Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 427-436.

Decomposition of Isometric Immersions between Warped Products

Michael Schaaf

Mathematisches Institut der Universität zu Köln Weyertal 86-90, D-50931 Köln, Germany

Abstract. A decomposition theorem for isometric immersions between two arbitrary warped products is derived. Thereby the well known decomposition theorems of Moore and Nölker are generalized.

Introduction

One is often interested in the decomposition of mathematical objects as a first step for a classification. The decomposition of riemannian manifolds into riemannian products is well described by the theorem of de Rham, see [7]. Moore proved in [4, Lemma] the decomposition of an isometric immersion f : ∏_{i=0}^{k}M_{i} → ℝ^{n} from a riemannian product into the euclidean n-space ℝ^{n}, which is based on the fact that there exist many product decompositions ℝ^{n} ≃ ℝ^{n_{0}} × ⋯ × ℝ^{n_{k}} . This theorem was an essential step in Ferus’ classification of symmetric submanifolds of ℝ^{n}, see [3]. The real space forms N_{κ}^{n} of constant curvature κ ≠ 0 do not admit local representations by riemannian products. In order to generalize Ferus’ result to N_{κ}^{n} Backes and Reckziegel introduced extrinsic products Φ : ∏_{i=0}^{k}N_{i} → N_{κ}^{n}, special isometric immersions of higher codimension, see [1]; and Takeuchi made use of a generalization of Moore’s lemma to the Lorentz space, see [10, Lemma 1.4]. But Nölker proved in [6, Theorem 7] that the best substitute for the product representations of ℝ^{n} are the local representations of N_{κ}^{n} by warped products (see Definition 2). Herewith, he could generalize Moore’s result to isometric immersions of warped products, see [6, Theorem 16]:

Theorem. Let f : M → N_{κ}^{n} be an isometric immersion of a connected warped product M = M_{0} ×_{ρ} ∏_{i=1}^{k}M_{i} into a real space form N_{κ}^{n} of constant curvature κ, and suppose that the second fundamental form of f satisfies the condition h|E_{i} ×_{M} E_{j} ≡ 0 for i ≠ j, where

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

the E_{i} are the subbundles of the tangent bundle T M tangential to the factors M_{i}. Then there exist an isometry Φ : N_{0} ×_{σ} ∏_{i=1}^{k}N_{i} → G from a warped product N_{0} ×_{σ} ∏_{i=1}^{k}N_{i} onto an open, dense subset G ⊂ N_{κ}^{n} and isometric immersions f_{i} : M_{i} → N_{i} such that

f = Φ ∘ (f_{0} × … × f_{k}) .

Using this result, for instance, Dillen and Nölker could classify normally flat semi-parallel submanifolds of real space forms, see [2].

This article is devoted to generalizations of Nölker’s result, where the ambient space is an arbitrary pseudoriemannian manifold. To formulate them we make use of the notions of WP-nets and net morphisms, the first of which has been introduced in [5] and specializes the notion of nets.

Definition 1. 1. A family E = (E_{i})_{i=0,...,k} of totally integrable subbundles E_{i} of the tangent bundle T M of a manifold M is called a net on M if and only if T M is the direct sum ⨁_{i=0}^{k}E_{i}. The foliation induced by E_{i} shall be denoted by L_{i}^{E} and its leaf through some point p ∈ M by L_{i}^{E}(p) . If in this situation M is pseudoriemannian, each subbundle E_{i} is non-degenerate, E_{i} ⊥ E_{j} for i ≠ j and moreover for every i = 1, …, k the subbundle E_{i} is spherical^{1} and its orthogonal complement E_{i}^{⊥} is autoparallel^{2} then the net E is called a WP-net on M and (M, E) a WP-netted manifold.

2. A C^{∞}-map f : M → N is called a net morphism with respect to some nets E = (E_{i})_{i=0,...,k} on M and F = (F_{i})_{i=0,...,k} on N if and only if f_{∗}E_{i}(p) ⊂ F_{i}(f(p)) for every p ∈ M , which is equivalent to the fact that f maps every leaf of the foliation L_{i}^{E} differentiably into a leaf of the foliation L_{i}^{F} (i = 0, …, k).

It was proved in [5, Proposition 4] that the product net of a warped product is a WP-net, and in [5, Corollary 1], conversely, that a WP-netted manifold (M, E) is locally a warped product, that means for every p ∈ M there exist an open neighbourhood U of p in M , a warped product N = N_{0} ×_{ρ} ∏_{i=1}^{k}N_{i} and an isometry Φ : N → U which is a net morphism with respect to the product net of N and E . Therefore, isometric net morphisms between WP-netted manifolds can locally be decomposed in the sense of the following Proposition 1, which motivates us to derive conditions for their existence.

