ISOMETRIC IMMERSIONS IN CODIMENSION TWO OF WARPED PRODUCTS INTO SPACE FORMS

MARCOS DAJCZER AND RUY TOJEIRO

Abstract. We provide a local classification of isometric immersions f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+k} in codimensions k = 1, 2 of warped products of Riemannian manifolds into space forms, under the assumptions that n ≥ k + 1 and that N^{p+n} = L^{p} ×_{ρ} M^{n} has no points with the same constant sectional curvature c as the ambient space form.

1. Introduction

A basic decomposition theorem due to Nölker [16] states that an isometric immersion f: N^{p+n} = L^{p} ×_{ρ} M^{n} → ℚ_{c}^{ℓ} of a warped product of connected Riemannian manifolds with warping function ρ ∈ C^{∞}(L^{p}) into a complete simply-connected space form of constant sectional curvature c is a warped product of isometric immersions (see [16] or Section 1 for the precise definition of this concept) whenever its second fundamental form α: T N ×T N → T^{⊥}N satisfies

α(X, V) = 0 for all X ∈ T L and V ∈ T M.

This generalizes a well-known result for isometric immersions of Riemannian products into Euclidean space due to Moore [14] as well as its extension by Molzan [13] for nonflat ambient space forms; see also [18].

It is a natural problem to understand the possible cases in which the isometric immersion f may fail, locally or globally, to be a warped product of isometric immersions. In high codimensions the warped product structure of the manifold does not seem to place enough restrictions on the isometric immersion in order to make possible a complete classification in either case. Even in the much more restrictive situation of Riemannian products, a successful local analysis has only been carried out in the case in which the codimension is two and the first factor is one-dimensional; see [3]. This was used therein to characterize isometric immersions f: L^{p} × M^{n} → ℝ^{p+n+2} of complete Riemannian products none of whose factors is everywhere flat (see Remark 35 below), carrying through the global results previously obtained by Moore [14] and Alexander-Maltz [1]. Based on earlier work due to Moore, such submanifolds were shown in [1] to split as products of hypersurfaces under the global assumption that they do not carry an Euclidean strip.

1991 Mathematics Subject Classification. 53B25.

The main goal of this paper is to provide a local classification of isometric immersions f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+k} in codimensions k = 1, 2, under the assumptions that n ≥ k + 1 and that N^{p+n} = L^{p} ×_{ρ} M^{n} has no points with the same constant sectional curvature c as the ambient space form. In the case of codimension k = 1, we prove that f must be a warped product of isometric immersions. In codimension k = 2 we show that only two other possibilities may arise. Namely, either f is a composition of a warped product of isometric immersions into ℚ_{c}^{p+n+1} with a local isometric immersion of ℚ_{c}^{p+n+1} into ℚ_{c}^{p+n+2} or N^{p+n} is a Riemannian manifold of a special type that admits a second decomposition as a warped product with respect to which f splits as a warped product of isometric immersions. We leave the precise statement for Section 3, where we also state its corresponding version for the case of Riemannian products.

We give examples showing that the restriction on the dimension of M^{n} is necessary. As for the hypothesis that N^{p+n} has no points with constant sectional curvature c, we observe that for manifolds with constant sectional curvature the assumption that they be warped products places no further restrictions on them since any such manifold can be realized as a warped product in many possible ways; see the discussion on warped product representations of space forms in Section 1. Therefore, for Riemannian manifolds with constant sectional curvature our problem reduces to classifying all isometric immersions in codimension two of such manifolds into space forms. In this regard, recall that if f: U ⊂ ℚ n ; c̃ → ℚ_{c}^{n+2}, n ≥ 4, is an isometric immersion, then c̃ ≥ c and, for c̃ > c, away from the set of umbilical points the immersion must be locally a composition of the umbilical inclusion of ℚ n ; c̃ into ℚ_{c}^{n+1} with a local isometric immersion of ℚ_{c}^{n+1} into ℚ_{c}^{n+2} (see [10], [12] and [7]). In fact, the latter statement can be derived from our main theorem, but in that result we exclude from our analysis the case of local isometric immersions of ℚ_{c}^{n} into ℚ_{c}^{n+2} with the same constant sectional curvature. A local description of these isometric immersions when c = 0 was given in [6].

As striking applications of our main result, we obtain that if L^{p} is a Riemannian manifold no open subset of which can be isometrically immersed into ℚ_{c}^{p+1}, then any isometric immersion f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 3, is either a cylindrical submanifold of Euclidean space or a rotational submanifold. Moreover, if L^{p} cannot be locally isometrically immersed in ℚ_{c}^{p+2}, then L^{p}×_{ρ}M^{n} cannot be locally isometrically immersed in ℚ_{c}^{p+n+2} either for whatever Riemannian manifold M^{n} of dimension n ≥ 3 and warping function ρ.

Acknowledgment. We are greatly indebted to the referee whose many suggestions and comments have decisively contributed for significant improvements in both the presentation and the mathematical content of this article.

2. Preliminaries

In this section we establish our notation and state some basic facts on warped products of Riemannian manifolds and their isometric immersions into the standard real space forms.

Given a a vector bundle E over a Riemannian manifold N, we denote by Γ(E) the set of all locally defined smooth sections of E. If N = L × M is a product manifold, we denote by H and V the horizontal and vertical subbundles of T N, that is, the distributions on N correspondent to the product foliations determined by L and M, respectively. Elements of Γ(H) will always be denoted by the letters X, Y, Z, whereas those in Γ(V) by the letters U, V, W. The same applies to individual tangent vectors. A vector field X ∈ Γ(H) (resp., V ∈ Γ(V)) is said to be the lift of a vector field X̃ ∈ Γ(T L) (resp., Ṽ ∈ Γ(T M)) if π_{L}_{∗}X = X̃ ∘ π_{L} (resp., π_{M}_{∗}V = Ṽ ∘ π_{M}), where π_{L}: L × M → L (resp., π_{M}: L × M → M) is the canonical projection onto L (resp., M). We denote the set of all lifts of vector fields in L (resp., M) by L(L) (resp., L(M)), and always denote vector fields in L and M with a tilde and use the same letters without the tilde to represent their lifts to N.

If L and M are Riemannian manifolds with Riemannian metrics ⟨ , ⟩_{L} and ⟨ , ⟩_{M}, respectively, the warped product N = L ×_{ρ} M with warping function ρ ∈ C^{∞}(L) is the product manifold L × M endowed with the warped product metric

⟨ , ⟩ = π_{L}^{∗}⟨ , ⟩_{L} + (ρ ∘ π_{L})^{2}π_{M}^{∗}⟨ , ⟩_{M}.

We always assume N to be connected. The Levi-Civita connections of N, L and M are related by (cf. [17])

(1)  ∇_{X}Y is the lift of ∇L ; X̃ Ỹ ,
(2)  ∇_{X}V = ∇_{V} X = −⟨X, η⟩V,
(3)  (∇_{V} W)_{V} is the lift of ∇M ; Ṽ W̃ ,
(4)  (∇_{V} W)_{H} = ⟨V, W ⟩η,

where X, Y ∈ L(L), V, W ∈ L(M) and η = − grad (log ρ∘π_{L}). Here and in the sequel, writing a vector field with a vector subbundle as a subscript indicates taking the section of that vector subbundle obtained by orthogonally projecting the vector field pointwise onto the corresponding fiber of the subbundle. Observe that the formula ∇_{X}V = −⟨X, η⟩V (resp., ∇_{V} X = −⟨X, η⟩V) is tensorial in X (resp., V), hence it also holds for horizontal (resp., vertical) vector fields that are not necessarily lifts. On the other hand, it characterizes vertical (resp., horizontal) vector fields that are lifts.

Recall that a vector subbundle E of T N is called totally geodesic or autoparallel if ∇_{X}Y ∈ Γ(E) for all X, Y ∈ Γ(E). It is called totally umbilical if there exists a vector field η ∈ Γ(E^{⊥}) such that (∇_{X}Y)_{E^{⊥}} = ⟨X, Y ⟩η for all X, Y ∈ Γ(E). If, in addition, the so called mean curvature normal η of E satisfies (∇_{X}η)_{E^{⊥}} = 0 for all X ∈ Γ(E), then E is said to be spherical. A totally umbilical vector subbundle E of T N is automatically integrable, and its leaves are totally umbilical submanifolds of N. If E is totally geodesic or spherical then the leaves are totally geodesic or spherical submanifolds of N, respectively. By a spherical submanifold we mean a totally umbilical submanifold whose mean curvature vector field is parallel in the normal connection.

It follows from (1) and (4), respectively, that H is totally geodesic and that V is totally umbilical with mean curvature normal η = − grad (log ρ ∘ π_{L}). Moreover, since η is a gradient vector field and H is totally geodesic we have

⟨∇_{V} η, X⟩ = ⟨∇_{X}η, V ⟩ = 0

for all X ∈ Γ(H) and V ∈ Γ(V), and hence V is spherical. The following extension due to Hiepko of the well-known decomposition theorem of de Rham shows that these properties characterize warped products.

Theorem 1. ([11]) Let N be a Riemannian manifold and let T N = H ⊕ V be an orthogonal decomposition into nontrivial vector subbundles such that H is totally geodesic and V is spherical. Then, for every point z_{0} ∈ N there exist an isometry Ψ of a warped product L ×_{ρ} M onto a neighborhood of z_{0} in N such that Ψ(L × {x}) and Ψ({y} × M) are integral manifolds of H and V, respectively, for all y ∈ L and x ∈ M. Moreover, if N is simply connected and complete then the isometry Ψ can be taken onto all of N.

Given a warped product N = L ×_{ρ} M, the lift of the curvature tensor ^{M}R of M to N is the tensor whose value at E_{1}, E_{2}, E_{3} ∈ T_{z}N is the unique vector in V_{z} that projects to ^{M}R(π_{M}_{∗}E_{1}, π_{M}_{∗}E_{2})π_{M}_{∗}E_{3} in T_{π_{M} (z)}M. The lift of the curvature tensor ^{L}R of L is similarly defined. Then the curvature tensors of L, M and N are related by

(5)  R(X, Y)Z = ^{L}R(X, Y)Z,
(6)  R(X, Y)V = R(V, W)X = 0,
(7)  R(X, U)V = ⟨U, V ⟩(∇_{X}η − ⟨η, X⟩η),
(8)  R(V, W)U = ^{M}R(V, W)U − ∥η∥^{2}(⟨W, U⟩V − ⟨V, U⟩W).

Since ∇_{X}η − ⟨η, X⟩η ∈ H because H is totally geodesic, all the information of (7) is contained in

(9)  ⟨R(X, V)W, Y ⟩ = ⟨V, W ⟩⟨∇_{X}η − ⟨X, η⟩η, Y ⟩.

The starting point for the proof of the main results of this paper is the observation that the curvature relations (5) to (8) impose several restrictions on the second fundamental form α: T N × T N → T^{⊥}N of an isometric immersion f: N → ℚ_{c}^{ℓ} when combined with the Gauss equation for f.

