Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 1, 125-134.

On Generalised Chen and k-minimal Immersions

Sheila Carter Uǧur Dursun^{1}

School of Mathematics, The University of Leeds Leeds LS2 9JT, U.K.

0. Introduction

B. Rouxel [5] and S. J. Li and C. S. Houh [3] have given extensions of the notion of an A-submanifold (Chen submanifold) to an A_{k}-submanifold. Here we investigate the relation between their definitions.

We introduce the idea of a k-minimal submanifold which satisfies a stronger condition than that of an A_{k}-submanifold. By generalising a construction from [1] we give a method for constructing k-minimal submanifolds.

1. A_{k}-immersions

Let f : M^{m} → ℝ^{m+d} be a smooth immersion of a connected, m-dimensional manifold without boundary into Euclidean m + d-space. For x ∈ M, the normal plane to f at x is defined by N_{x}(f) = {n ∈ ℝ^{m+d} : <n, df_{x}X > = 0, ∀ X ∈ T_{x}M}, where df_{x} : T_{x}M → {f(x)} × ℝ^{m+d} ≡ ℝ^{m+d} is the tangent mapping induced from f. For n ∈ N_{x}(f), the shape operator A_{n} : T_{x}M → T_{x}M is defined to be the linear mapping such that for any tangent vector field X : M → T M, df_{x}(A_{n}X) is the tangent component of the directional derivative of n in direction X. The mean curvature vector is defined, at x ∈ M, by H = \frac{1}{m} ∑_{i=0}^{d−1}(traceA_{e_{i}})e_{i} where {e_{0}, …, e_{d−1}} is an orthonormal basis for N_{x}(f).

In [3], S. J. Li and C. S. Houh define normal fields A_{1}(H), A_{2}(H), …, A_{d−1}(H) on M. At x ∈ M, if H ≠ 0, A_{1}(H) = ∑_{i=1}^{d−1}(traceA_{H}A_{e_{i}})e_{i} where {e_{0}, …, e_{d−1}} is an orthonormal basis for N_{x}(f) with e_{0} = H/∥H∥. If H = 0, then A_{1}(H) is defined to be 0. So A_{1}(H) is orthogonal to H.

1The second author is supported by Istanbul Technical University 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

126 Sh. Carter, U. Dursun: On Generalised Chen and k-minimal Immersions

Now assume that A_{1}(H), …, A_{d−1}(H) have been defined on M and are pairwise orthogonal and such that at each x ∈ M, A_{k−1}(H) ≠ 0 ⇒ A_{j}(H) ≠ 0 for j = 1, …, k −2. Then, at x ∈ M, if A_{k−1}(H) ≠ 0 define A_{k}(H) = ∑_{i=k}^{d−1}(traceA_{A_{k−1}(H)}A_{e_{i}})e_{i} where {e_{0}, …, e_{d−1}} is an orthonormal basis for N_{x}(f) with e_{0} = H/∥H∥ and e_{i} = A_{i}(H)/∥A_{i}(H)∥, i = 1, …, k − 1. If A_{k−1}(H) = 0, define A_{k}(H) = 0. Then A_{k}(H) is orthogonal to each of H, A_{1}(H), …, A_{k−1}(H). This definition is independent of the choice of basis for N_{x}(f).

In this way the normal fields A_{1}(H), A_{2}(H), …, A_{d−1}(H) are defined. For k ≥ d, we make the convention that A_{k}(H) ≡ 0. Also, we write A_{0}(H) = H.

Following [3], f is said to be an A_{k}-immersion if A_{k}(H) ≡ 0 and is said to be a k-Chen immersion if A_{k}(H) ≡ 0 and A_{k−1}(H) ≢ 0. So A_{0}-immersion is minimal. A_{1}-immersions are also known as Chen immersions. Note that f : M^{m} → ℝ^{m+d} is a k-Chen for some k ≤ d. Further, if f has flat normal bundle and m < d then it follows from the next theorem that f is k-Chen for some k ≤ m.

Theorem 1.1 Let f : M^{m} → ℝ^{m+d} be an immersion with flat normal bundle. Then A_{k}(H) ≡ 0 for some k ≤ min{d, m}.

We first prove a lemma :

Lemma 1.2 Let N_{x}(f) = Λ⊕Λ^{⊥} where Λ^{⊥} is the orthogonal complement of Λ in N_{x}(f), and suppose that ∀ u ∈ Λ, ∀ n ∈ Λ^{⊥}, traceA_{n} = 0 and traceA_{u}A_{n} = 0 then, at x, A_{k}(H) = 0 for some k ≤ dimΛ.

Proof. Let dimΛ = ℓ. Take an orthonormal basis {e_{0}, …, e_{d−1}} for N_{x}(f) with e_{0}, …, e_{ℓ−1}∈ Λ and e_{ℓ}, …, e_{d−1} ∈ Λ^{⊥}. So traceA_{e_{j}} = 0 for j = ℓ, …, d − 1 and traceA_{e_{i}}A_{e_{j}} = 0 for i = 0, …, ℓ − 1, j = ℓ, …, d − 1.