In general the occurance of WP-nets cannot be guaranteed; for instance on an irreducible simply connected riemannian symmetric space of non-constant curvature there exist no WPnets, see [9, Theorem 15]. Hence, to get a chance that Nölker’s theorem keeps valid for arbitrary pseudoriemannian manifolds N instead of N_{κ}^{n}, we have to suppose the existence of a WP-net on N , which is adapted to the immersion f at least at one point. In fact, this is the crucial point as is proved by the following main results of this article.

1A non-degenerate subbundle E of T M is said to be spherical if and only if there exists a section H^{E} ∈ Γ(E^{⊥}) with the following properties: for every X, Y ∈ Γ(E) we have (∇_{X}Y)^{E^{⊥}} = ⟨X, Y ⟩ · H^{E} and (∇_{X}H)^{E^{⊥}} = 0 , where (·)^{E^{⊥}} denotes the orthogonal projection T M → E^{⊥}; H^{E} is called the mean curvature normal of E.

2A non-degenerate subbundle E of T M is said to be autoparallel if and only if ∇_{X}Y ∈ Γ(E) for every X, Y ∈ Γ(E), that means if and only it is spherical with vanishing mean curvature normal.

Theorem 1. Let M = M_{0} ×_{ρ} ∏_{i=1}^{k}M_{i} be a connected warped product with product net E = (E_{i})_{i=0,...,k}, N a pseudoriemannian manifold, which is equipped with a WP-net F = (F_{i})_{i=0,...,k}, and p̄ ∈ M some point. Furthermore, let f : M → N be an isometric immersion satisfying the following conditions

h|E_{i} ×_{M} E_{j} ≡ 0 for i ≠ j ,  (D)
f(L_{i}^{E}(p̄)) ⊂ L_{i}^{F}(f(p̄)) for every i = 0, …, k.  (I)

Then f is a net morphism, and therefore f can locally be decomposed in the manner of Proposition 1.

Combining Theorem 1 with Proposition 1 we immediately obtain:

Corollary 1. Let f : M → N be an isometric immersion between two warped products M = M_{0} ×_{ρ} ∏_{i=1}^{k}M_{i} and N = N_{0} ×_{σ} ∏_{i=1}^{k}N_{i}. If M is connected, f satisfies the conditions (D) and (I) of Theorem 1 with respect to the product nets E and F of M and N , respectively, and some point p̄ ∈ M and if the representations of M and N as warped products are normalized with respect to p̄ and its image f(p̄) (see Definition 2), respectively, then there exist isometric immersions f_{i} : M_{i} → N_{i} for i = 0, …, k such that

f = f_{0} × ⋯ × f_{k} .

As Nölker’s examples show, the domain of a WP-net often is only an open subset of the considered manifold. The following result enables us to handle such situations in general.

Theorem 2. Let N be a pseudoriemannian manifold, Φ : N_{0} ×_{σ} ∏_{i=1}^{k}N_{i} → G an isometry from a warped product onto an open subset G of N and F the WP-net on G induced by Φ . Furthermore, let M := M_{0} ×_{ρ} ∏_{i=1}^{k}M_{i} be a connected warped product with product net E and f : M → N an isometric immersion satisfying the conditions (D) and (I) of Theorem 1 with respect to the given nets E and F and some point p̄ ∈ M . Then the image f(M) lies completely in G; if moreover the representations of M and G as warped products are normalized with respect to p̄ and Φ^{−1}(f(p̄)) , respectively, then there exist isometric immersions f_{i} : M_{i} → N_{i} for i = 0, …, k such that

f = Φ ∘ (f_{0} × ⋯ × f_{k}) .

Acknowledgements. The author would like to express his heartiest thanks to Professor H. Reckziegel for his encouragement for this article and valuable hints and comments.

1. Warped product nets and net morphisms

Firstly, we recall the definition of warped products and show some basic results.

Definition 2. Let M_{0}, …, M_{k} be manifolds, M := ∏_{i=0}^{k}M_{i} the product and π_{i} : M → M_{i} the canonical projections (i = 0, …, k). A pseudoriemannian metric ⟨·, ·⟩ on M is said to be a warped product metric if there exist pseudoriemannian metrics ⟨·, ·⟩_{0}, …, ⟨·, ·⟩_{k} on M_{0}, …, M_{k}, respectively, and a C^{∞}-function ρ := (ρ_{1}, …, ρ_{k}) : M_{0} → ℝ_{+}^{k} such that for every X, Y ∈ 𝔛(M) we have