Proposition 2. Let f: L×_{ρ} M → ℚ_{c}^{ℓ} be an isometric immersion of a warped product. Then the curvature-like tensor

C(E_{1}, E_{2}, E_{3}, E_{4}) : = ⟨R(E_{1}, E_{2})E_{3}, E_{4}⟩ − c⟨(E_{1} ∧ E_{2})E_{3}, E_{4}⟩
(10)  = ⟨α(E_{1}, E_{4}), α(E_{2}, E_{3})⟩ − ⟨α(E_{1}, E_{3}), α(E_{2}, E_{4})⟩

satisfies

(11)  C(X, V, W, Y) = ⟨V, W ⟩⟨∇_{X}η − ⟨X, η⟩η − cX, Y ⟩,
(12)  C(X, Y, V, Z) = 0,
(13)  C(X, Y, V, W) = 0,
(14)  C(X, U, V, W) = 0.

We now introduce the notion of a warped product of isometric immersions into ℚ_{c}^{ℓ} which plays a fundamental role in this paper. This relies on the warped product representations of ℚ_{c}^{ℓ}, that is, isometries of warped products onto open subsets of ℚ_{c}^{ℓ}. All such isometries were described by Nölker for warped products with arbitrarily many factors; see [16] for details. In particular, any isometry of a warped product with two factors onto an open subset of ℚ_{c}^{ℓ} arises as a restriction of an explicitly constructible isometry

Ψ: V^{ℓ−m}(⊂ ℚ_{c}^{ℓ−m}) ×_{σ} ℚ m ; c̃ → ℚ_{c}^{ℓ}

onto an open dense subset of ℚ_{c}^{ℓ}, where ℚ m ; c̃ is a complete spherical submanifold of ℚ_{c}^{ℓ} and V^{ℓ−m} is an open subset of the unique totally geodesic submanifold ℚ_{c}^{ℓ−m} of ℚ_{c}^{ℓ} (of constant sectional curvature c if ℓ − m ≥ 2) whose tangent space at some point z̄ ∈ ℚ m ; c̃ is the orthogonal complement of the tangent space of ℚ m ; c̃ at z̄. The isometry Ψ is, in fact, completely determined by the choice of ℚ m ; c̃ and of a point z̄ ∈ ℚ m ; c̃, and it is called the warped product representation of ℚ_{c}^{ℓ} determined by (z̄, ℚ m ; c̃). If c ≠ 0, we consider the standard model of ℚ_{c}^{ℓ} as a complete spherical submanifold of 𝕆^{ℓ+1}, where 𝕆^{ℓ+1} denotes either Euclidean space ℝ^{ℓ+1} if c > 0 or Lorentzian space 𝕃^{ℓ+1} if c < 0. Then, for c ≠ 0 the warping function σ is the restriction to V^{ℓ−m} of the height function z ↦ ⟨z, a⟩ in 𝕆^{ℓ+1}, where −a is the mean curvature vector of ℚ m ; c̃ in 𝕆^{ℓ+1} at z̄. Similarly, if c = 0 then σ(z) = 1 + ⟨z − z̄, a⟩, where −a is the mean curvature vector of ℚ m ; c̃ in ℚ_{c}^{ℓ} = ℝ^{ℓ} at z̄. In every case ⟨a, a⟩ = c̃.

Definition. Let Ψ: V^{ℓ−m} ×_{σ} ℚ m ; c̃ → ℚ_{c}^{ℓ} be a warped product representation, let h_{1} : L → V^{ℓ−m} and h_{2} : M → ℚ m ; c̃ be isometric immersions, and let ρ = σ ∘h_{1}. Then the isometric immersion f = Ψ∘(h_{1} ×h_{2}): N = L×_{ρ} M → ℚ_{c}^{ℓ} is called the warped product of the isometric immersions h_{1} and h_{2} determined by Ψ.

[figure: L ×_{ρ=σ∘h_{1}} M; h_{1}; ℚ_{c}^{ℓ}; h_{2}; ⟳; f = Ψ ∘ (h_{1} × h_{2}); V^{ℓ−m} ×_{σ} ℚ m ; c̃; Ψ]

Example 3. If N = L×_{ρ}M is not a Riemannian product and h_{2} is an isometry, then f is called a rotational submanifold with profile h_{1}. This means that V^{ℓ−m} is a half-space of a totally geodesic submanifold ℚ_{c}^{ℓ−m} ⊂ ℚ_{c}^{ℓ} bounded by a totally geodesic submanifold ℚ_{c}^{ℓ−m−1} and f(N) is the submanifold of ℚ_{c}^{ℓ} generated by the action on h_{1}(L) of the subgroup of isometries of ℚ_{c}^{ℓ} that leave ℚ_{c}^{ℓ−m−1} invariant.

Example 4. If N = L×_{ρ}M is not a Riemannian product and h_{1} : L → V^{ℓ−m} is a local isometry then, for c = 0, we have that f(N) is contained in the product of an Euclidean factor ℝ^{ℓ−m−1} with a cone in ℝ^{m+1} over h_{2}. If c ≠ 0, then f(N) is the union of open subsets of the totally geodesic submanifolds of ℚ_{c}^{ℓ} through the points of h_{2}(M) ⊂ ℚ m ; c̃ whose tangent spaces at the points of h_{2}(M) are the normal spaces of ℚ m ; c̃ in ℚ_{c}^{ℓ}.

Notice that any warped product of isometric immersions in codimension one must be as in one of the preceding examples. In codimension two only a third possibility arises, namely, the case in which both h_{1} and h_{2} are hypersurfaces.

Important special cases of warped products of isometric immersions arise as follows. Let ℚ_{c_{1}}^{ℓ_{1}} and ℚ_{c_{2}}^{ℓ_{2}} be complete spherical submanifolds of ℚ_{c}^{ℓ} through a fixed point z̄ ∈ ℚ_{c}^{ℓ} whose tangent spaces at z̄ are orthogonal and whose mean curvature vectors ψ_{1} and ψ_{2} at z̄ satisfy ⟨ψ_{1}, ψ_{2}⟩ = −c and ψ_{1} (resp., ψ_{2}) is orthogonal to Tz̄ℚ_{c_{2}}^{ℓ_{2}} (resp., Tz̄ℚ_{c_{1}}^{ℓ_{1}}). Let Ψ: V^{ℓ−ℓ_{2}} ×_{σ} ℚ_{c_{2}}^{ℓ_{2}} → ℚ_{c}^{ℓ} be the warped product representation of ℚ_{c}^{ℓ} determined by (z̄, ℚ_{c_{2}}^{ℓ_{2}}). Then ℚ_{c_{1}}^{ℓ_{1}} ⊂ V^{ℓ−ℓ_{2}} and σ ∘ i = 1, where i: ℚ_{c_{1}}^{ℓ_{1}} → V^{ℓ−ℓ_{2}} is the inclusion map. The warped product Ψ ∘ (i × id): ℚ_{c_{1}}^{ℓ_{1}} × ℚ_{c_{2}}^{ℓ_{2}} → ℚ_{c}^{ℓ} of the inclusion and the identity map is an isometric embedding called the isometric embedding of the Riemannian product ℚ_{c_{1}}^{ℓ_{1}} × ℚ_{c_{2}}^{ℓ_{2}} into ℚ_{c}^{ℓ} as an extrinsic Riemannian product.

The structure of the second fundamental form of a warped product of isometric immersions is described in the following result.

Proposition 5. Let N = L ×_{ρ} M and N̄ = L̄ ×ρ̄ M̄ be warped product manifolds, and let F: L → L̄ and G: M → M̄ be isometric immersions with ρ = ρ̄ ∘ F. Then f = F × G: N → N̄ is an isometric immersion and at z = (y, x) ∈ N we have

(i) πL̄_{∗}f_{∗}T_{z}N = F_{∗}T_{y}L, πL̄_{∗}T_{z}^{⊥}N = T_{y}^{⊥}L, πM̄_{∗}f_{∗}T_{z}N = G_{∗}T_{x}M, πM̄_{∗}T_{z}^{⊥}N = T_{x}^{⊥}M.

(ii) (grad ρ̄(F (y)))_{T_{y}^{⊥}L} = grad ρ̄(F (y)) − F_{∗}grad ρ (y).

(iii) The second fundamental forms of F, G and f are related by

(15)  πM̄_{∗}α^{f}(E_{1}, E_{2}) = α^{G}(π_{M}_{∗}E_{1}, π_{M}_{∗}E_{2}),

(16)  \begin{matrix}πL̄_{∗}α^{f}(E_{1}, E_{2}) = α^{F} (π_{L}_{∗}E_{1}, π_{L}_{∗}E_{2})− \\ ρ (y)⟨π_{M}_{∗}E_{1}, π_{M}_{∗}E_{2}⟩(grad ρ̄(F (y)))_{T_{y}^{⊥}L}. \end{matrix}

Given an isometric immersion f: N → N̄, a normal vector ζ ∈ T_{z}^{⊥}N is called a principal curvature normal vector at z if the subspace

Δ_{ζ}(z) = {T ∈ T_{z}N : α(T, E) = ⟨T, E⟩ζ for all E ∈ T_{z}N}

is nontrivial. In this case Δ_{ζ}(z) is called the eigenspace correspondent to ζ. If ζ = 0 then Δ(z) := Δ_{0}(z) is called the relative nullity subspace of f at z.

Corollary 6. Let f be an isometric immersion as in Proposition 5. Then at z = (y, x) ∈ N we have

(i) V_{z} ⊂ Δ_{ζ}(z) for a principal curvature normal vector ζ ∈ T_{z}^{⊥}N if and only if G is umbilical at x with mean curvature vector πM̄_{∗}ζ and πL̄_{∗}ζ = −ρ^{−1}(grad ρ̄)_{T_{y}^{⊥}L}. In particular, we have that V_{z} ⊂ Δ(z) if and only if G is totally geodesic at x and (grad ρ̄)_{T_{y}^{⊥}L} = 0.

(ii) H_{z} ⊂ Δ(z) if and only if F is totally geodesic at y.

Given a vector a ≠ 0 in either ℝ^{ℓ} or 𝕆^{ℓ+1}, according as c = 0 or c ≠ 0, let U be the vector field on ℚ_{c}^{ℓ} defined by U_{z} = a − c⟨a, z⟩z and let F^{a} be the 1-dimensional totally geodesic distribution generated by U on the open dense subset W^{a} = {z ∈ ℚ_{c}^{ℓ} : U_{z} ≠ 0}. Notice that ℚ_{c}^{ℓ} \ W^{a} is empty for c = 0 as well as for (c < 0, ⟨a, a⟩ ≥ 0), it contains one point for (c < 0, ⟨a, a⟩ < 0) and two points for c > 0. Observe also that for c = 0 (resp., c ≠ 0) the vector field U is the gradient of the function σ: ℚ_{c}^{ℓ} → ℝ given by σ(z) = 1 + ⟨z − z̄, a⟩ for a fixed z̄ ∈ ℚ_{c}^{ℓ} (resp., σ(z) = ⟨z, a⟩), which was used in the definition of a warped product representation of ℚ_{c}^{ℓ}. We say that an isometric immersion g: L^{p} → ℚ_{c}^{ℓ} is cylindrical with respect to a if g(L) ⊂ W^{a} and F^{a} is everywhere tangent to g(L), or equivalently, if U_{g(y)} = grad σ(g(y)) is nonzero and tangent to g(L) for any y ∈ L. The last assertion in Corollary 6-(i) yields the following.