Now A_{0}(H) = H = \frac{1}{m} ∑_{i=0}^{d−1}(traceA_{e_{i}})e_{i} = \frac{1}{m} ∑_{i=0}^{ℓ−1}(traceA_{e_{i}})e_{i} ∈ Λ. Suppose that A_{0}(H), …, A_{r−1}(H) are nonzero and belong to Λ. We choose the basis for N_{x}(f) so that e_{i} = A_{i}(H)/∥A_{i}(H)∥, i = 0, … r − 1. Then

A_{r}(H) = ∑_{i=r}^{d−1}(traceA_{A_{r−1}(H)}A_{e_{i}})e_{i} = ∑_{i=r}^{ℓ−1}(traceA_{A_{r−1}(H)}A_{e_{i}})e_{i} ∈ Λ.

So, ∀ r, A_{r}(H) ≠ 0 ⇒ A_{r}(H) ∈ Λ.

Hence, at x, A_{k}(H) = 0 for some k ≤ dimΛ. □

Proof of Theorem 1.1. As remarked above, A_{k}(H) ≡ 0 for some k ≤ d. Let x ∈ M, then since f has flat normal bundle {A_{n} : n ∈ N_{x}(f)} is a family of self-adjoint commuting operators on N_{x}(f) and for some t ∈ ℤ, 1 ≤ t ≤ m, there exist v_{1}, …, v_{t} ∈ N_{x}(f) and α_{1}, …, α_{t} ∈ ℤ with ∑_{i=1}^{t} α_{i} ≤ m such that ∀ n ∈ N_{x}(f),

det(1 − A_{n}) = ∏_{i=1}^{t}(1− <v_{i}, n>)^{α_{i}}, [4].

Put Λ = span{v_{1}, …, v_{t}}. Now if n ∈ N_{x}(f) with ∥n∥ = 1, the principal curvatures in direction n are <v_{1}, n>, …, <v_{t}, n> where <v_{i}, n> has multiplicity α_{i}, i = 1, …, t, together with a zero principal curvature of multiplicity m − ∑_{i=1}^{t} α_{i}. So, for any n ∈ Λ^{⊥} the principal curvatures in direction n/∥n∥ are all zero and therefore A_{n/∥n∥}, and hence A_{n}, is the zero operator.

It follows from Lemma 1.2 that A_{k}(H) = 0 for some k ≤ dimΛ ≤ t ≤ m. □

B. Rouxel used a different approach in [5]. He considered the quadratic form on N_{x}(f) defined by Q(u, v) = (traceA_{u})(traceA_{v}) − trace(A_{u}A_{v}). With respect to the inner product on N_{x}(f) induced from ℝ^{m+d}, Q(u, v) = <u, Qv> where Q : N_{x}(f) → N_{x}(f) is a linear map. Let λ_{1}, …, λ_{s} be the distinct eigenvalues of this linear map with corresponding eigenspaces E_{1}, …, E_{s}, so that N_{x}(f) = E_{1}⊕⋯⊕E_{s}. Let q(H) denote the number of nonzero projections of H on the eigenspaces E_{1}, …, E_{s}. Clearly q(H) = 0 if and only if H = 0. It is shown in [5] that q(H) = 1 if and only if H ≠ 0 and A_{1}(H) = 0. Here we will show that f is an A_{k}-immersion if and only if q(H) ≤ k at all points of M, and f is k-Chen if and only if f is A_{k} and q(H) = k at some x ∈ M. This follows from the next theorem.

Theorem 1.3 Let f : M^{m} → ℝ^{m+d} be an immersion and let k ≥ 1. Then, at x ∈ M, q(H) = k if and only if A_{k−1}(H) ≠ 0 and A_{k}(H) = 0.

The proof of this theorem depends on the following lemmas:

Lemma 1.4 At x ∈ M, let A_{k−1}(H) ≠ 0 and A_{k}(H) = 0, k ≥ 1. Then 1 ≤ q(H) ≤ k.

Proof. Let {e_{0}, …, e_{d−1}} be an orthonormal basis for N_{x}(f) with e_{i} = A_{i}(H)/∥A_{i}(H)∥, i = 0, …, k − 1. Then, writing u = ∑_{i=0}^{d−1} u_{i}e_{i}, v = ∑_{i=0}^{d−1} v_{i}e_{i},

Q(u, v) = ∑_{i,j=0}^{d−1} u_{i}v_{j}(traceA_{e_{i}}traceA_{e_{j}} − traceA_{e_{i}}A_{e_{j}}).

Now let Q denote the (d × d)-matrix with ij-th term

[Q]_{ij} = traceA_{e_{i}}traceA_{e_{j}} − traceA_{e_{i}}A_{e_{j}}.

From the definitions of A_{0}(H), …, A_{k−1}(H),

traceA_{e_{j}} = 0 for j = 1, …, d − 1 and traceA_{e_{i}}A_{e_{j}} = 0 for i = 0, …, k − 2, j ≥ k.