⟨X, Y ⟩ = ⟨π_{0∗}X, π_{0∗}Y ⟩_{0} + ∑_{i=1}^{k}(ρ_{i} ∘ π_{0})^{2} · ⟨π_{i∗}X, π_{i∗}Y ⟩_{i} .  (1)

In this situation the pseudoriemannian manifold (M, ⟨·, ·⟩) is called a warped product; we denote it by M_{0} ×_{ρ} ∏_{i=1}^{k}M_{i} and call ρ its warping function. If we fix some point p̄ ∈ M , then we can choose the representing data ⟨·, ·⟩_{i} and ρ_{i} of ⟨·, ·⟩ in such a way that we have ρ(p̄_{0}) = (1, …, 1) ; in this situation we say that the representation of M as a warped product is normalized with respect to p̄.

Remark 1. In [5, Theorem 2] there are derived sufficient conditions for a WP-netted manifold to be a warped product.

In the proof of the main results we make use of the following rules for the Levi-Civita connection of a WP-netted manifold:

Lemma 1. Let E = (E_{0}, E_{1}) be a WP-net on a pseudoriemannian manifold M , H ∈ Γ(E_{0}) the mean curvature normal of E_{1} and T M → E_{i} , v ↦ v^{i} the orthogonal projection (i = 0, 1). Then the following assertions are true.

(R1) ∀X ∈ Γ(E_{0}), Y ∈ Γ(E_{1}) : ∇_{X}Y ∈ Γ(E_{1})
(R2) ∀X ∈ 𝔛(M), Y ∈ Γ(E_{1}) : (∇_{X}Y)^{0} = ⟨X, Y ⟩ H
(R3) ∀X ∈ Γ(E_{1}), Y ∈ Γ(E_{0}) : (∇_{X}Y)^{1} = −⟨Y, H⟩ X

(R4) If α : I → M is an E_{1}-integral C^{∞}-curve, t_{0} ∈ I and X ∈ 𝔛_{α}(M) fulfills the

equation ∇_{∂}X = −⟨X, H ∘ α⟩ α̇ with X(t_{0}) ∈ E_{0}(α(t_{0})) , then X ∈ Γ_{α}(E_{0}) . (R5) If α : I → M is an E_{0}-integral C^{∞}-curve, t_{0} ∈ I and X ∈ 𝔛_{α}(M) is a ∇-parallel

vector field along α with X(t_{0}) ∈ E_{1}(α(t_{0})) , then X ∈ Γ_{α}(E_{1}) .

Proof. We define a linear connection ˜∇ on M by putting

˜∇_{X}Y := ∇_{X}Y − ⟨X^{1}, Y^{1}⟩ · H + ⟨X, H⟩ · Y^{1} + ⟨Y, H⟩ · X^{1}  (2)

for every X, Y ∈ 𝔛(M) . In the case that M is the warped product M = M_{0} ×_{ρ} M_{1} , ˜∇ is the Levi-Civita connection of the ordinary pseudoriemannian product M_{0} × M_{1} (see Proposition 1 in [5]). As M locally is always of this form, we obtain that each subbundle E_{i} is ˜∇-parallel. Herewith, the rules (R1)–(R3) follow immediately.

For (R4): As (∇_{∂}X^{0})^{1} =^{(R3)} −⟨X, H ∘ α⟩ α̇ = (∇_{∂}X)^{1} , we get (∇_{∂}X^{1})^{1} = 0 , that means

∇_{∂}X^{1} ∈ Γ_{α}(E_{0}) . Thus, we have ∇_{∂}X^{1} =^{(R2)} ⟨α̇, X^{1}⟩ H ∘ α, that means X^{1} fulfills some linear differential equation (more exactly: X^{1} is a parallel vectorfield along α with respect to the linear connection ∇′ := ∇ − ⟨·, ·⟩ H) with X^{1}(t_{0}) = 0 . Therefore, we get X^{1} ≡ 0 , that means X ∈ Γ(E_{0}) .

For (R5): Since ∇_{∂}X^{0} ∈ Γ_{α}(E_{0}) because of (2) and ∇_{∂}X^{1} ∈ Γ_{α}(E_{1}) because of (R1), we obtain that also X^{0} is parallel along α. As X^{0}(t_{0}) = 0 , we therefore get X^{0} ≡ 0 , that means X ∈ Γ_{α}(E_{1}) . □

We will list the main properties of isometric net morphisms between warped products in the following proposition. It can locally be applied to isometric net morphisms between arbitrary WP-netted manifolds; thus, it is of interest in combination with Theorem 1.