Corollary 7. Let f = Ψ ∘ (F × G): N = L ×_{ρ} M → ℚ_{c}^{ℓ} be a warped product of isometric immersions, where Ψ: V^{ℓ−m} ×_{σ} ℚ m ; c̃ → ℚ_{c}^{ℓ} is a warped product representation determined by (z̄, ℚm ; c̃). If G is totally geodesic and F is cylindrical with respect to the mean curvature vector −a of ℚm ; c̃ at z̄ in either ℝ^{ℓ} or 𝕆^{ℓ+1}, according as c = 0 or c ≠ 0, then the vertical subbundle of T N is contained in the relative nullity subbundle of f. Conversely, if the the vertical subbundle of T N is contained in the relative nullity subbundle of f then G is totally geodesic and F |_{U} is cylindrical with respect to the mean curvature vector −a of ℚm ; c̃ at z̄, where U is the open subset of L where grad ρ does not vanish.

If f: L ×_{ρ} M → ℚ_{c}^{ℓ} is a warped product of isometric immersions, then it follows from Proposition 5-(ii) that at any point z ∈ L × M its second fundamental form satisfies

(17)  α(X, V) = 0 for all X ∈ H_{z} and V ∈ V_{z}.

The following theorem due to Nölker states that the converse is also true. Recall that the spherical hull of an isometric immersion G: M → ℚ_{c}^{ℓ} is the complete spherical submanifold of least dimension that contains G(M).

Theorem 8. ([16]) Let f: L ×_{ρ} M → ℚ_{c}^{ℓ} be an isometric immersion of a warped product whose second fundamental form satisfies condition (17) everywhere. For a fixed point (ȳ, x̄) ∈ L ×_{ρ} M with ρ(ȳ) = 1, let F: L → ℚ_{c}^{ℓ} and G: M → ℚ_{c}^{ℓ} be given by F (y) = f(y, x̄) and G(x) = f(ȳ, x), and let ℚ m ; c̃ be the spherical hull of G. Then (f(ȳ, x̄), ℚ m ; c̃) determines a warped product representation Ψ: V^{ℓ−m} ×_{σ} ℚ m ; c̃ → ℚ_{c}^{ℓ} such that F (L) ⊂ V^{ℓ−m} and f = Ψ ∘ (F × G), where in the last equation F and G are regarded as maps into V^{ℓ−m} and ℚ m ; c̃, respectively.

The preceding theorem is also valid for isometric immersions of warped products with arbitrarily many factors (see [16]). It contains as a particular case the following result due to Molzan (cf. Corollary 17 of [16]), which is an extension to nonflat ambient space forms of the main lemma in [14].

Corollary 9. ([13]) Let f: L × M → ℚ_{c}^{ℓ} be an isometric immersion of a Riemannian product whose second fundamental form satisfies condition (17) everywhere. For a fixed point (ȳ, x̄) ∈ L×M define F: L → ℚ_{c}^{ℓ} and G: M → ℚ_{c}^{ℓ} by F (y) = f(y, x̄) and G(x) = f(ȳ, x), and denote by ℚ_{c_{1}}^{ℓ_{1}} and ℚ_{c_{2}}^{ℓ_{2}} the spherical hulls of F (L) and G(M), respectively. Then F and G are isometric immersions and there exists an isometric embedding Φ: ℚ_{c_{1}}^{ℓ_{1}} × ℚ_{c_{2}}^{ℓ_{2}} → ℚ_{c}^{ℓ} as an extrinsic Riemannian product such that f = Φ ∘ (F × G), where in the last equation F and G are regarded as maps into ℚ_{c_{1}}^{ℓ_{1}} and ℚ_{c_{2}}^{ℓ_{2}}, respectively.

In applying Theorem 8 one must often be able to determine the dimension of the spherical hull of G. In the remaining of this section we develop a tool for computing that dimension.

Given an isometric immersion g: M^{n} → ℚ_{c}^{ℓ}, a subbundle Z̃ of the normal bundle of g is called umbilical if there exists θ ∈ Γ(Z̃) such that

(α^{g}(E_{1}, E_{2}))Z̃ = ⟨E_{1}, E_{2}⟩θ

for all E_{1}, E_{2} ∈ Γ(T M). We say that θ is the principal curvature normal of Z̃. If n ≥ 2 and the subbundle Z̃ is parallel in the normal connection, then the Codazzi equations of g imply that the vector field θ is also parallel in the normal connection. In particular, it has constant length. If g(M^{n}) is contained in a complete spherical submanifold ℚm ; c̃ of ℚ_{c}^{ℓ} with dimension m and constant sectional curvature c̃, then the pulled-back subbundle Z̃ = g^{∗}T^{⊥}ℚm ; c̃, where T^{⊥}ℚm ; c̃ is the normal bundle of ℚm ; c̃ in ℚ_{c}^{ℓ}, is an umbilical parallel subbundle of T^{⊥}M of rank ℓ − m. Conversely, we have the following result due to Yau.

Proposition 10. ([20]) Let g: M^{n} → ℚ_{c}^{ℓ}, n ≥ 2, be an isometric immersion. Assume that there exists an umbilical parallel subbundle Z̃ of T^{⊥}M with principal curvature normal θ and rank ℓ − m. Then there exists a complete spherical submanifold ℚm ; c̃ of ℚ_{c}^{ℓ} with dimension m and constant sectional curvature c̃ = c + ∥θ∥^{2} such that g(M^{n}) ⊂ ℚm ; c̃.

As a consequence, the dimension of the spherical hull of an isometric immersion can be characterized as follows.

Corollary 11. Let g: M^{n} → ℚ_{c}^{ℓ}, n ≥ 2, be an isometric immersion. Then the dimension of the spherical hull of g is m if and only if ℓ−m is the maximal rank of an umbilical parallel subbundle Z̃ of T^{⊥}M. Moreover, the spherical hull of g has constant sectional curvature c̃ = c+∥θ∥^{2}, where θ is the principal curvature normal of Z̃.

Corollary 12. Let f: N^{p+n} = L^{p} ×_{ρ} M^{n} → ℚ_{c}^{ℓ}, n ≥ 2, be an isometric immersion of a warped product whose second fundamental form satisfies (17) everywhere. Given ȳ ∈ L with ρ(ȳ) = 1, let G: M → ℚ_{c}^{ℓ} be defined by G = f ∘ iȳ, where iȳ: M^{n} → N^{p+n} given by iȳ(x) = (ȳ, x) is the (isometric) inclusion of M^{n} into N^{p+n} as a leaf of the vertical subbundle V. Then the spherical hull of G has dimension m = ℓ − p − k, where k is the maximal rank of a parallel subbundle Z of i∗ ; ȳT^{⊥}N such that

(18)  α^{f}(iȳ_{∗}V, iȳ_{∗}W)_{Z} = ⟨V, W ⟩θ

for some θ ∈ Γ(Z) and for all V, W ∈ Γ(T M). If Z is such a subbundle, then θ ∈ Γ(Z) is parallel, hence has constant length. Moreover, the spherical hull of G has constant sectional curvature c + ∥θ∥^{2} + ∥grad logρ(ȳ)∥^{2}.

Proof: The normal bundle of iȳ is i∗ ; ȳH, where H is the horizontal subbundle of T N, hence the normal bundle of G splits as

T_{G}^{⊥}M = i∗ ; ȳT^{⊥}N ⊕ f_{∗}i∗ ; ȳH

and the second fundamental form of G splits accordingly as

(19)  α^{G}(V, W) = α^{f}(iȳ_{∗}V, iȳ_{∗}W) + ⟨V, W ⟩f_{∗}(η ∘ iȳ),

where η = − grad (log ρ ∘ π_{L}) is the mean curvature normal of V. In particular, it follows that f_{∗}i∗ ; ȳH is an umbilical subbundle of T_{G}^{⊥}M with principal curvature normal f_{∗}(η ∘ iȳ). Moreover, using that the second fundamental form of f satisfies (17) it follows that f_{∗}i∗ ; ȳH is parallel in the normal connection of G. It is now easily seen that a subbundle Z of i∗ ; ȳT^{⊥}N is parallel and satisfies (18) if and only if Z ⊕ f_{∗}i∗ ; ȳH is a parallel umbilical subbundle of T_{G}^{⊥}M with principal curvature normal θ + f_{∗}(η ∘ iȳ). The conclusion follows from Corollary 11.

In the sequel only the following two special cases of Corollary 12 will be needed, in which the assumptions in part (i) (resp., (ii)) easily imply that the vector subbundle Z equals i∗ ; ȳT^{⊥}N (resp., {0}).

Corollary 13. Under the assumptions of Corollary 12 we have

(i) If the vertical subbundle V is contained in the eigendistribution correspondent to a principal curvature normal ζ of f, then the spherical hull of G has dimension m = n and constant sectional curvature c̃ = c + ∥ζ ∘ iȳ∥^{2} + ∥grad log ρ(ȳ)∥^{2}.

(ii) If there exists no local vector field ξ̄ ∈ Γ(i∗ ; ȳT^{⊥}N) such that Af ; ξ̄ ∘ iȳ_{∗} = λiȳ_{∗} for some λ ∈ C^{∞}(M), then the spherical hull of G has dimension m = ℓ − p and constant sectional curvature c̃ = c + ∥grad log ρ(ȳ)∥^{2}.

3. The results

Our main result provides a complete local classification of isometric immersions f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2} of a warped product under the assumptions that n ≥ 3 and that N^{p+n} = L^{p}×_{ρ}M^{n} is free of points with constant sectional curvature c. Here and in the sequel it is always assumed that p, n ≥ 1, and only further restrictions on those dimensions are explicitly stated.

Theorem 14. Assume that a warped product N^{p+n} = L^{p} ×_{ρ} M^{n} with n ≥ 3 is free of points with constant sectional curvature c. Then for any isometric immersion f: N^{p+n} → ℚ_{c}^{p+n+2} there exists an open dense subset of N^{p+n} each of whose points lies in an open product neighborhood U = L_{0}^{p} × M_{0}^{n} ⊂ L^{p} × M^{n} such that one of the following possibilities holds:

(i) f|_{U} is a warped product of isometric immersions with respect to a warped product representation Ψ: V^{p+k_{1}} ×_{σ}ℚ n+k_{2} ; c̃ → ℚ_{c}^{p+n+2}, k_{1} + k_{2} = 2.