Further A_{k}(H) = 0 ⇒ traceA_{e_{k−1}}A_{e_{j}} = 0 for j ≥ k.

Thus [Q]_{ij} = 0 for i = 0, …, k − 1, j ≥ k. Therefore the matrix Q has the form (\begin{matrix}B & 0 \\ 0 & C \end{matrix}) where B is a (k × k)-matrix.

Now if λ is an eigenvalue of Q with corresponding eigenspace E and λ is not an eigenvalue of B then E ⊂ span{e_{k}, …, e_{d−1}} and, since H = ∥H∥e_{0}, H has zero projection on E. Hence q(H) is at most the number of distinct eigenvalues of B. So 1 ≤ q(H) ≤ k. □

Lemma 1.5 At x ∈ M, let H, A_{1}(H), …, A_{k−1}(H) be nonzero, let v ∈ N_{x}(f), v ≠ 0, with v orthogonal to span{H, A_{1}(H), …, A_{k−1}(H)} and traceA_{A_{k−1}(H)}A_{v} = 0. Then v is orthogonal to A_{k}(H).

Proof. Let {e_{0}, …, e_{d−1}} be an orthonormal basis for N_{x}(f) with e_{i} = A_{i}(H)/∥A_{i}(H)∥, i = 0, …, k−1 and e_{k} = v/∥v∥. Then A_{k}(H) = ∑_{i=k}^{d−1}(traceA_{A_{k−1}(H)}A_{e_{i}})e_{i}. So the projection of A_{k}(H) in direction e_{k} is traceA_{A_{k−1}(H)}A_{e_{k}} = \frac{1}{∥v∥}traceA_{A_{k−1}(H)}A_{v} = 0. Therefore A_{k}(H) is orthogonal to v. □

We now take a different point of view. Suppose Q has distinct eigenvalues λ_{1}, …, λ_{s}, 1 ≤ s ≤ d, with corresponding eigenspaces E_{1}, …, E_{s} ⊂ N_{x}(f). For u ∈ N_{x}(f), write u = u_{1}+⋯+u_{s} where u_{j} is the projection of u on E_{j}, j = 1, …, s.

By a change of basis for N_{x}(f), if necessary, we can assume that Q(u, v) = ∑_{j=1}^{s} λ_{j} <u_{j}, v_{j}>.

Lemma 1.6 At x ∈ M, let q(H) = k, then A_{k}(H) = 0.

Proof. By reordering the eigenspaces if necessary we can suppose that H = H_{1} + ⋯ + H_{k} ∈ E_{1} ⊕ ⋯ ⊕ E_{k} with H_{1}, …, H_{k} nonzero. Put H^{∗} = span{H_{1}, …, H_{k}}. We show that ∀ r = 0, 1, …, A_{r}(H) ∈ H^{∗}. Now A_{0}(H) = H ∈ H^{∗}. Suppose that A_{r−1}(H) ∈ H^{∗}. If A_{r−1}(H) = 0, then A_{r}(H) = 0 ∈ H^{∗}. If A_{r−1}(H) ≠ 0 then A_{r−1}(H) = α_{1}H_{1} + ⋯ + α_{k}H_{k} for some α_{1}, …, α_{k} ∈ ℝ. Let v ∈ N_{x}(f), v ≠ 0 with v orthogonal to H^{∗}, say v = v_{1} + ⋯ + v_{s} ∈ E_{1} ⊕ ⋯ ⊕ E_{s}. Then

Q(A_{r−1}(H), v) = ∑_{j=1}^{s} λ_{j} <(A_{r−1}(H))_{j}, v_{j}>

= ∑_{j=1}^{k} λ_{j}α_{j} <H_{j}, v_{j}> = 0

since for j = 1, …, k, <H_{j}, v_{j}> = <H_{j}, v> = 0. Also

Q(A_{r−1}(H), v) = traceA_{A_{r−1}(H)}traceA_{v} − traceA_{A_{r−1}(H)}A_{v}
= −traceA_{A_{r−1}(H)}A_{v}, since traceA_{v} = 0 as <H, v> = 0.

Therefore traceA_{A_{r−1}(H)}A_{v} = 0. So, by Lemma 1.5, v is orthogonal to A_{r}(H). This holds for all v ∈ N_{x}(f) with v orthogonal to H^{∗}. Hence A_{r}(H) ∈ H^{∗}.

Since dimH^{∗} = k it follows that A_{k}(H) = 0. □

Proof of Theorem 1.3. The theorem is true for k = 1, since by Lemma 1.4, A_{0}(H) ≠ 0 and A_{1}(H) = 0 ⇒ q(H) = 1. Also, q(H) = 1 ⇒ H ≠ 0 and, by Lemma 1.6, q(H) = 1 ⇒ A_{1}(H) = 0.

Now assume that the statement of the theorem holds for k ≤ r − 1. Then by Lemma 1.4, A_{r−1}(H) ≠ 0 and A_{r}(H) = 0 ⇒ q(H) ≤ r. But, by the induction hypothesis, q(H) < r ⇒ A_{r−1}(H) = 0. Hence q(H) = r.