Proposition 1. Let M := M_{0} ×_{ρ} ∏_{i=1}^{k}M_{i} and N := N_{0} ×_{σ} ∏_{i=1}^{k}N_{i} be two warped products with their product nets E and F , respectively. Furthermore, let f : M → N be an isometric immersion, which at the same time is a net morphism with respect to E and F . Then f has a representation f = f_{0} × ⋯ × f_{k} with C^{∞}-maps f_{i} : M_{i} → N_{i} and the following assertions are true:

1. f_{0} : M_{0} → N_{0} is an isometric immersion.

2. For every i = 1, …, k the map f_{i} : M_{i} → N_{i} is a homothetical immersion, that means: there exists a constant c_{i} ∈ ℝ_{+} such that ⟨f_{i∗}X, f_{i∗}Y ⟩_{i} = c_{i}^{2} · ⟨X, Y ⟩_{i} for all X, Y ∈ 𝔛(M_{i}) . Moreover, we have ρ_{i} ≡ c_{i} · σ_{i} ∘ f_{0} . If the representations of M and N as warped products are normalized with respect to some point p̄ and its image f(p̄) , then c_{1} = ⋯ = c_{k} = 1 ; in this way all immersions f_{i} become isometric.

3. f satisfies the condition (D) with respect to E .

Proof. As f(L_{i}^{E}(p)) ⊂ L_{i}^{F}(f(p)) for every p ∈ M , i = 0, …, k, it is easy to obtain the representation f = f_{0} × ⋯ × f_{k} (see the proof of [8, Lemma 1]). The assertions 1. and 2. follow immediately from the special kind of warped product metrics and the fact that f is isometric; and 3. follows from the fact that the second fundamental form of f can be expressed in terms of the second fundamental forms of f_{0}, …, f_{k} (see [6, Lemma 12]). □

Remark 2. If (M, E) and (N, F) are WP-netted manifolds and f : M → N is an isometric net morphism with respect to E and F , then we can apply locally Proposition 1 and get that f satisfies the condition (D) with respect to E . Hence, condition (D) in Theorem 1 is necessary.

2. Proof of the main results

Proof of Theorem 1. From condition (I) we get

f_{∗}E_{i}(p) ⊂ F_{i}(f(p)) for every p ∈ L_{i}^{E}(p̄), i = 0, …, k.  (3)

We give a proof by induction. Therefore, we start with the case k = 1 . Let ∇^{M} resp. ∇^{N} denote the Levi-Civita connections of M and N , H ∈ Γ(E_{0}) resp. H̃ ∈ Γ(F_{0}) the mean curvature normals of E_{1} and F_{1} , and T M → E_{i} , v ↦ v^{i} resp. T N → F_{i} , w ↦ w^{i} the orthogonal projections (i = 0, 1). We show step by step

∀p ∈ M : f_{∗}E_{1}(p) ⊂ F_{1}(f(p)),  (4)
∀p ∈ L_{0}^{E}(p̄) : H̃(f(p)) − f_{∗}H(p)∈ ⊥_{p} f,  (5)
∀p ∈ M : H̃(f(p)) − f_{∗}H(p)∈ ⊥_{p} f,  (6)
∀p ∈ M : f_{∗}E_{0}(p) ⊂ F_{0}(f(p)) .  (7)

Because of (4) and (7) we then get that f is a net morphism with respect to E and F .

For (4): Let p = (p_{0}, p_{1}) ∈ M be an arbitrary point. Then the point q := (p̄_{0}, p_{1}) satisfies q ∈ L_{1}^{E}(p̄) ∩ L_{0}^{E}(p) . Hence, there exists a C^{∞}-curve α : [0, 1] → L_{0}^{E}(p) such that α(0) = p and α(1) = q. For a fixed vector v ∈ E_{1}(p) let X be the ∇^{M}-parallel vector field along α with X(0) = v. From rule (R5) (see Lemma 1) we get X ∈ Γ_{α}(E_{1}) , hence, f_{∗}X(1) ∈ F_{1}(f(q)) because of (3). Therefore, we have ∇^{N}_{∂}f_{∗}X = f_{∗}∇^{M}_{∂}X+h(α̇, X) = 0 and

0 = (∇^{N}_{∂}f_{∗}X)^{0} = (∇^{N}_{∂}(f_{∗}X)^{0})^{0} + (∇^{N}_{∂}(f_{∗}X)^{1})^{0} =^{(R2)} (∇^{N}_{∂}(f_{∗}X)^{0})^{0} + ⟨f_{∗}α̇, (f_{∗}X)^{1}⟩ H̃ ∘ f ∘ α.