[figure: L_{0}^{p} ×_{σ∘h_{1}} M_{0}^{n}; h_{1}; ℚ_{c}^{p+n+2}; h_{2}; ⟳; f|_{U} = Ψ ∘ (h_{1} × h_{2}); V^{p+k_{1}} ×_{σ} ℚ n+k_{2} ; c̃; Ψ]

(ii) f|_{U} is a composition H ∘ g of isometric immersions where g is a warped product of isometric immersions g = Ψ ∘ (h_{1} × h_{2}) determined by a warped product representation Ψ: V^{p+k_{1}} ×_{σ} ℚ n+k_{2} ; c̃ → ℚ_{c}^{p+n+1} with k_{1} + k_{2} = 1, and H: W → ℚ_{c}^{p+n+2} is an isometric immersion of an open subset W ⊃ g(U) of ℚ_{c}^{p+n+1}.

[figure: L_{0}^{p} ×_{σ∘h_{1}} M_{0}^{n}; h_{1}; ℚ_{c}^{p+n+2}; h_{2}; ⟳; f|_{U} = H ∘ Ψ ∘ (h_{1} × h_{2}); V^{p+k_{1}} ×_{σ} ℚ n+k_{2} ; c̃; Ψ; W ⊂ ℚ_{c}^{p+n+1}; H] (iii) There exist open intervals I, J ⊂ ℝ such that L_{0}^{p}, M_{0}^{n}, U split as

L_{0}^{p} = L_{0}^{p−1} ×_{ρ_{1}} I, M_{0}^{n} = J ×_{ρ_{2}} M_{0}^{n−1} and

U = L_{0}^{p−1} ×_{ρ_{1}} ((I ×_{ρ_{3}} J) ×ρ̄ M_{0}^{n−1}),

where ρ_{1} ∈ C^{∞}(L_{0}^{p−1}), ρ_{2} ∈ C^{∞}(J), ρ_{3} ∈ C^{∞}(I) and ρ̄ ∈ C^{∞}(I × J) satisfy

ρ = (ρ_{1} ∘ π_{L_{0}^{p−1}})(ρ_{3} ∘ π_{I}) and ρ̄ = (ρ_{3} ∘ π_{I})(ρ_{2} ∘ π_{J}),

and there exist warped product representations

Ψ_{1} : V^{p−1} ×_{σ_{1}} ℚ n+3 ; c̃ → ℚ_{c}^{p+n+2} and Ψ_{2} : W^{4} ×_{σ_{2}} ℚ n−1 ; c̄ → ℚ n+3 ; c̃,

an isometric immersion g: I ×_{ρ_{3}} J → W^{4} and isometries i_{1} : L_{0}^{p−1} →

W ^{p−1} ⊂ V ^{p−1} ⊂ ℚ_{c}^{p−1} and i_{2} : M^{n−1} → W ^{n−1} ⊂ ℚ n−1 ; c̄ onto open

subsets such that f|_{U} = Ψ_{1} ∘ (i_{1} × (Ψ_{2} ∘ (g × i_{2}))), ρ̄ = σ_{2} ∘ g and ρ_{1} = σ_{1} ∘i_{1}. Moreover, L_{0}^{p} has constant sectional curvature c if p ≥ 2.

[figure: W ^{4} ×_{σ_{2}} ℚ n−1 ; c̄; i_{1}; i_{2}; g; ∪; W ^{n−1}; Ψ_{2}; f|_{U} = Ψ_{1} ∘ (i_{1} × (Ψ_{2} ∘ (g × i_{2}))); V^{p−1} ×_{σ_{1}} ℚ n+3 ; c̃; ∪; W ^{p−1}; ℚ_{c}^{p+n+2}; L_{0}^{p−1}×_{ρ_{1}=σ_{1}∘i_{1}}((I×_{ρ_{3}} J)×_{ρ̄=σ_{2}∘g} M_{0}^{n−1}) = U = L_{0}^{p} ×_{ρ} M_{0}^{n}; Ψ_{1}; ⟳]

In case (iii) the isometric immersion g: I ×_{ρ_{3}} J → W^{4} is neither a warped product g = Ψ_{3} ∘ (α × β), where Ψ_{3} : V^{1+k_{1}} ×_{σ_{3}} ℚ 1+k_{2} ; ĉ → ℚ 4 ; c̃ is a warped product representation with k_{1} + k_{2} = 2 and α: I → V^{1+k_{1}} and β: J → ℚ 1+k_{2} ; ĉ are unit speed curves with ρ_{3} = σ_{3} ∘ α, nor a composition H ∘ G of such a warped product G = Ψ_{3} ∘ (α × β), determined by a warped product representation Ψ_{3} : V^{1+k_{1}} ×_{σ_{3}} ℚ 1+k_{2} ; ĉ → ℚ 3 ; c̃ as before with k_{1} + k_{2} = 1, and an isometric immersion H of an open subset W ⊃ G(I × J) into ℚ 4 ; c̃. It would be interesting to exhibit an explicit example of such an isometric immersion. Notice that it must satisfy the additional condition σ_{2} ∘ g = (ρ_{3} ∘ π_{I})(ρ_{2} ∘ π_{J}) for some ρ_{2} ∈ C^{∞}(J).

Cases (i) − (iii) are disjoint. In fact, we will prove that under the assumptions of the theorem there are three distinct possible structures for the second fundamental form of f, each of which corresponds to one of the cases in the statement.

Notice that the conclusion of the theorem remains unchanged under the apparently weaker assumption that the subset of points of N^{p+n} with constant sectional curvature c has empty interior.

Theorem 14 does not hold without the assumption that n ≥ 3. In fact, we argue next that local isometric immersions of the round three-dimensional sphere 𝕊^{3} into ℝ^{5} are generically as in neither of the cases in the statement with respect to any local decomposition of 𝕊^{3} as a warped product.

Example 15. It was shown in [8] (cf. Corollary 4 in [9]) that local isometric immersions of 𝕊^{3} into ℝ^{5} that are nowhere compositions (i.e., on no open subset they are compositions of the umbilical inclusion into ℝ^{4} with a local isometric immersion of ℝ^{4} into ℝ^{5}) are in correspondence with solutions (V, h) on open simply connected subsets U_{0} ⊂ ℝ^{3} of the nonlinear system of PDE’s

(I) \{ \begin{matrix}(i) \frac{∂V_{ir}}{∂u_{j}} = h_{ji}V_{jr}, (ii) \frac{∂h_{ik}}{∂u_{j}} = h_{ij}h_{jk}, \\ (iii) \frac{∂h_{ij}}{∂u_{i}} + \frac{∂h_{ji}}{∂u_{j}} + ∑_{k} h_{ki}h_{kj} + V_{i3}V_{j3} = 0, i ≠ j ≠ k ≠ i, \end{matrix}

called the generalized elliptic sinh-Gordon equation. Here V : U_{0} → 𝕆_{1}(3) is a smooth map taking values in the group of orthogonal matrices with respect to the Lorentz metric of signature (+, +, −) and x ∈ U_{0} ↦ h(x) is a smooth map such that h(x) is an off-diagonal (3 × 3)-matrix for every x ∈ U_{0}. More precisely, for any such isometric immersion there exist a local system of coordinates (u_{1}, u_{2}, u_{3}), an orthonormal normal frame {ξ_{1}, ξ_{2}} and matrix functions V and h as above such that

(20)  A_{ξ_{r}}X_{i} = V_{i3}^{−1}V_{ir}X_{i}, 1 ≤ r ≤ 2, 1 ≤ i ≤ 3,

and

(21)  ∇_{∂/∂u_{i}}X_{j} = h_{ji}X_{i}, 1 ≤ i ≠ j ≤ 3,

where X_{i} is a unit vector field with ∂/∂u_{i} = V_{i3}X_{i}. The compatibility equations for f are equivalent to system (I). Conversely, any solution (V, h) of system (I) on an open simply connected subset U_{0} ⊂ ℝ^{3} gives rise to such an isometric immersion by means of the fundamental theorem of submanifolds.

By a theorem of Bourlet (see [4], Théorème VIII), there exists one and only one analytic solution (V, h) of system (I) in a neighborhood of an initial value u_{0} = (u_{1}^{0}, u_{2}^{0}, u_{3}^{0}) such that V (u_{0}) ∈ 𝕆_{1}(3) and such that V_{i}^{k} and h_{ij}, i < j (resp., i > j) reduce to an arbitrarily given analytic function of u_{i} (resp., u_{j}) when the remaining variables take their initial values. Thus, for a generic local analytic solution (V, h) the functions h_{ij} are nowhere vanishing; see the last section of [9] for explicit isometric immersions with this property.

It follows easily from (20) that no such isometric immersion admits a normal vector field whose shape operator has rank one. In particular, it can not be as in case (ii). Also, if f is as in case (i) with respect to a decomposition U = L^{k_{1}} ×_{ρ} M^{k_{2}} of U as a warped product, then we must have that k_{2} = 1. In fact, otherwise the second fundamental form of f would be given by

α(Y, Z) = ⟨Y_{1}, Z_{1}⟩η_{1} + ⟨Y_{2}, Z_{2}⟩η_{2}

for some normal vector fields η_{1}, η_{2} satisfying ⟨η_{1}, η_{2}⟩ = 1 = ∥η_{2}∥, where Y_{i}, Z_{i}, 1 ≤ i ≤ 2, are the components of Y, Z according to the product decomposition of U. This easily implies that the shape operator with respect to a normal vector field orthogonal to η_{2} has rank one. Thus, the distribution tangent to the second factor is one-dimensional and invariant by all shape operators of f, and hence it must be spanned by one of the vector fields X_{i}, 1 ≤ i ≤ 3, say, X_{3}. In particular, this implies that the distribution spanned by X_{1}, X_{2} is totally geodesic, and hence the functions h_{31} and h_{32} vanish everywhere by (21). Finally, we claim that the same holds if f is as in case (iii). In effect, in this case U splits as a warped product

U = L^{1} ×_{ρ} ℚ 2 ; c̃ = L^{1} ×_{ρ} (J ×ρ̄ M^{1}) = (L^{1} ×_{ρ} J) ×_{(ρ∘π_{L^{1}} ) ρ̄} M^{1},

and f is a warped product f = g × i with respect to the last decomposition. Thus, we have again that the one-dimensional distribution tangent to M^{1} is invariant by all shape operators of f, and the same argument used in the preceding case proves our claim. It follows that f is generically as in neither of the cases in Theorem 14 with respect to any local decomposition of 𝕊^{3} as a warped product.

We now discuss some further results. The case of hypersurfaces is interesting in its own right. Although it can be proved as a corollary of Theorem 14, it is easier to derive it as an immediate consequence of Theorem 8 and Proposition 23 of Section 3.

Theorem 16. Assume that a warped product L^{p} ×_{ρ} M^{n}, n ≥ 2, has no points with constant sectional curvature c. Then any isometric immersion f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+1} is a warped product f = Ψ ∘ (F × G), where Ψ: V^{p+k_{1}} ×_{σ} ℚ n+k_{2} ; c̃ → ℚ_{c}^{p+n+1} is a warped product representation with k_{1} + k_{2} = 1 and F: L^{p} → V^{p+k_{1}}, G: M^{n} → ℚ n+k_{2} ; c̃ are isometric immersions.