Conversely, if q(H) = r then, by Lemma 1.6, A_{r}(H) = 0. Again, by the induction hypothesis, if A_{r−1}(H) = 0 then q(H) < r and hence A_{r−1}(H) ≠ 0. Hence the statement of the theorem holds for all k ≥ 1. □

2. k-minimal immersions

In this section we give methods for constructing A_{k}-immersions by generalising a construction from [1]. Let f : M^{m} → ℝ^{p} be an immersion such that for some s, 2 ≤ s ≤ p − m, f has an orthonormal set of parallel normal fields {u_{1}, …, u_{s}} and let g : W^{w} → ℝ^{s} be an immersion. We consider h : M × W → ℝ^{p} defined by h(x, y) = f(x) + ∑_{i=1}^{s} g_{i}(y)u_{i}(x), where g(y) = (g_{1}(y), …, g_{s}(y)). It was proved in Theorem 1.1 of [1] that

(1) h is an immersion if and only if, ∀ x ∈ M, ∀ y ∈ W, the linear map (1−∑_{i=1}^{s} g_{i}(y)A_{u_{i}}) : T_{x}M → T_{x}M is nonsingular,

(2) ξ ∈ N_{(x,y)}(h) if and only if ξ = n + ∑_{i=1}^{s} η_{i}u_{i}(x) where n ∈ N_{x}(f), <n, u_{i}(x)> = 0 for i = 1, …, s and η = (η_{1}, …, η_{s}) ∈ N_{y}(g),

(3) the shape operator of h, C_{ξ} : T_{(x,y)}(M × W) → T_{(x,y)}(M × W), is given in terms of the shape operator of f, A_{n} : T_{x}M → T_{x}M, and the shape operator of g, B_{η} : T_{y}W → T_{y}W by the formula

C_{ξ}(X, Y) = ((1 − ∑_{i=1}^{s} g_{i}(y)A_{u_{i}})^{−1}A_{ξ}X, B_{η}Y)

where X : M → T M, Y : W → T W are tangent fields. We will use this construction to give examples of A_{k}-immersions for different values of k.

In [1] we introduced the idea of a 1-minimal immersion. We generalise this to a k-minimal immersion. We consider a parallel, flat k-plane subbundle Λ(f) of the normal bundle N(f) of an immersion f, that is, locally there is a set of k parallel normal fields which at each point x determines an orthonormal basis for the fibre Λ_{x}(f) ⊂ N_{x}(f).

Definition 2.1 Let f : M^{m} → ℝ^{p} be an immersion with a parallel, flat k-plane subbundle, Λ(f), of N(f). We say that f is k-minimal with respect to Λ(f) if ∀x ∈ M, ∀n ∈ N_{x}(f) with n orthogonal to Λ_{x}(f), ∀u ∈ Λ_{x}(f) with 1 − A_{u} nonsingular, trace(1 − A_{u})^{−1}A_{n} = 0.

From Proposition 2.2 of [1], f is k-minimal with respect to Λ(f) if and only if ∀x ∈ M, ∀n ∈ N_{x}(f) with n orthogonal to Λ_{x}(f), ∀u ∈ Λ_{x}(f), ∀ℓ = 0, 1, … trace(A_{u})^{ℓ}A_{n} = 0.

For most applications we will need Λ(f) to satisfy the global condition that there exist a set of parallel normal fields u_{i} : M^{m} → ℝ^{p}, i = 1, …, k such that ∀x ∈ M, {u_{1}(x), …, u_{k}(x)} is an orthonormal basis for Λ_{x}(f). In this case we will say Λ(f) is global parallel flat. In the definition of 1-minimal in [1] we only considered this stronger condition.

Proposition 2.2 Let f : M^{m} → ℝ^{p} be k-minimal with respect to Λ(f), then f is an A_{k}- immersion.

Proof. Since, ∀x ∈ M, ∀n ∈ N_{x}(f) with n orthogonal to Λ_{x}(f), ∀u ∈ Λ_{x}(f) traceA_{n} = 0 and traceA_{u}A_{n} = 0, this result follows immediately from Lemma 1.2. □

A relation between k-minimal and minimal immersions is given in the next theorem.

Theorem 2.3 Let f be an immersion and let Λ(f) be a parallel, flat k-plane subbundle of N(f). Let Ω = {(x, u) ∈ Λ(f) : 1 − A_{u} is nonsingular at x} and let F : Ω → ℝ^{p} be defined by F (x, u) = f(x) + u(x). Then f is k-minimal with respect to Λ(f) if and only if F is minimal.