Since moreover

0 = ⟨α̇, X⟩ = ⟨f_{∗}α̇, f_{∗}X⟩ = ⟨(f_{∗}α̇)^{0}, (f_{∗}X)^{0}⟩ + ⟨(f_{∗}α̇)^{1}, (f_{∗}X)^{1}⟩,

we obtain

(∇^{N}_{∂}(f_{∗}X)^{0})^{0} = ⟨(f_{∗}α̇)^{0}, (f_{∗}X)^{0}⟩ H̃ ∘ f ∘ α.

Thus, (f_{∗}X)^{0} is a solution of some linear differential equation (more exactly: (f_{∗}X)^{0} is a parallel section in the vector bundle F_{0} along f ∘ α with respect to the linear connection ∇′ given by ∇′_{Y} Z := (∇^{N}_{Y} Z)^{0}−⟨Y, Z⟩ H̃ for every Y ∈ 𝔛(N), Z ∈ Γ(F_{0}) ) with (f_{∗}X)^{0}(1) = 0 . Therefore, we get (f_{∗}X)^{0} ≡ 0, in particular f_{∗}v ∈ F_{1}(f(p)) . Herewith, (4) is proved.

We prepare the proof of (5) by the following

Lemma 2. Let M, N be two pseudoriemannian manifolds, L ⊂ M, L̃ ⊂ N two pseudoriemannian submanifolds and g : M → N an isometric immersion, such that g|L is a C^{∞}-map in L̃. Then for every p ∈ L the following implication holds true^{3}:

g_{∗}(⊥_{p} L)⊂ ⊥_{g(p)} L̃ ⇒ ∀u, v ∈ T_{p}L : hL̃(g_{∗}u, g_{∗}v) − g_{∗}h^{L}(u, v)∈ ⊥_{p} g.  (8)

Proof of Lemma 2. By assumption there exists an isometric immersion g̃ : L → L̃ such that (L̃ ↪ N) ∘ g̃ = g ∘ (L ↪ M) , thus hL̃(g̃_{∗}u, g̃_{∗}v) + hg̃(u, v) = h^{g}(u, v) + g_{∗}h^{L}(u, v) for all p ∈ L and u, v ∈ T_{p}L. Therefore, it suffices to conclude from the left hand side of (8) that hg̃(u, v)∈ ⊥_{p} g. Now we can decompose the tangent space T_{g(p)}N as

T_{g(p)}N = \underbrace{g_{∗}T_{p}M}_{=g_{∗}T_{p}L⊖g_{∗}⊥_{p} L} ⊖ ⊥_{p} g = \underbrace{T_{g̃(p)}L̃}_{=g̃_{∗}T_{p}L⊖⊥_{p}g̃} ⊖ ⊥_{g(p)} L̃ .

Since g_{∗}T_{p}L = g̃_{∗}T_{p}L we therefore obtain from g_{∗} ⊥_{p} L ⊂ ⊥_{g(p)} L̃ immediately ⊥_{p} g̃ ⊂ ⊥_{p} g, thus in particular hg̃(u, v)∈ ⊥_{p} g. □

To show now (5) let p ∈ L_{0}^{E}(p̄) be an arbitrary point. Then f_{∗}E_{0}(p) ⊂ F_{0}(f(p)) because of (3) and f_{∗}E_{1}(q) ⊂ F_{1}(f(q)) for every q ∈ L_{1}^{E}(p) because of (4). We apply Lemma 2 with g = f, L := L_{1}^{E}(p) and L̃ := L_{1}^{F}(f(p)) ; in this situation ⊥_{p} L = E_{0}(p), ⊥_{f(p)} L̃ = F_{0}(f(p)) and f|L is a C^{∞}-map in L̃. Since hL̃(f_{∗}u, f_{∗}v) = ⟨f_{∗}u, f_{∗}v⟩ H̃(f(p)) = ⟨u, v⟩ H̃(f(p)) and f_{∗}h^{L}(u, v) = ⟨u, v⟩ f_{∗}H(p) for every u, v ∈ T_{p}L, we conclude with u = v ≠ 0 from Lemma 2: H̃(f(p)) − f_{∗}H(p)∈ ⊥_{p} f .