Again, the preceding result is false if the assumption that n ≥ 2 is dropped; rotation surfaces in ℝ^{3} admit many isometric deformations into nonrotational surfaces (cf. [2]).

In deriving Theorem 14 we also obtain the following result for the case of Riemannian products, which extends Theorem 1 in [14] in the case of products with two factors. Therein, isometric immersions of Riemannian products with arbitrarily many factors into Euclidean space were shown to split as a product of isometric immersions under the assumptions that no factor has an open subset of flat points and that the codimension equals the number of factors. We point out that in the case of Riemannian products the factors may change the roles. This observation is applied several times throughout the paper.

Theorem 17. Let f: L^{p} × M^{n} → ℚ_{c}^{p+n+2} be an isometric immersion of a Riemannian product. If c = 0 assume that either L^{p} or M^{n} has dimension at least two and is free of flat points. If c ≠ 0 assume that either n ≥ 3 or p ≥ 3. Then there exists an open dense subset of L^{p} × M^{n} each of whose points lies in an open product neighborhood U = L_{0}^{p} × M_{0}^{n} ⊂ L^{p} × M^{n} such that one of the following possibilities holds: Case c = 0.

(i) There exist an orthogonal decomposition ℝ^{p+n+2} = ℝ^{p+k_{1}} × ℝ^{n+k_{2}} with k_{1} + k_{2} = 2 and isometric immersions h_{1} : L_{0}^{p} → ℝ^{p+k_{1}} and h_{2} : M_{0}^{n} → ℝ^{n+k_{2}} such that f|_{U} = h_{1} × h_{2}.

[figure: L_{0}^{p} × M_{0}^{n}; h_{1}; h_{2}; f|_{U} = h_{1} × h_{2}; ℝ^{p+k_{1}} × ℝ^{n+k_{2}} = ℝ^{p+n+2}]

(ii) There exist an orthogonal decomposition ℝ^{p+n+1} = ℝ^{p+k_{1}} × ℝ^{n+k_{2}}, k_{1} + k_{2} = 1, and isometric immersions h_{1} : L_{0}^{p} → ℝ^{p+k_{1}}, h_{2} : M_{0}^{n} → ℝ^{n+k_{2}} and H: W → ℝ^{p+n+2} of an open subset W ⊃ (h_{1} × h_{2})(U) of ℝ^{p+n+1} such that f|_{U} = H ∘ (h_{1} × h_{2}).

[figure: L_{0}^{p} × M_{0}^{n}; h_{1}; ℝ^{p+n+2}; h_{2}; ⟳; f|_{U} = H ∘ (h_{1} × h_{2}); ℝ^{p+k_{1}} × ℝ^{n+k_{2}} ⊃W; H] Case c ≠ 0.

(i) There exist an embedding Φ: ℚ_{c_{1}}^{p+k_{1}} ×ℚ_{c_{2}}^{n+k_{2}} → ℚ_{c}^{p+n+2} as an extrinsic Riemannian product with k_{1} + k_{2} = 1, and isometric immersions h_{1} : L_{0}^{p} → ℚ_{c_{1}}^{p+k_{1}} and h_{2} : M_{0}^{n} → ℚ_{c_{2}}^{n+k_{2}} such that f|_{U} = Φ∘(h_{1}×h_{2}).

[figure: L_{0}^{p} × M_{0}^{n}; h_{1}; h_{2}; f|_{U} = Φ ∘ (h_{1} × h_{2}); ℚ_{c_{1}}^{p+k_{1}} ×ℚ_{c_{2}}^{n+k_{2}}; ⟳; Φ; ℚ_{c}^{p+n+2}]

(ii) There exist an embedding Φ: ℚ_{c_{1}}^{p} × ℚ_{c_{2}}^{n} → ℚ_{c}^{p+n+1} as an extrinsic Riemannian product, local isometries i_{1} : L_{0}^{p} → ℚ_{c_{1}}^{p} and i_{2} : M_{0}^{n} → ℚ_{c_{2}}^{n}, and an isometric immersion H: W → ℚ_{c}^{p+n+2} of an open subset W ⊃ Φ ∘ (i_{1} × i_{2})(U) of ℚ_{c}^{p+n+1} such that f|_{U} = H ∘ Φ ∘ (i_{1} × i_{2}).

[figure: L_{0}^{p} × M_{0}^{n}; i_{1}; ℚ_{c}^{p+n+2}; i_{2}; f|_{U} = H ∘ Φ ∘ (i_{1} × i_{2}); ℚ_{c_{1}}^{p} × ℚ_{c_{2}}^{n}; W ⊂ ℚ_{c}^{p+n+1}; H; Φ; ⟳]

As an example showing that for c ≠ 0 the assumption that either n ≥ 3 or p ≥ 3 is indeed necessary, we may take any local isometric immersion of ℝ^{3} into 𝕊^{5} that is not a product α × g: I × V → 𝕊^{1}(r_{1}) × 𝕊^{3}(r_{2}), r_{1}^{2} + r_{2}^{2} = 1, where α: I → ℝ^{2} is a unit speed parametrization of an open subset of a circle of radius r_{1} and g: V → 𝕊^{3}(r_{2}) is an isometric immersion of an open subset V ⊂ ℝ^{2}. Recall that local isometric immersions of ℝ^{3} into 𝕊^{5} were shown in [19] to be in correspondence with solutions on simply connected open subsets of ℝ^{3} of the so-called generalized wave equation. As in the previous discussion on local isometric immersions of 𝕊^{3} into ℝ^{5}, one may easily argue that the class of local isometric immersions of ℝ^{3} into 𝕊^{5} that are given as products as just described is only a rather special subclass of the whole class of such isometric immersions.

We now give precise statements of the applications of Theorem 14 referred to at the end of the introduction. Recall that an isometric immersion F: L^{p} → ℚ_{c}^{p+m} is said to be locally rigid if it is rigid restricted to any open subset of L^{p}.

Corollary 18. Let L^{p} be a Riemannian manifold no open subset of which can be isometrically immersed in ℚ_{c}^{p+1}. If f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 3, is an isometric immersion, then there exist a warped product representation Ψ: V^{p+2}×_{σ}ℚ n ; c̃ → ℚ_{c}^{p+n+2}, an isometric immersion F: L^{p} → V^{p+2} and a local isometry i: M^{n} → ℚ n ; c̃ such that f = Ψ ∘ (F × i). In particular, if L^{p} is a Riemannian manifold that admits a locally rigid isometric immersion F: L^{p} → ℚ_{c}^{p+2}, then the preceding conclusion holds and, in addition, the isometric immersion f is also locally rigid.

Proof: By Theorem 14, any isometric immersion f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 3, must be locally as in one of the three cases described in its statement. However, under the assumption that L^{p} has no open subset that can be isometrically immersed in ℚ_{c}^{p+1}, it follows that f|_{U} can not be as in case (ii) on any open subset U = L_{0}^{p} × M_{0}^{n} ⊂ L^{p} × M^{n}, for there can not exist by that assumption any isometric immersion h_{1} : L_{0}^{p} → ℚ_{c}^{p+k_{1}} with 0 ≤ k_{1} ≤ 1. Moreover, f|_{U} can not be as in case (iii) either on any such open subset, for in that case L_{0}^{p} would have constant sectional curvature c, and hence it would admit locally an isometric immersion into ℚ_{c}^{p+1}. Therefore f must be globally as in case (i). The last assertion is now clear.

We say that a Riemannian manifold can be locally isometrically immersed in ℚ_{c}^{ℓ} if each point has an open neighborhood that admits an isometric immersion into ℚ_{c}^{ℓ}. Arguing in a similar way as in the proof of Corollary 18 yields the following.

Corollary 19. Let L^{p} be a Riemannian manifold that cannot be locally isometrically immersed in ℚ_{c}^{p+2}. Then L^{p} ×_{ρ} M^{n} can not be locally isometrically immersed in ℚ_{c}^{p+n+2} for any Riemannian manifold M^{n} of dimension n ≥ 3 and any warping function ρ.

In view of Nölker’s result it is also natural to study isometric immersions of warped products into space forms in the light of the following definition.

Definition. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{ℓ} be an isometric immersion of a warped product. Given z ∈ L^{p} ×_{ρ} M^{n}, we denote

α(H_{z}, V_{z}) = span{α(Y, V) : Y ∈ H_{z} and V ∈ V_{z}}.

We say that the immersion f at z is of type (A) if dim α(H_{z}, V_{z}) = 0, i.e., α(Y, V) = 0 for all Y ∈ H_{z} and V ∈ V_{z}, (B) if dim α(H_{z}, V_{z}) = 1, (C) if dim α(H_{z}, V_{z}) ≥ 2.

Notice that type A is the case of Nölker’s decomposition theorem. Therefore, it is a natural problem to determine the isometric immersions that are everywhere of type B. In the following section we obtain a complete solution to this problem in the codimension two case under the assumption that n ≥ 3 (for the case of Riemannian products it is enough to assume that p + n ≥ 3); see the two paragraphs before Proposition 27. The proof of Theorem 14 is then accomplished as follows: type C is excluded by Proposition 36, type A corresponding to Theorem 8 gives case (i), and type B splits into two subcases B_{1} and B_{2} handled in Propositions 27 and 31, respectively, which correspond to the cases (ii) and (iii). Similarly for the proof of Theorem 17: type C is excluded by Corollary 37, type A corresponding to Theorem 9 gives subcase (i) in both cases c = 0 and c ≠ 0, type B_{1} handled in Corollary 30 gives subcase (ii) in either case, and type B_{2} is excluded in either case by Corollary 34 and Corollary 32, respectively.

4. Immersions of type B

Our main goal in this section is to provide a complete local classification of isometric immersions f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2} that are everywhere of type B under the assumption that n ≥ 3. A similar classification for the special case of isometric immersions of Riemannian products is also given, for which it is enough to assume p + n ≥ 3.

First we determine the pointwise structure of the second fundamental forms of isometric immersions of type B, starting with some general facts that are valid in arbitrary codimension.

Lemma 20. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{ℓ} be an isometric immersion of a warped product. Assume that f is not of type A at a point z ∈ N^{p+n} = L^{p} ×_{ρ} M^{n} and that for every Y ∈ H_{z} the linear map

(22)  B_{Y} : V_{z} → T_{z}^{⊥}N, V ↦ α(Y, V),

satisfies rank B_{Y} ≤ 1. Then there exists a unit vector e ∈ V_{z}, uniquely determined up to its sign, such that

(23)  α(Y, V) = ⟨V, e⟩α(Y, e)

and

(24)  α(V, W) − ⟨V, e⟩⟨W, e⟩α(e, e) ⊥ α(H_{z}, V_{z})

for all Y ∈ H_{z} and V, W ∈ V_{z}.