Proof. In a neighbourhood V of x_{0} ∈ M we can take a set of parallel normal fields {u_{1}, …, u_{k}} which at each point of V determines an orthonormal basis for the fibre of Λ(f). Then, locally, Ω can be identified with an open subset of V ×ℝ^{k}, by (x, ∑_{i=1}^{k} t_{i}u_{i}(x)) ≡ (x, t), where t = (t_{1}, …, t_{k}) and F (x, t) = f(x) + ∑_{i=1}^{k} t_{i}u_{i}(x). Now use the construction described at the start of this section with M replaced by V, s = k, g : ℝ^{k} → ℝ^{k} the identity function and h = F. So F is an immersion and n ∈ N_{(x,t)}(F) if and only if n is orthogonal to Λ_{x}(f) in N_{x}(f). Further if C_{n} denotes the shape operator of F then, at (x, t), traceC_{n} = trace(1 − ∑_{i=1}^{k} t_{i}A_{u_{i}})^{−1}A_{n} at x. Writing u = ∑_{i=1}^{k} t_{i}u_{i}(x), traceC_{n} = trace(1 − A_{u})^{−1}A_{n}.

Now F is minimal if and only if ∀x ∈ M, ∀n orthogonal to Λ_{x}(f), traceC_{n} = 0. This holds if and only if f is k-minimal with respect to Λ(f). □

In [1] we constructed many examples of 1-minimal immersions. The next theorem gives a way of using these examples to build up examples of k-minimal (and hence A_{k}-) immersions for different values of k > 1.

Theorem 2.4 Let f : M^{m} → ℝ^{p} be an immersion with an orthonormal set of parallel normal fields {u_{1}, …, u_{k}} for some 1 ≤ k ≤ p − m, and let g : W^{w} → ℝ^{q}, q − w ≥ k be an immersion with an orthonormal set of parallel normal fields {ψ_{1}, …, ψ_{q−w}} and such that ∀x ∈ M, 1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}} is nonsingular at x, where g(y) = (g_{1}(y), …, g_{q}(y)). Let {e_{1}, …, e_{q−k}} be the standard basis for ℝ^{q−k} and let h : M × W → ℝ^{p+q−k} ≡ ℝ^{p} × ℝ^{q−k} be defined by

h(x, y) = (f(x) + ∑_{i=1}^{k} g_{i}(y)u_{i}(x), ∑_{i=1}^{q−k} g_{k+i}(y)e_{i}).

Then

(1) h is an immersion.

(2) ξ ∈ N_{(x,y)}(h) if and only if ξ = (n + ∑_{i=1}^{k} η_{i}u_{i}(x), ∑_{i=1}^{q−k} η_{k+i}e_{i}) ∈ N_{x}(f) × ℝ^{q−k}, for some n ∈ N_{x}(f) with n orthogonal to span{u_{1}(x), …, u_{k}(x)}, η = (η_{1}, …, η_{q}) ∈ N_{y}(g).

(3) If μ_{j} : M × W → ℝ^{p} × ℝ^{q−k} is defined by μ_{j}(x, y) = (∑_{i=1}^{k} ψ_{j,i}(y)u_{i}(x), ∑_{i=1}^{q−k} ψ_{j,k+i}(y)e_{i}), where ψ_{j}(y) = (ψ_{j,1}(y), …, ψ_{j,q}(y)), j = 1, …, q − w, then (μ_{1}, …, μ_{q−w}) is an orthonormal set of parallel normal fields for h.

(4) C_{ξ}(X, Y) = ((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v}X, B_{η}Y) and traceC_{ξ} = trace((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v}) + traceB_{η}, where ξ = (n + ∑_{i=1}^{k} η_{i}u_{i}(x), ∑_{i=1}^{q−k} η_{k+i}e_{i}), v = n + ∑_{i=1}^{k} η_{i}u_{i}(x), C_{ξ} : T_{(x,y)}(M × W) → T_{(x,y)}(M × W) is the shape operator of h and B_{η} : T_{y}W → T_{y}W is the shape operator of g.

(5) Let Λ(f) be the global parallel flat subbundle of N(f) with fibre Λ_{x}(f) = span{u_{1}(x), …, u_{k}(x)} and let Λ̃(h) be the global parallel flat subbundle of N(h) with fibre Λ̃_{(x,y)}(h) = span{μ_{1}(x, y), …, μ_{q−w}(x, y)}. Then if f is k-minimal with respect to Λ(f), h is k-minimal with respect to Λ̃(h).

Proof. Define f̂ : M^{m} → ℝ^{p+q−k} ≡ ℝ^{p} × ℝ^{q−k} and û_{i} : M^{m} → ℝ^{p+q−k} ≡ ℝ^{p} × ℝ^{q−k}, i = 1, …, k by f̂(x) = (f(x), 0) and û_{i}(x) = (u_{i}(x), 0). For i = 1, …, q − k put ê_{i} = (0, e_{i}) ∈ ℝ^{p} × ℝ^{q−k}, where {e_{1}, …, e_{q−k}} is the standard basis for ℝ^{q−k}. Then

h(x, y) = f̂(x) + ∑_{i=1}^{k} g_{i}(y)û_{i}(x) + ∑_{i=1}^{q−k} g_{k+i}(y)ê_{i}.

and

μ_{j}(x, y) = ∑_{i=1}^{k} ψ_{j,i}(y)û_{i}(x) + ∑_{i=1}^{q−k} ψ_{j,k+i}(y)ê_{i}, j = 1, …, q − w.