For (6): Let p = (p_{0}, p_{1}) ∈ M be an arbitrary point. Then the point q := (p_{0}, p̄_{1}) satisfies q ∈ L_{0}^{E}(p̄) ∩ L_{1}^{E}(p) . Hence, there exists a C^{∞}-curve α : [0, 1] → L_{1}^{E}(p) such that α(0) = p, α(1) = q. For an arbitrary vector v ∈ E_{1}(p) we have f_{∗}v ∈ F_{1}(f(p)) , thus ⟨H̃(f(p)), f_{∗}v⟩ = 0 . Moreover ⟨f_{∗}H(q), f_{∗}v⟩ = ⟨H(p), v⟩ = 0 . Therefore, it suffices to show ⟨H̃(f(p)) − f_{∗}H(p), f_{∗}v⟩ = 0 for every v ∈ E_{0}(p) . Let v ∈ E_{0}(p) be an arbitrary vector and X the vector field along α, which satisfies ∇_{∂}X = −⟨X, H ∘ α⟩ and X(0) = v. Then we have X ∈ Γ_{α}(E_{0}) because of (R4) and ∇^{N}_{∂}f_{∗}X = f_{∗}∇^{M}_{∂}X + h^{f}(α̇, X) = −⟨H ∘ α, X⟩ f_{∗}α̇. As ⟨H̃ ∘ f ∘ α − f_{∗}H ∘ α, f_{∗}X⟩(1) = 0 because of (5), it suffices to show

3We denote the normal bundle of an isometric immersion f resp. submanifold L by ⊥f resp. ⊥L.

∂⟨H̃ ∘ f ∘ α − f_{∗}H ∘ α, f_{∗}X⟩ = 0.  (9)

Now we regard the splitting

∂⟨H̃ ∘ f ∘ α − f_{∗}H ∘ α, f_{∗}X⟩ = \underbrace{⟨∇^{N}_{∂}(H̃ ∘ f ∘ α), f_{∗}X⟩}_{=:A}

− \underbrace{⟨∇^{N}_{∂}(f_{∗}H ∘ α), f_{∗}X⟩}_{=:B} + \underbrace{⟨H̃ ∘ f ∘ α − f_{∗}H ∘ α, ∇^{N}_{∂}f_{∗}X⟩,}_{=:C}

with A, B, C ∈ C^{∞}([0, 1]) . Since

C = ⟨H̃ ∘ f ∘ α − f_{∗}H ∘ α, −⟨H ∘ α, X⟩ f_{∗}α̇⟩

and α is E_{1}-integral, thus f ∘ α is F_{1}-integral, we obtain C ≡ 0 .

Furthermore ∇^{N}_{∂}(f_{∗}H ∘ α) = f_{∗}∇^{M}_{∂}(H ∘ α) + h(α̇, H ∘ α) = f_{∗}∇^{M}_{∂}(H ∘ α) , thus B = ⟨f_{∗}∇^{M}_{∂}(H ∘ α), f_{∗}X⟩. Since H ∘ α is parallel in E_{0} , we have ∇^{M}_{∂}(H ∘ α) ∈ Γ_{α}(E_{1}) and hence B ≡ 0 .

Since H̃ ∘ f ∘ α is parallel in F_{0}, we have ∇^{N}_{∂}(H̃ ∘ f ∘ α) ∈ Γ_{f∘α}(F_{1}), thus A = ⟨∇^{N}_{∂}(H̃ ∘ f ∘ α), (f_{∗}X)^{1}⟩. Because of ⟨H̃ ∘ f ∘ α, (f_{∗}X)^{1}⟩ ≡ 0 we obtain

0 =∂⟨H̃ ∘ f ∘ α, (f_{∗}X)^{1}⟩
=⟨∇^{N}_{∂}(H̃ ∘ f ∘ α), (f_{∗}X)^{1}⟩ + ⟨H̃ ∘ f ∘ α, ∇^{N}_{∂}(f_{∗}X)^{1}⟩.  (10)

Since moreover f ∘ α is F_{1}-integral, we get

(∇^{N}_{∂}(f_{∗}X)^{1})^{0} =^{(R2)} ⟨f_{∗}α̇, f_{∗}X⟩ H̃ ∘ f ∘ α = 0 ,

and with (10) we conclude immediately A ≡ 0 . Herewith, (9) and therefore also (6) is proved completely.

For (7): Let p = (p_{0}, p_{1}) ∈ M be an arbitrary point. Then the point q := (p_{0}, p̄_{1}) satisfies q ∈ L_{0}^{E}(p̄) ∩ L_{1}^{E}(p) . Hence, there exists a C^{∞}-curve α : [0, 1] → L_{1}^{E}(p) such that α(0) = p, α(1) = q. Let v ∈ E_{0}(p) be an arbitrary vector and X ∈ 𝔛_{α}(M) be the vector field along α which satisfies ∇_{∂}X = −⟨X, H ∘α⟩ and X(0) = v. Then we have X ∈ Γ_{α}(E_{0}) because of (R4). Using (6) we obtain

∇^{N}_{∂}f_{∗}X = f_{∗}∇_{∂}X + h(α̇, X) = −⟨X, H ∘ α⟩ f_{∗}α̇
= −⟨f_{∗}X, f_{∗}H ∘ α⟩ f_{∗}α̇ = −⟨f_{∗}X, H̃ ∘ f ∘ α⟩ f_{∗}α̇ ∈ Γ_{f∘α}(F_{1}) .