Proof: Let X ∈ H_{z} be such that rank B_{X} = 1. Then D(X) := ker B_{X} has codimension 1 in V_{z}. Let e ∈ V_{z} be one of the unit vectors perpendicular to D(X) and write B_{X}e = λξ where λ ≠ 0 and ξ ∈ T_{z}^{⊥}N is a unit vector. Let Y ∈ H_{z} and V ∈ D(X) be arbitrary vectors. Then (13) implies

(25)  \begin{matrix}⟨B_{X}e, B_{Y} V ⟩ = ⟨α(X, e), α(Y, V)⟩ = ⟨α(X, V), α(Y, e)⟩ \\ = ⟨B_{X}V, α(Y, e)⟩ = 0. \end{matrix}

Now consider the linear map B_{X+tY} for arbitrary t ∈ ℝ. By assumption its rank is at most 1. Therefore the vectors B_{X+tY} e = λξ +tB_{Y} e and B_{X+tY} V = tB_{Y} V are linearly dependent, and hence

⟨B_{X+tY} e, B_{X+tY} e⟩⟨B_{X+tY} V, B_{X+tY} V ⟩ − ⟨B_{X+tY} e, B_{X+tY} V ⟩^{2} = 0.

As the left hand side of this equation is a polynomial ∑_{i=2}^{4} a_{i}t^{i}, its coefficients must vanish; in particular, because of (25) we obtain 0 = a_{2} = λ^{2}∥B_{Y} V ∥^{2}. Hence B_{Y} |_{D(X)} = 0, and (23) follows.

By means of (14) we derive

⟨α(V, W), α(Y, e)⟩ = ⟨α(Y, V), α(e, W)⟩ = ⟨V, e⟩⟨α(e, W), α(Y, e)⟩.

Applying this result to α(W, e) instead of α(V, W) we obtain

⟨α(V, W) − ⟨V, e⟩⟨W, e⟩α(e, e), α(Y, e)⟩ = 0,

which implies (24) in view of (23).

Lemma 21. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{ℓ} be an isometric immersion of a warped product. Assume that f is of type B at z ∈ N^{p+n} = L^{p} ×_{ρ} M^{n}. Then there exist unique, up to their signs, unit vectors X ∈ H_{z}, e ∈ V_{z} and ξ ∈ T_{z}^{⊥}N, and β, λ, γ ∈ ℝ with λ ≠ 0 such that

(26)  ⟨α(Y, Z), ξ⟩ = β⟨Y, X⟩⟨Z, X⟩,
(27)  α(Y, V) = λ⟨Y, X⟩⟨V, e⟩ξ,
(28)  ⟨α(V, W), ξ⟩ = γ⟨V, e⟩⟨W, e⟩,
(29)  \begin{matrix}⟨P̃α(Y, Z), P̃α(V, W) − ⟨V, W ⟩P̃α(e, e)⟩ \\ = (βγ − λ^{2})⟨Y, X⟩⟨Z, X⟩⟨P V, P W ⟩, \end{matrix}

where P̃: T_{z}^{⊥}N → T_{z}^{⊥}N and P: V_{z} → V_{z} denote the orthogonal projections onto the subspaces {ξ}^{⊥} ⊂ T_{z}^{⊥}N and {e}^{⊥} ⊂ V_{z}, respectively. Moreover, if N^{p+n} = L^{p} × M^{n} is a Riemannian product then

(30)  \begin{matrix}⟨P̃α(Y, Z), P̃α(V, W)⟩ + (βγ − λ^{2})⟨Y, X⟩⟨Z, X⟩⟨V, e⟩⟨W, e⟩ \\ +c⟨Y, Z⟩⟨V, W ⟩ = 0. \end{matrix}

Proof: Let ξ ∈ T_{z}^{⊥}N be a unit vector such that α(H_{z}, V_{z}) = ℝ ξ. Given Y ∈ H_{z}, then B_{Y} takes its values in ℝ ξ, and hence rank B_{Y} ≤ 1. Thus, we may apply Lemma 20. On the other hand, since the linear map H_{z} → T_{z}^{⊥}N defined by Y ↦ α(Y, e) also takes its values in ℝ ξ, it follows that, up to sign, there exists exactly one unit vector X ∈ H_{z} perpendicular to its kernel. Set γ = ⟨α(e, e), ξ⟩, λ = ⟨α(X, e), ξ⟩ and β = ⟨α(X, X), ξ⟩. Notice that λ ≠ 0 because f is of type B at z. We obtain (28) from (24), whereas (27) follows from (23) and α(Y, e) = ⟨Y, X⟩α(X, e) = λ⟨Y, X⟩ξ. Using (12) we obtain

λ⟨α(Y, Z), ξ⟩ = ⟨α(Y, Z), α(X, e)⟩ = ⟨α(X, Z), α(Y, e)⟩ = λ⟨Y, X⟩⟨α(X, Z), ξ⟩,

and applying this result to α(Z, X) instead of α(Y, Z) we end up with (26). We obtain from (11), (26), (27) and (28) that

⟨V, W ⟩⟨∇_{Y} η − ⟨Y, η⟩η − cY, Z⟩

= ⟨α(Y, Z), α(V, W)⟩ − ⟨α(Y, W), α(Z, V)⟩

= (βγ − λ^{2})⟨Y, X⟩⟨Z, X⟩⟨V, e⟩⟨W, e⟩ + ⟨P̃α(Y, Z), P̃α(V, W)⟩.

This yields (30) if N^{p+n} = L^{p} × M^{n} is a Riemannian product. In the general case, putting W = V = e we get

⟨∇_{Y} η − ⟨Y, η⟩η − cY, Z⟩ = (βγ − λ^{2})⟨Y, X⟩⟨Z, X⟩ + ⟨P̃α(Y, Z), P̃α(e, e)⟩.

The two preceding equations yield

\begin{matrix}⟨P̃α(Y, Z), P̃α(V, W) − ⟨V, W ⟩P̃α(e, e)⟩ \\ = (βγ − λ^{2})⟨Y, X⟩⟨Z, X⟩(⟨V, W ⟩ − ⟨V, e⟩⟨W, e⟩), \end{matrix}

which coincides with (29).

Remark and Definition 22. Equations (26), (27) and (28) are equivalent to

(31)  A_{ξ}Y = ⟨Y, X⟩(βX + λe), A_{ξ}V = ⟨V, e⟩(λX + γe),

and

(32)  P̃α(Y, V) = 0.

In particular, it follows from (31) that the rank of A_{ξ} at z is either 1 or 2, according to βγ−λ^{2} being zero or not. We say accordingly that f is of type B_{1} or of type B_{2} at z.

We now show that in the case of hypersurfaces f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+1}, n ≥ 2, only types A and B_{1} can occur pointwise.

Proposition 23. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+1}, n ≥ 2, be an isometric immersion of a warped product. Then, at any point z ∈ N^{p+n} = L^{p} ×_{ρ} M^{n} either f is of type A or of type B_{1}. Moreover, in the latter case N^{p+n} has constant sectional curvature c at z.

Proof: Assume that f is not of type A at z. Since n ≥ 2, we may choose a unit vector V ∈ {e}^{⊥} ⊂ V_{z}. Applying (29) for W = V and Z = Y = X, and using that P̃ = 0, it follows that βγ − λ^{2} = 0. Therefore f is of type B_{1} at z. The last assertion follows from the Gauss equation of f.

Theorem 16 now follows by putting together the preceding result and Theorem 8. For Riemannian products we obtain the following corollary.

Corollary 24. Let f: L^{p} × M^{n} → ℚ_{c}^{p+n+1} be an isometric immersion of a Riemannian product. Assume that p + n ≥ 3 and, if c = 0, that either L^{p} or M^{n}, say, the latter, has dimension at least two and is free of flat points. Then f is of type A everywhere and we have:

(i) If c = 0 there exist an orthogonal decomposition ℝ^{p+n+1} = ℝ^{p}⊕ℝ^{n+1}, a local isometry i: L^{p} → ℝ^{p} and an isometric immersion h: M^{n} → ℝ^{n+1} such that f = i × h.

(ii) If c ≠ 0 there exist an embedding Φ: ℚ_{c_{1}}^{p} × ℚ_{c_{2}}^{n} → ℚ_{c}^{p+n+1} as an extrinsic Riemannian product and local isometries i_{1} : L^{p} → ℚ_{c_{1}}^{p} and i_{2} : M^{n} → ℚ_{c_{2}}^{n} such that f = Φ ∘ (i_{1} × i_{2}).

From now on we consider isometric immersions f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}. Assume that f is of type B at a point z ∈ N^{p+n} = L^{p} ×_{ρ} M^{n} and let X, e, ξ, β, λ and γ be as in Lemma 21. Choose one of the unit vectors ξ̃ ∈ T_{z}^{⊥}N perpendicular to ξ, and define the symmetric bilinear forms

\begin{matrix}β̃: H_{z} × H_{z} → ℝ, (Y, Z) ↦ ⟨α(Y, Z), ξ̃⟩, \\ γ̃: V_{z} × V_{z} → ℝ, (V, W) ↦ ⟨α(V, W), ξ̃⟩. \end{matrix}

Set also β̃_{0} = β̃(X, X), γ̃_{0} = γ̃(e, e) and δ̃_{0} := β̃_{0}γ̃_{0}+βγ−λ^{2}. Then Lemma 21 can be strengthened as follows.

Proposition 25. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2} be an isometric immersion. Assume that f is of type B at z ∈ N^{p+n} = L^{p} ×_{ρ} M^{n}. With the preceding notations we have:

(i) If f is of type B_{1} at z, then one of the following (not exclusive) possibilities holds:

γ̃(V, W) = ⟨V, W ⟩γ̃_{0} or β̃ = 0.

(ii) If n ≥ 2 and f is of type B_{2} at z, then

(33)  β̃(Y, Z) = ⟨Y, X⟩⟨Z, X⟩β̃_{0} with β̃_{0} ≠ 0, and
(34)  β̃_{0} γ̃(V, W) = ⟨V, W ⟩δ̃_{0} − (βγ − λ^{2})⟨V, e⟩⟨W, e⟩.

Proof: Equation (29) now reads

(35) β̃(Y, Z)(γ̃(V, W) − ⟨V, W ⟩γ̃_{0}) = (βγ − λ^{2})⟨Y, X⟩⟨Z, X⟩⟨P V, P W ⟩.

If βγ − λ^{2} = 0 then the preceding equation proves assertion (i). Choosing Y = Z = X in (35) yields

(36)  β̃_{0} (γ̃(V, W) − ⟨V, W ⟩γ̃_{0}) = (βγ − λ^{2})⟨P V, P W ⟩.

Using (36), n ≥ 2 and βγ − λ^{2} ≠ 0 we derive β̃_{0} ≠ 0 and from (35) it follows that β̃(Y, Z) = ⟨Y, X⟩⟨Z, X⟩β̃_{0}. Finally, (36) also yields (34).

Taking into account (30) we have the following additional information in the case of Riemannian products.

Corollary 26. Let f: L^{p} × M^{n} → ℚ_{c}^{p+n+2} be an isometric immersion of a Riemannian product. Assume that f is of type B at z ∈ L^{p} ×M^{n}. Then, with the preceding notations, we have that δ̃_{0} = −c and, in addition:

(i) If f is of type B_{1} at z, then

β̃(Y, Z) = ⟨Y, Z⟩β̃_{0}, γ̃(V, W) = ⟨V, W ⟩γ̃_{0} and β̃_{0}γ̃_{0} + c = 0.