Now h is defined from f̂ and g as in the construction described at the start of this section, since {û_{1}, …, û_{k}, ê_{1}, …, ê_{q−k}} is an orthonormal set of parallel normal fields for f̂. For (n, z) ∈ N_{x}(f̂) ≡ N_{x}(f) × ℝ^{q−k}, the shape operator of f̂ in the direction (n, z), Â_{(n,z)} : T_{x}M → T_{x}M is given by Â_{(n,z)} = A_{n}, the shape operator of f in the direction n. So ∀i = 1, …, k, Âû_{i} = A_{u_{i}}, and ∀i = 1, …, q − k, Âê_{i} = 0 is the zero map.

(1): By Theorem 1.1 of [1], h is an immersion, since 1 − ∑_{i=1}^{k} g_{i}(y)Âû_{i} − ∑_{i=1}^{q−k} g_{k+i}(y)Âê_{i} = 1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}}, and this map is nonsingular, by hypothesis.

(2): Again using Theorem 1.1 of [1], ξ ∈ N_{(x,y)}(h) if and only if ξ = n̂ + ∑_{i=1}^{k} η_{i}û_{i}(x) + ∑_{i=1}^{q−k} η_{k+i}ê_{i}, where n̂ ∈ N_{x}(f̂), <n̂, û_{i}(x)> = 0 ∀i = 1, …, k and <n̂, ê_{i}> = 0 ∀i = 1, …, q−k, and η = (η_{1}, …, η_{q}) ∈ N_{y}(g). Now n̂ = (n, z) for some n ∈ N_{x}(f), z ∈ ℝ^{q−k}. So the given condition on n̂ is satisfied if and only if n̂ = (n, 0) and <n, u_{i}(x)> = 0 for i = 1, …, k. Hence ξ = (n + ∑_{i=1}^{k} η_{i}u_{i}(x), ∑_{i=1}^{q−k} η_{k+i}e_{i}) ∈ N_{x}(f) × ℝ^{q−k}.

(3): From (2), for j = 1, …, q − w, μ_{j}(x, y) ∈ N_{(x,y)}(h). Also μ_{j} is parallel in the normal bundle of h since u_{1}, …, u_{k} are parallel in the normal bundle of f and ψ_{1}, …, ψ_{q−w} are parallel in the normal bundle of g. Further <μ_{j}(x, y), μ_{i}(x, y)> = ∑_{ℓ=1}^{q}(ψ_{j,ℓ}(y), ψ_{i,ℓ}(y)) = <ψ_{j}(y), ψ_{i}(y)> = δ_{ji}. Therefore {μ_{1}, …, μ_{q−w}} is an orthonormal set of parallel normal fields for h.

(4): Now ξ is as in (2), and Theorem 1.1 of [1] gives

C_{ξ}(X, Y) = ((1 − ∑_{i=1}^{k} g_{i}(y)Âû_{i} − ∑_{i=1}^{q−k} g_{k+i}(y)Âê_{i})^{−1}Â_{v}X, B_{η}Y)

= ((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v}X, B_{η}Y)
and as in [1], traceC_{ξ} = trace((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v}) + traceB_{η}.

(5): Note that (3) shows that Λ̃(h) is a global parallel flat (q − w)-plane subbundle of the normal bundle of h. Let μ ∈ Λ̃_{(x,y)}(h) = span{μ_{1}(x, y), …, μ_{q−w}(x, y)}. So μ = (∑_{i=1}^{k} η_{i}u_{i}(x), ∑_{i=1}^{q−k} η_{k+i}e_{i}) for some η = (η_{1}, …, η_{q}) ∈ N_{y}(g). Therefore from (4), C_{μ}(X, Y) = ((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v}X, B_{η}Y), where v = ∑_{i=1}^{k} η_{i}u_{i}. Now ξ is orthogonal to Λ̃_{(x,y)}(h) if and only if ξ = n̂ = (n, 0) ∈ N_{x}(f) × ℝ^{q−k} and from (4),

Cn̂(X, Y) = ((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{n}X, 0).

Hence, for ℓ = 1, 2, …,

(C_{μ})^{ℓ}Cn̂ = ((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v})^{ℓ}(1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{n}), 0).