Since f_{∗}X(1) ∈ f_{∗}E_{0}(q) ⊂ F_{0}(f(q)) because of (3), we therefore obtain from (R4): f_{∗}X ∈ Γ_{f∘α}(F_{0}) , in particular f_{∗}v ∈ F_{0}(f(p)) .

Herewith, the theorem in the case k = 1 is proved completely.

In the proof by induction for the cases k > 1 in two situations we make use of the following lemma, which is a certain associativity of the property that a map is a net morphism, see [8, Proposition 2]. For that we define that a net E = (E_{i})_{i=0,...,k} on a manifold M is said to be locally decomposable if for every p ∈ M there exists an open neighbourhood U of p in M and a diffeomorphism Φ : U → N onto a product manifold N = ∏_{i=0}^{k}N_{i} which is a net morphism with respect to E|_{U} and the product net of N ; this is satisfied, for instance, if (M, E) is a WP-netted manifold.

Lemma 3. Let M̂ := ∏_{i=0}^{k}M̂_{i} be a connected product manifold with product net Ê := (Ê_{i})_{i=0,...,k} and p̂ ∈ M̂ some point. Let N be another manifold which is equipped with a net F̂ = (F̂_{i})_{i=0,...,k}. Suppose further that there exist two locally decomposable nets Ê′ = (Ê′_{j})_{j=0,...,l}, F̂′ = (F̂′_{j})_{j=0,...,l}, which are refinements of Ê and F̂, that means there exists a partition {0, …, l} = ˙∪_{i=0}^{k}I_{i} such that Ê_{i} = ⊕_{j∈I_{i}} Ê′_{j} and F̂_{i} = ⊕_{j∈I_{i}} F̂′_{j} for every i = 0, …, k. If in this situation f : M → N is a net morphism with respect to Ê and F̂ and moreover

∀i = 0, …, k, p ∈ LÊ ; i(p̂), j ∈ I_{i} : f_{∗}Ê′_{j}(p) ⊂ F̂′_{j}(f(p)) ,  (11)

then f is a net morphism with respect to the refinements Ê′ and F̂′.

Assuming that the theorem is true for some k ≥ 1 let a connected warped product M = M_{0} ×_{ρ} ∏_{i=1}^{k+1}M_{i} with product net E = (E_{i})_{i=0,...,k+1} , a WP-net F = (F_{i})_{i=0,...,k+1} on N and an isometric immersion f : M → N with the quoted properties be given. Then we define the connected warped product M̃ := M_{0} ×_{(ρ_{1},...,ρ_{k})} ∏_{i=1}^{k}M_{i}, its product net Ẽ = (Ẽ_{i})_{i=0,...,k}, the point p̃:= (p̄_{0}, …, p̄_{k}) and the totally geodesic immersion φ : M̃ → M , (p_{0}, …, p_{k}) ↦ (p_{0}, …, p_{k}, p̄_{k+1}) . The second fundamental form h̃ of the isometric immersion f̃:= f ∘ φ : M̃ → N is given by h̃ = h ∘ (φ_{∗} × φ_{∗}) . As φ_{∗}Ẽ_{i}(p) = E_{i}(φ(p)) for every p ∈ M̃ and i = 0, …, k, f̃ satisfies the condition (D) with respect to Ẽ. Further, we define the WP-

net F̃ = (F_{0} ⊕ F_{k+1}, F_{1}, …, F_{k}) on N . As φ(LẼ ; i(p̃)) = L_{i}^{E}(p̄) for every i = 0, …, k,

f̃ satisfies the condition (I) with respect to p̃ and the nets Ẽ and F̃. Therefore, we can apply the theorem in the case k to f̃ and obtain that f̃ is a net morphism with respect to Ẽ and F̃. If we define the WP-net Ẽ′ := (Ẽ_{0}, …, Ẽ_{k}, 0) on M̃ (where 0 represents the trivial subbundle of M̃ ), then we can apply Lemma 3 using the partition {0, …, k + 1} = {0, k + 1} ˙∪{1} ⋯ ˙∪{k} and the dictionary

Lemma 3 M̂ p̂ f Ê F̂ Ê′ F̂′ ; Here  M̃ p̃ f̃ Ẽ F̃ Ẽ′ F

and obtain that f̃ is a net morphism with respect to Ẽ′ and F ; notice that (11) is satisfied in this situation because f̃_{∗}Ẽ_{i}(p) = f_{∗}E_{i}(φ(p)) ⊂ F_{i}(f̃(p)) for every p ∈ LẼ ; i(p̃) and i = 0, …, k. In particular we now even know that for every point p ∈ M̃ we have