(ii) If p, n ≥ 2 and f is of type B_{2} at z, then c = 0,

(37) β̃(Y, Z) = ⟨Y, X⟩⟨Z, X⟩β̃_{0}, and γ̃(V, W) = ⟨V, e⟩⟨W, e⟩γ̃_{0}.

In the remaining of this section we make a detailed study of isometric immersions f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 2, that are everywhere of type B. In this case we may choose smooth unit vector fields X, e and ξ (and hence a smooth unit normal vector field ξ̃ orthogonal to ξ), and smooth functions β, λ and γ that satisfy pointwise the conditions of Lemma 21. In Propositions 27 and 31 below we classify isometric immersions of types B_{1} and B_{2}, respectively, the latter only for n ≥ 3. In Corollary 33 we determine the special subclass of isometric immersions of type B_{2} for which δ̃_{0} in (34) is everywhere vanishing. Isometric immersions of type B_{1} of Riemannian products with dimension p + n ≥ 3 are classified in Corollary 30. In Corollary 32 we show that there exists no isometric immersion f: L^{p} × M^{n} → ℚ_{c}^{p+n+2}, p + n ≥ 3, of type B_{2} if c ≠ 0 and in Corollary 34 we classify such isometric immersions for c = 0. This yields a local classification of isometric immersions of type B in codimension two of warped products L^{p} ×_{ρ} M^{n} for which n ≥ 3, as well as of Riemannian products L^{p} × M^{n} for which p + n ≥ 3.

Proposition 27. Let f: N^{p+n} = L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 2, be an isometric embedding of type B_{1}. Then f is a composition H ∘ g of isometric immersions, where g = Ψ ∘ (h_{1} × h_{2}) is a warped product of isometric immersions determined by a warped product representation Ψ: V^{p+k_{1}} ×_{σ} ℚ n+k_{2} ; c̃ → ℚ_{c}^{p+n+1}, k_{1} +k_{2} = 1, and H: W → ℚ_{c}^{p+n+2} is an isometric immersion of an open subset W ⊃ g(N^{p+n}) of ℚ_{c}^{p+n+1} (see the diagram in Theorem 14-(ii)).

Proof: Set E_{0} = (βX + λe)/∥βX + λe∥. Since βγ − λ^{2} = 0 we have that λ(βX + λe) = β(λX + γe), and hence (31) yields

(38)  A_{ξ}E = (β + γ)⟨E, E_{0}⟩E_{0} for all E ∈ Γ(T N).

Observe that β + γ ≠ 0 because βγ − λ^{2} > 0. The Gauss equation for f and the fact that A_{ξ} has rank 1 imply that Aξ̃ satisfies the Gauss equation for an isometric immersion of N^{p+n} into ℚ_{c}^{p+n+1}. We claim that it also satisfies the Codazzi equation for such an isometric immersion. Define a connection one-form ω on N^{p+n} by ω(E) = ⟨∇_{E}^{⊥}ξ, ξ̃⟩. By the Codazzi equation for f we have

(39)  (∇_{E_{1}}Aξ̃)E_{2} − (∇_{E_{2}}Aξ̃)E_{1} = ω(E_{2})A_{ξ}E_{1} − ω(E_{1})A_{ξ}E_{2}.

The following fact and (38) imply that the right hand side of (39) vanishes, and the claim follows.

Fact 28. The one-form ω satisfies ω(E) = ⟨E, E_{0}⟩ω(E_{0}) for all E ∈ Γ(T N), or equivalently, ω(E) = 0 for all E ∈ Γ(ker A_{ξ}).

In proving Fact 28 it is useful to observe that

(40) ker A_{ξ(z)} = {E_{0}(z)}^{⊥} = span{λ⟨V, e⟩Y − β⟨Y, X⟩V : Y ∈ H_{z}, V ∈ V_{z}}
(41)  = span{λ⟨Y, X⟩V − γ⟨V, e⟩Y : Y ∈ H_{z}, V ∈ V_{z}}.

By Proposition 25-(i), at each point z ∈ N^{p+n} either Aξ̃|_{V_{z}} = γ̃_{0} id, where id denotes the identity tensor, or Aξ̃|_{H_{z}} = 0. Since ω(E) is a continuous function, it suffices to prove that ω(E)(z) = 0 at points z ∈ N^{p+n} that are contained in an entire neighborhood U ⊂ N^{p+n} in which one of the preceding possibilities holds everywhere.

Case Aξ̃|_{V} = γ̃_{0} id. For Y ∈ L(L^{p}) and V, W ∈ L(M^{n}) we obtain using (2) and (4) that

(42) ⟨(∇_{Y} Aξ̃)V − (∇_{V} Aξ̃)Y, W ⟩ = ⟨∇_{Y} Aξ̃V − ∇_{V} Aξ̃Y − Aξ̃[Y, V], W ⟩
(43)  = ⟨∇_{Y} (γ̃_{0}V), W ⟩ + ⟨Aξ̃Y, ∇_{V} W ⟩
(44)  = (Y (γ̃_{0}) + ⟨(Aξ̃ − γ̃_{0} id)Y, η⟩)⟨V, W ⟩.

On the other hand, by the Codazzi equation we have

⟨(∇_{Y} Aξ̃)V − (∇_{V} Aξ̃)Y, W ⟩ = ⟨ω(V)A_{ξ}Y − ω(Y)A_{ξ}V, W ⟩
= ⟨W, e⟩ω(λ⟨Y, X⟩V − γ⟨V, e⟩Y),

where in the second equality we have used (31). Thus

(45) (Y (γ̃_{0}) + ⟨(Aξ̃ − γ̃_{0} id)Y, η⟩)⟨V, W ⟩ = ⟨W, e⟩ω(λ⟨Y, X⟩V − γ⟨V, e⟩Y)

for all Y ∈ L(L^{p}) and V, W ∈ L(M^{n}). As these equations are tensorial, they are also valid for arbitrary horizontal (resp., vertical) vector fields Y (resp., V, W). In particular, if we apply (45) for W = V orthogonal to e we obtain that the expression between parenthesis on the left-hand-side vanishes. Then, for W = e this yields

ω(λ⟨Y, X⟩V − γ⟨V, e⟩Y) = 0,

thus proving Fact 28 by (41) in this case.

Case Aξ̃|_{H} = 0. Using (1) we obtain for Y, Z ∈ L(L^{p}) and V ∈ L(M^{n}) that

⟨(∇_{Y} Aξ̃)V, Z⟩ = ⟨∇_{Y} Aξ̃V, Z⟩ − ⟨Aξ̃∇_{Y} V, Z⟩
= Y ⟨Aξ̃V, Z⟩ − ⟨Aξ̃V, ∇_{Y} Z⟩ − ⟨∇_{Y} V, Aξ̃Z⟩ = 0

and, analogously, that

⟨(∇_{V} Aξ̃)Y, Z⟩ = 0.

Using again that these equations are tensorial, we can replace Z by the vector field X. We obtain from (31) and the Codazzi equation that

ω(λ⟨V, e⟩Y − β⟨Y, X⟩V) = 0,

which by (40) proves Fact 28 also in this case.

It follows from Theorem 5’ in [7] and the assumption that f is an embedding that f is a composition f = H ∘ g, where g: N^{p+n} → ℚ_{c}^{p+n+1} is an isometric immersion such that A_{δ}^{g} = Aξ̃ for some unit normal vector field δ of g, and H: W → ℚ_{c}^{p+n+2} is an isometric immersion of an open subset W ⊂ ℚ_{c}^{p+n+1} containing g(N^{p+n}). Moreover, since A_{δ}^{g} = Aξ̃ satisfies A_{δ}^{g}|_{V_{z}} = γ̃_{0} id or A_{δ}^{g}|_{H_{z}} = 0 at any z ∈ N^{p+n}, it follows that g is of type A, and the conclusion follows from Theorem 8.

Remark 29. By Proposition 27, if f: N^{p+n} = L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 2, is an isometric embedding of type B_{1} then it must satisfy one of the conditions in Proposition 25-(i) everywhere. Moreover, for n ≥ 1 we have that both conditions hold simultaneously if and only if the isometric immersion g: N^{p+n} → ℚ_{c}^{p+n+1} satisfies A_{δ}^{g}|_{V} = γ̃_{0} id and A_{δ}^{g}|_{H} = 0. By Corollary 6, this is the case if and only if g = Ψ ∘ (h_{1} × h_{2}) with h_{1} totally geodesic and h_{2} a local isometry, where Ψ: V^{p+1} ×_{σ} ℚ n ; c̃ → ℚ_{c}^{p+n+1} is a warped product representation determined by (ℚ n ; c̃, z̄). Furthermore, if in addition γ̃_{0} = 0, that is, g is totally geodesic, and grad ρ has no zeros, then Corollary 7 implies that h_{1} must be cylindrical with respect to a, where −a is the mean curvature vector of ℚ n ; c̃ at z̄ in either ℝ^{p+n+1} or 𝕆^{p+n+2}, according as c = 0 or c ≠ 0. Conversely, if h_{1} is totally geodesic and cylindrical with respect to a and h_{2} is a local isometry, then g = Ψ ∘ (h_{1} × h_{2}) is totally geodesic.

By using the main lemma in [14] or Corollary 9, according as c = 0 or c ≠ 0, instead of Theorem 8, we obtain the following for isometric immersions of Riemannian products.

Corollary 30. Let f: L^{p} × M^{n} → ℚ_{c}^{p+n+2}, p + n ≥ 3, be an isometric embedding of type B_{1} of a Riemannian product.

(i) If c = 0, then there exist an orthogonal decomposition ℝ^{p+n+2} = ℝ^{p+k_{1}} × ℝ^{n+k_{2}} with k_{1} + k_{2} = 1 and isometric immersions h_{1} : L_{0}^{p} → ℝ^{p+k_{1}}, h_{2} : M_{0}^{n} → ℝ^{n+k_{2}} and H: W → ℝ^{p+n+2} of an open subset W ⊃ (h_{1} × h_{2})(U) of ℝ^{p+n+1} such that f|_{U} = H ∘ (h_{1} × h_{2}) (see the diagram in Case c = 0 - (ii) of Theorem 17 ).

(ii) If c ≠ 0 there exist an isometric embedding Φ: ℚ_{c_{1}}^{p} × ℚ_{c_{2}}^{n} → ℚ_{c}^{p+n+1} as an extrinsic Riemannian product, local isometries i_{1} : L^{p} → ℚ_{c_{1}}^{p} and i_{2} : M^{n} → ℚ_{c_{2}}^{n}, and an isometric immersion H: W → ℚ_{c}^{p+n+2} of an open subset W ⊃ Φ∘(i_{1}×i_{2})(L^{p}×M^{n}) of ℚ_{c}^{p+n+1} such that f = H ∘ Φ ∘ (i_{1} × i_{2}) (see the diagram in Case c ≠ 0 - (ii) of Theorem 17 ).