So, as in [1], trace(C_{μ})^{ℓ}Cn̂ = trace((1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{v})^{ℓ}(1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}A_{n}. Now A_{v} = ∑_{i=1}^{k} η_{i}A_{u_{i}}, therefore, as trace is linear, trace(C_{μ})^{ℓ}Cn̂ is a sum of terms of the form α(i_{1}, …, i_{k}) ∑ traceDA_{u_{j_{1}}}DA_{u_{j_{2}}} ⋯ DA_{u_{j_{k}}} DA_{n}, where D = (1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}})^{−1}, i_{1}, …, i_{k} ∈ {1, …, k}, α(i_{1}, …, i_{k}) ∈ ℝ and the sum is taken over all permutations j_{1}, …, j_{k} of i_{1}, …, i_{k}. But, up to a constant multiple, this term is \frac{∂^{ℓ}}{∂t_{i_{1}}···∂t_{i_{k}}} trace(1−∑_{i=1}^{k} t_{i}A_{u_{i}})^{−1}A_{n} at (t_{1}, …, t_{k}) = (g_{1}(y), …, g_{k}(y)), and, since f is k-minimal with respect to Λ(f), trace(1−∑_{i=1}^{k} t_{i}A_{u_{i}})^{−1}A_{n}= 0, for all (t_{1}, …, t_{k}) for which (1 − ∑_{i=1}^{k} t_{i}A_{u_{i}}) is nonsingular. Hence trace(C_{μ})^{ℓ}Cn̂ = 0 and therefore h is (q − k)-minimal with respect to Λ̃(h). □

If M and W are compact, f : M^{m} → ℝ^{p} is a 1-minimal immersion with respect to a parallel normal field and g : W^{w} → ℝ^{w+k} has codimension k and flat normal bundle then, replacing g by ϵg for some sufficiently small ϵ > 0 if necessary, as in [1], to ensure that the map h : M × W → ℝ^{p+w} is an immersion it follows from Theorem 2.4 that h is k-minimal with respect to a global parallel flat normal subbundle. This immersion can then be used as the base immersion f for further constructions. For example, if f is k-minimal and g has codimension k then the constructed h will be k-minimal.

3. Umbilical normal directions

Immersions with an umbilical normal direction can be used to construct A_{k}-immersions and k-minimal immersions for different values of k.

Let f : M^{m} → ℝ^{p} be an immersion and let v : M^{m} → ℝ^{p} be a parallel normal field for f such that A_{v} = λ1, λ : M → ℝ. Let g : W^{w} → ℝ^{q} be an immersion such that, ∀ x ∈ M, 1 − g_{1}(y)A_{v} is nonsingular at x, and let h : M × W → ℝ^{p+q−1} be defined by

h(x, y) = (f(x) + g_{1}(y)v(x), ∑_{i=1}^{q−k} g_{1+i}(y)e_{i})

where {e_{1}, …, e_{q−1}} is the standard basis for ℝ^{q−1}. By Theorem 2.4, h is an immersion.

Let Π be a linear subspace of N_{x}(f) = Π ⊕ Π^{⊥} and, for (x, y) ∈ M × W, let Π̃ = {(u + η_{1}v(x), ∑_{i=1}^{q−k} η_{1+i}e_{i}) : u ∈ Π, <u, v(x)> = 0, η ∈ N_{y}(g)}. Then, by Theorem 2.4, Π̃ ⊂ N_{(x,y)}(h) = Π̃ ⊕ Π̃^{⊥}.

Theorem 3.1 With f, v, g, h, Π, Π̃ as above, for x ∈ M let the shape operators of f satisfy the condition :

∀u ∈ Π, ∀n ∈ Π^{⊥}, traceA_{u}^{r}A_{n} = 0 for r = 0, 1, …, ℓ.

Then, for (x, y) ∈ M × W the shape operators of h satisfy the condition :

∀ũ ∈ Π̃, ∀ñ ∈ Π̃^{⊥}, traceCr ; ũCñ = 0 for r = 0, 1, …, ℓ.

Proof. Let ũ = (u + η_{1}v(x), ∑_{i=1}^{q−1} η_{1+i}e_{i}) ∈ Π̃, where u ∈ Π, <u, v(x)> = 0, η ∈ N_{y}(g)}. As in Theorem 2.4,

Cũ(X, Y) = ((1 − g_{1}(y)A_{v})^{−1}A_{u+η_{1}v}X, B_{η}Y) = (\frac{1}{1 − λg_{1}(y)}A_{u+η_{1}v}X, B_{η}Y).

Let ñ ∈ Π̃^{⊥}, then ñ = (n, 0) for some n ∈ Π^{⊥}. So

Cñ(X, Y) = ((1 − g_{1}(y)A_{v})^{−1}A_{n}X, 0) = (\frac{1}{1 − λg_{1}(y)}A_{n}X, 0).

Hence

traceCℓ ; ũCñ =\frac{1}{(1 − λg_{1}(y))^{ℓ+1}} traceA_{u+η_{1}v}^{ℓ}A_{n}
=\frac{1}{(1 − λg_{1}(y))^{ℓ+1}} ∑_{r=0}^{ℓ} \binom{ℓ}{r}(λη_{1})^{ℓ−r}traceA_{u}^{r}A_{n} = 0 since u ∈ Π, n ∈ Π^{⊥}. □

Corollary 3.2 With f, v, g, h as at the start of this section, let f be an A_{k}-immersion. Then h is an A_{k+q−w}-immersion. Further, if ∀x ∈ M, v(x) ∈ span{A_{0}(H), …, A_{k−1}(H)}, where H is the mean curvature vector of f, then h is an A_{k+q−w−1}-immersion.