∀i = 0, …, k : f_{∗}E_{i}(φ(p)) = f̃_{∗}Ẽ_{i}(p) ⊂ F_{i}(f̃(p)) ,  (12)

hence f̃_{∗}T_{p}M̃ ⊂ F_{k+1}^{⊥}(f̃(p)) . If now E′ and F′ are the WP-nets on M resp. N given by E′ = (E′_{0}, E′_{1}) = (E_{k+1}^{⊥}, E_{k+1}) and F′ = (F′_{0}, F′_{1}) = (F_{k+1}^{⊥}, F_{k+1}), then we therefore conclude f(L_{0}^{E′}(p̄)) = f̃(M̃) ⊂ L_{0}^{F′}(f(p̄)) . By assumption of the theorem we also have f(L_{1}^{E′}(p̄)) = f(L_{k+1}^{E}(p̄)) ⊂ L_{k+1}^{F}(f(p̄)) = L_{1}^{F′}(f(p̄)) . Furthermore, f trivially satisfies also condition (D) with respect to p̄ and the net E′, and E′ in fact is the product net of the WP-representation M = M̃ ×_{ρ′} M_{k+1} of M , where ρ′ is given by ρ′(p_{0}, …, p_{k}) := ρ_{k+1}(p_{0}) . Thus, we can apply the theorem in the case k = 1 to f and obtain that f is a net morphism with respect to E′ and F′. Since because of (12) and (3) we moreover have f_{∗}E_{i}(p) ⊂ F_{i}(f(p)) for every p ∈ L_{0}^{E′}(p̄) = φ(M̃), i = 0, …, k and f_{∗}E_{k+1}(p) ⊂ F_{k+1}(f(p)) for every p ∈ L_{1}^{E′}(p̄) = L_{k+1}^{E}(p̄) , we can apply Lemma 3 again using now the partition {0, …, k + 1} = {0, …, k} ˙∪{k + 1} and the dictionary

Lemma 3  M̂  p̂ f Ê F̂ Ê′ F̂′ ; Here  M̃ ×_{ρ′} M_{k+1} p̄ f E′ F′ E F

and obtain that f is a net morphism with respect to E and F . □

Proof of Theorem 2. The crucial point is the proof of the inclusion f(M) ⊂ G. All other assertions are an immediate consequence of Corollary 1 applied to the map Φ^{−1} ∘f . But this inclusion has already been proved in the setting of arbitrary nets in [8, Proposition 3], if two hypotheses are satisfied: The first hypothesis (i) is identical with condition (I); the second hypothesis (ii) means that for every family (U_{i})_{i=0,...,k} of connected open subsets U_{i} ⊂ M_{i} with p̄ ∈ U := ∏_{i=0}^{k}U_{i} and f(U) ⊂ G the restriction f|U : U → G is a net morphism with respect to E|_{U} and F ; this is satisfied in our situation because of Theorem 1. □

References

[1] Backes, E.; Reckziegel,H.: On symmetric submanifolds of spaces of constant curvature. Math. Ann. 263 (1984), 419–433.

[2] Dillen, F.; Nölker, S.: Semi-parallelity, multi-rotation surfaces and the helix property. J. reine angew. Math. 435 (1993), 33–63.

[3] Ferus, D.: Produkt-Zerlegungen von Immersionen mit paralleler zweiter Fundamentalform. Math. Ann. 211 (1974), 1–5.

[4] Moore, J. D.: On isometric immersions of Riemannian products. J. Differential Geom. 5 (1971), 159–168.

[5] Meumertzheim, M.; Reckziegel, H.; Schaaf, M.: Decomposition of twisted and warped product nets. To appear in Results Math.

[6] Nölker, S.: Isometric immersions of warped products. Diff. Geom. Appl. 6 (1996), 1–30.

[7] de Rham, G.: Sur la réductibilité d’un espace de Riemann. Comment. Math. Helv. 26 (1952), 328–344.

[8] Reckziegel, H.; Schaaf, M.: De Rham decomposition of netted manifolds. Results Math. 35 (1999), 175–191.

[9] Schaaf, M.: On decomposition theorems in differential geometry with special regard to the theory of warped products. Doctoral Thesis, Cologne 1998.

436 M. Schaaf: Decomposition of Isometric Immersions between Warped Products

[10] Takeuchi, M.: Parallel submanifolds of space forms. Manifolds and Liegroups, Papers in honor of Y. Matsushima, Birkhäuser 1981, 429–447.

Received October 10, 1999