve φ ∘ σ satisfy the following conditions:

(10)  γ_{h} ∈ ⨁_{j=0}^{q} N^{2j+1} M , if h is even , γ_{h} ∈ ⨁_{j=0}^{p} N^{2j}M , if h is odd ,

where q = [\frac{d − 1}{2}] and p = [\frac{d}{2}], then φ(M) is a 2-symmetric submanifold.

Proof. Put X(s) = σ̇(s). We note that for γ_{1} = X and γ_{2} = \frac{s^{0}(X, X)}{k_{1}} the conditions (10)

are always satisfied.

Sakamoto [7] calculated the Frenet vectors of the curve γ.

Let \overline{∇}^{i} s^{0} be denote the i-th order derivative of the second fundamental form (see (5)) we have that

(11)  γ_{j} = (k_{1}k_{2} … k_{j−1})^{−1} ∑_{i} a_{ji}(\overline{∇}^{i−2} s^{0})(X^{i})

for j = 2, …, d where the index i runs over the range {2, 4, …, j} if j is even, and {3, 5, …, j} if j is odd. The coefficients a_{ji} are polynomials of the k_{1}^{2}, …, k_{j−2}^{2} [7].

A. Carfagna D’Andrea: Immersions into Spheres with Parallel … 85

We will show that the conditions (10) are equivalent to

(12)  \{ \begin{matrix}(\overline{∇}^{i} s^{0})(X^{i+2}) ∈ ⨁_{j=0}^{p} N^{2j}M , if i is odd, \\ (\overline{∇}^{i} s^{0})(X^{i+2}) ∈ ⨁_{j=0}^{q} N^{2j+1}M , if i is even, \\ for 1 ≤ i ≤ h . \end{matrix}

In fact, for h = 3 the equivalence is true since γ_{3} = (k_{1}k_{2})^{−1}(\overline{∇} s^{0})(X, X, X). So γ_{3} ∈ ⨁_{j=0}^{p} N^{2j}M if and only if \overline{∇} s^{0} ∈ ⨁_{j=0}^{p} N^{2j}M. We proceed by induction on h = 3, 4, …

If h is odd, then γ_{h} ∈ ⨁_{j=0}^{p} N^{2j}M if and only if (12) is true for 1 ≤ i ≤ h. By (11) it follows that

γ_{h+1} = (k_{1}k_{2} … k_{h})^{−1}(a_{h+1,2} s^{0}(X, X) + a_{h+1,4}(\overline{∇}^{2} s^{0})(X^{4}) + … + (\overline{∇}^{h−1} s^{0})(X^{h+1}) ,

so (\overline{∇}^{h−1} s^{0})(X^{h+1}) ∈ ⨁_{j=0}^{q} N^{2j+1} M if and only if γ_{h+1} satisfies the condition (10). For h even the proof is the same.

Now we can prove the statement of the theorem if we show the equivalence of the conditions (12) with \overline{∇}^{i} S^{i−1} = 0 for all 1 ≤ i ≤ h. For i = 1, since (\overline{∇} s^{0})(X^{3}) = (∇^{1} s^{0})(X^{3}) + S^{1}(X^{3}), we have

(\overline{∇} s^{0})(X^{3}) ∈ ⨁_{j=0}^{p} N^{2j}M ⟺ (∇^{1} s^{0})(X^{3}) = 0 .

For i = 2

(\overline{∇}^{2} s^{0})(X^{4}) = − A^{2}(X, S^{1}(X^{3})) + S^{2}(X^{4}) + ∇^{2} S^{1}(X^{4})

and (\overline{∇}^{2} s^{0})(X^{4}) ∈ ⨁_{j=0}^{q} N^{2j+1} M if and only if (∇^{2} S^{1})(X^{4}) = 0. If we calculate the other

derivatives, since ∇^{h+1} S^{h} = 0 is equivalent to ∇^{h−1} A^{h} = 0 for all h, [1], the component of the i-th order derivative in the normal space N^{i}M is given by ∇^{i} S^{i−1}(X^{i+2}). So we have the equivalence of the conditions (12) with the (\overline{∇}^{i} S^{i−1})(X^{i+2}) = 0. Since \overline{∇}^{i} S^{i−1} is symmetric by Codazzi’s generalized equations [8], it follows that \overline{∇}^{i} S^{i−1} = 0. By theorem 3.2 we have the statement.

References

[1] Carfagna D’Andrea, A.; Mazzocco, R.; Romani, G.: Some characterizations of 2-symmetric submanifolds in spaces of constant curvature. Czechoslovak Math. Journal 44 (119) (1994), 691–711.

[2] Ferus, D.: Symmetric submanifolds of Euclidean spaces. Math. Ann. 247 (1980), 81–83.

[3] do Carmo, M.; Wallach, N.: Minimal immersions of spheres into spheres. Annals of Math. 93 (1971), 43–62.

[4] Ho Kim, Y.: Surfaces of a Euclidean space with helical or planar geodesics through a point. Ann. Mat. Pura e Appl. IV, CLXIV (1993), 1–35.

[5] Kowalski, O.; Külich, I.: Generalized symmetric submanifolds of Euclidean spaces. Math. Ann. 277 (1987), 67–78.

[6] O’Neill, B.: Isotropic and Kähler Immersions. Canadian Journal of Math. 17 (1965), 909–915.

[7] Sakamoto, K.: Helical immersions into a unit sphere. Math.Ann. 261 (1982), 63–80.

[8] Spivak, M.: A comprehensive introduction to differential geometry. (Publish or Perish Inc., 1979).

[9] Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18 (1968), 203–215.

[10] Tsukada, K.: Helical geodesic immersions of compact rank one symmetric spaces into spheres. Tokyo J. Math. 6 (1983), 267–290.

[11] Tsukada, K.: Isotropic minimal immersions of spheres into spheres. J. Math. Soc. Japan, 35 (1983), 355–379.

Received June 30, 1995