Proof. At x ∈ M, put Π = span{A_{0}(H), …, A_{k−1}(H), v(x)}. Then for u ∈ Π, u = ζ +αv(x), where ζ ∈ span{A_{0}(H), …, A_{k−1}(H)}, α ∈ ℝ. For n ∈ Π^{⊥}, traceA_{n} = 0, since <n, H> = 0 and traceA_{u}A_{n} = traceA_{ζ}A_{n} + λα traceA_{n} = 0, since ζ ∈ span{A_{0}(H), …, A_{k−1}(H)} and A_{k}(H) = 0. Therefore by Theorem 3.1 with ℓ = 1, ∀ũ ∈ Π̃, ñ ∈ Π̃^{⊥}, traceCñ = 0 and traceCũCñ = 0.

Hence by Lemma 1.2 h is an A_{k+q−w}-immersion, since, ∀x ∈ M, dimΠ̃ ≤ k + q − w. If, ∀x ∈ M, v(x) ∈ span{A_{0}(H), …, A_{k−1}(H)} then, ∀x ∈ M, dimΠ̃ ≤ k + q − w − 1 and therefore h is an A_{k+q−w−1}-immersion. □

In the particular case where g is a codimension one immersion and, ∀x ∈ M, v(x) ∈ span{A_{0}(H), …, A_{k−1}(H)} then h is an A_{k}-immersion. If f is A_{1}, g has codimension one and ∀x ∈ M, v(x) ∉ {αH : α ∈ ℝ} then h is an A_{2}-immersion. For examples of A_{k}-immersions with umbilical normal directions, see [2]. A product of circles 𝕊^{1}(a) × 𝕊^{1}(b) × 𝕊^{1}(c) with radii a, b, c is 1-Chen if a = b = c, 2-Chen if two of a, b, c are equal and 3-Chen if no two of a, b, c are equal. This manifold lies on a 5-sphere in ℝ^{6} and the radial normal direction is umbilical. Taking the inclusion of this embedding into ℝ^{ℓ}, ℓ > 6, gives more umbilical parallel normal directions. These examples can also be used in the next Corollary.

Corollary 3.3 With f, v, g, h as at the start of this section, let f be k-minimal with respect to a parallel flat subbundle Λ(f) of N(f) and let g have flat normal bundle. If, ∀x ∈ M v(x) ∉ Λ_{x}(f) then h is (k+q−w)-minimal. If, ∀x ∈ M v(x) ∈ Λ_{x}(f) then h is (k+q−w−1)- minimal.

Proof. For x ∈ M, put Π_{x} = {ζ + αv(x) : ζ ∈ Λ_{x}(f), α ∈ ℝ}, then Π̃_{(x,y)} = {(u + η_{1}v(x), ∑_{i=1}^{q−1} η_{1+i}e_{i}) : u ∈ Π_{x}(f), η ∈ N_{y}(y)}. For u ∈ Π_{x}, u = ζ + αv(x), and n ∈ Π_{x}^{⊥}, traceA_{u}^{ℓ}A_{n} = ∑_{r=0}^{ℓ} \binom{ℓ}{r}(λα)^{ℓ−r}traceA_{ζ}^{r}A_{n} = 0 for ℓ = 0, 1, …, since ζ ∈ Λ_{x}(f) and n ∈ Λ_{x}^{⊥}(f) and f is k-minimal with respect to Λ(f).

Therefore, by Theorem 3.1, ∀ũ ∈ Π̃_{(x,y)}, ñ ∈ Π̃_{(x,y)}^{⊥}, traceCℓ ; ũCñ = 0 for ℓ = 0, 1, …. Let Π̃(h) denote the subbundle of h with fibre Π̃_{(x,y)} at (x, y) ∈ M × W. Note that dimΠ̃_{(x,y)} = k+q−w if v(x) ∉ Λ_{x}(f) and dimΠ̃_{(x,y)} = k+q−w−1 if v(x) ∈ Λ_{x}(f). Also Π̃(h) is a parallel flat subbundle of N_{x}(f). Hence h is (k + q − w)-minimal (resp. (k + q − w − 1)-minimal) with respect to Π̃(h) if, ∀x ∈ M, v(x) ∉ Λ_{x}(f) (resp. v(x) ∈ Λ_{x}(f)). □

References

[1] Carter, S.; Dursun, U.: Partial tubes and Chen submanifolds. University of Leeds, Pure Mathematics 1995 Preprint Series, No. 25.

[2] Dursun, U.: On product k-Chen submanifolds. University of Leeds, Pure Mathematics 1995 Preprint Series, No. 27.

[3] Li, S. J.; Houh, C. S.: Generalized Chen submanifolds. Journal of Geometry 48 (1993), 144–156.

[4] Palais, R. S.; Terng, C. L.: Critical point theory and submanifold geometry. Lecture Notes in Mathematics 1353, Springer, New York, 1988.

[5] Rouxel, B.: A-submanifolds in Euclidean space. Kodai Math. J. 4 (1981), 181–188.

Received December 7, 1995