We now consider isometric immersions f: L^{p}×_{ρ}M^{n} → ℚ_{c}^{p+n+2} of type B_{2}. In the following statement, in order not to have to consider separately the cases p = 1 and p ≥ 2, we agree that in the first case all information related to the splitting L^{p} = L^{p−1} ×_{ρ_{1}} I should be disregarded.

Proposition 31. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2} be an isometric immersion of type B_{2} and assume that n ≥ 3. Then locally we have: L^{p} and M^{n} split as warped products L^{p} = L^{p−1} ×_{ρ_{1}} I and M^{n} = J ×_{ρ_{2}} M^{n−1} where I, J ⊂ ℝ are open intervals, and

N^{p+n} = L^{p−1} ×_{ρ_{1}} ((I ×_{ρ_{3}} J) ×ρ̄ M^{n−1}),

where ρ_{1} ∈ C^{∞}(L^{p−1}), ρ_{2} ∈ C^{∞}(J), ρ_{3} ∈ C^{∞}(I) and ρ̄ ∈ C^{∞}(I × J) satisfy

ρ = (ρ_{1} ∘ π_{L^{p−1}})(ρ_{3} ∘ π_{I}) and ρ̄ = (ρ_{3} ∘ π_{I})(ρ_{2} ∘ π_{J}),

and there exist warped product representations

Ψ_{1} : V^{p−1} ×_{σ_{1}} ℚ n+3 ; c̃ → ℚ_{c}^{p+n+2} and Ψ_{2} : W^{4} ×_{σ_{2}} ℚ n−1 ; c̄ → ℚ n+3 ; c̃,

isometries i_{1} : L^{p−1} → W^{p−1} ⊂ V^{p−1} ⊂ ℚ_{c}^{p−1} and i_{2} : M^{n−1} → W ^{n−1} ⊂ ℚ n−1 ; c̄ onto open subsets, and an isometric immersion g: I ×_{ρ_{3}} J → W^{4} of type B_{2} such that ρ̄ = σ_{2} ∘ g, ρ_{1} = σ_{1} ∘ i_{1} and f = Ψ_{1} ∘ (i_{1} × (Ψ_{2} ∘ (g × i_{2}))). Moreover, L^{p} has constant sectional curvature c if p ≥ 2 (see the diagram in Theorem 14-(iii)).

Proof: We have by (32), (33) and (34) that

(46)  Aξ̃Y = β̃_{0}⟨Y, X⟩X and Aξ̃V = b̃V + (γ̃_{0} − b̃)⟨V, e⟩e,

where b̃ = β̃_{0}^{−1} δ̃_{0}. On the other hand, we have that A_{ξ} is given by (31). Thus, for the relative nullity subspace Δ(z) at z ∈ N^{p+n} there are two possibilities:

Δ(z) = \{ \begin{matrix}\{X\}^{⊥} ⊂ H if δ̃_{0}(z) ≠ 0, \\ (\{X\}^{⊥} ⊂ H) ⊕ (\{e\}^{⊥} ⊂ V) if δ̃_{0}(z) = 0. \end{matrix}

In the remaining of this proof the letters T and S will always denote vector fields in Γ({X}^{⊥} ⊂ H) and Γ({e}^{⊥} ⊂ V), respectively. We also denote by (A_{δ}, u, v, w) taking the w-component of the Codazzi equation for A_{δ} and the vectors u, v.

We first prove that if p ≥ 2 then L^{p} splits locally as L^{p} = L^{p−1}×_{ρ_{1}} I, where I is an open interval, and that ρ = (ρ_{1} ∘ π_{L^{p−1}})(ρ_{3} ∘ π_{I}) for some functions ρ_{1} ∈ C^{∞}(L^{p−1}) and ρ_{3} ∈ C^{∞}(I). We point out that this fact also holds if n = 1, 2. As a first step, we show that the vector field X is the lift of a vector field X̃ ∈ Γ(T L). For that, we must prove that

(47)  ∇_{V} X = −⟨X, η⟩V.

Notice that (Aξ̃, X, V, T) reads

⟨∇_{X}Aξ̃V − Aξ̃∇_{X}V − ∇_{V} Aξ̃X + Aξ̃∇_{V} X, T ⟩ = ⟨ω(V)A_{ξ}X − ω(X)A_{ξ}V, T ⟩.

Then, by means of (31) and (46) we obtain

(48)  ⟨∇_{V} X, T ⟩ = 0.

On the other hand, by means of (4),

(49)  ⟨∇_{V} X, W ⟩ = −⟨∇_{V} W, X⟩ = −⟨V, W ⟩⟨X, η⟩.

Using also that ⟨∇_{V} X, X⟩ = 0, for X has unit length, we obtain (47) from (48) and (49). Notice that X̃ has unit length, because

⟨X̃, X̃⟩_{L} ∘ π_{L} = ⟨π_{L}_{∗}X, π_{L}_{∗}X⟩_{L} = ⟨X, X⟩_{N} = 1.

We show next that the distribution {X̃}^{⊥} is totally geodesic in L^{p}. In effect, for any T̃_{1}, T̃_{2} ∈ Γ({X̃}^{⊥}) we have

⟨∇L ; T̃_{1} T̃_{2}, X̃⟩_{L} ∘ π_{L} = ⟨π_{L}_{∗}∇_{T_{1}}T_{2}, π_{L}_{∗}X⟩_{L} = ⟨∇_{T_{1}}T_{2}, X⟩_{N} = 0,

where the last equality follows from the fact that the relative nullity distribution Δ is totally geodesic and T_{1}, T_{2} ∈ Γ(Δ), whereas X ∈ Γ(Δ^{⊥}).

Our next step is to prove that the vector field ζ = η − ⟨X, η⟩X ∈ Γ({X}^{⊥}) is the lift of a vector field ζ̃ ∈ Γ({X̃}^{⊥}), i.e., ∇_{V} ζ = −⟨ζ, η⟩V. This follows from

⟨∇_{V} ζ, W ⟩ = −⟨ζ, ∇_{V} W ⟩ = −⟨V, W ⟩⟨η, ζ⟩, ⟨∇_{V} ζ, X⟩ = −⟨ζ, ∇_{V} X⟩ = 0,

where we have used (47), and from

⟨∇_{V} ζ, T ⟩ = ⟨∇_{V} η, T ⟩ − V (⟨X, η⟩)⟨X, T ⟩ − ⟨X, η⟩⟨∇_{V} X, T ⟩ = 0,

where we have used (47) for the last term and that V is spherical for the first term.

Our final step is to show that the distribution {X̃} is spherical with mean curvature vector ζ̃. We have from (A_{ξ}, X, e, T) and λ ≠ 0 that ⟨∇_{X}X, T ⟩ = ⟨∇_{e}e, T ⟩ = ⟨η, T ⟩, and hence

∇L ; X̃ X̃ ∘ π_{L} = π_{L}_{∗}∇_{X}X = π_{L}_{∗}ζ = ζ̃.

On the other hand, we obtain from (11) for Y = T and V = W ≠ 0 that

⟨∇_{X}η, T ⟩ = ⟨X, η⟩⟨T, η⟩.

Thus,

⟨∇_{X}ζ, T ⟩ = ⟨∇_{X}η, T ⟩ − ⟨X, η⟩⟨∇_{X}X, T ⟩ = 0,

and therefore,

⟨∇L ; X̃ ζ̃, T̃⟩_{L} ∘ π_{L} = ⟨π_{L}_{∗}∇_{X}ζ, π_{L}_{∗}T ⟩_{L} = ⟨∇_{X}ζ, T ⟩_{N} = 0 for T̃ ∈ Γ({X̃}^{⊥}),

which completes the proof of the step.

By Theorem 1, we have that locally L^{p} splits as L^{p} = L^{p−1} ×_{ρ_{1}} I, where I is an open interval and ζ̃ = −grad log(ρ_{1} ∘ π_{L^{p−1}}). In particular, the lift ζ of ζ̃ to N^{p+n} is

ζ = −grad log(ρ_{1} ∘ π_{L^{p−1}} ∘ π_{L^{p}}).

Since we also have η = −grad log(ρ ∘ π_{L}), we obtain ⟨X, η⟩X = η − ζ = −grad log(ρ̂ ∘ π_{L}) with ρ̂ = ρ(ρ_{1} ∘ π_{L^{p−1}})^{−1} ∈ C^{∞}(L^{p}). Moreover, since

(T̃(log ρ̂)) ∘ π_{L^{p}} = T (log(ρ̂ ∘ π_{L^{p}})) = −⟨X, η⟩⟨X, T ⟩ = 0,

it follows that there exists ρ_{3} ∈ C^{∞}(I) such that ρ̂ = ρ_{3} ∘ π_{I}.

Let us now prove that locally M^{n} also splits as M^{n} = J ×_{ρ_{2}} M^{n−1}, where J is an open interval. First, we obtain from (Aξ̃, Y, e, S) and b̃ ≠ γ̃_{0} that ⟨∇_{Y} e, S⟩ = 0. Since also ⟨∇_{Y} e, e⟩ = 0, for e has unit length, and ⟨∇_{Y} e, Z⟩ = −⟨∇_{Y} Z, e⟩ = 0, we have

(50)  ∇_{Y} e = 0.

It follows that

∇_{Y} (ρ ∘ π_{L})e = Y (ρ ∘ π_{L})e = −⟨Y, η⟩(ρ ∘ π_{L})e.

This implies that (ρ ∘ π_{L})e is the lift of a vector field ẽ ∈ Γ(M^{n}). Notice that ẽ is a unit vector field, for

⟨ẽ, ẽ⟩_{M} ∘ π_{M} = ⟨π_{M}_{∗}(ρe), π_{M}_{∗}(ρe)⟩_{M} = ρ^{−2}⟨ρe, ρe⟩_{N} = 1.

Thus, in order to show that locally M^{n} splits as claimed, by Theorem 1 it suffices to prove that the distribution {ẽ} is totally geodesic and that {ẽ}^{⊥} is spherical.

We obtain from (A_{ξ}, X, e, S) and (50) that

(51)  ⟨∇_{e}e, S⟩ = 0.

In particular, it follows that ∇_{e}e = η. We conclude that the distribution {ẽ} is totally geodesic from

⟨∇M ; ẽ ẽ, S̃⟩_{M} ∘ π_{M} = ⟨π_{M}_{∗}∇_{ρe}ρe, π_{M}_{∗}S⟩_{M} = ⟨∇_{e}e, S⟩_{N} = 0

for any S̃ ∈ Γ({ẽ}^{⊥}). On the other hand, we obtain from (Aξ̃, S_{1}, e, S_{2}) that

⟨∇_{S_{1}}S_{2}, e⟩ = φ⟨S_{1}, S_{2}⟩,

where φ = (b̃−γ̃_{0})^{−1}e(b̃). Thus, the distribution {e}^{⊥} ⊂ V is totally umbilical. Moreover,

\begin{matrix}⟨∇M ; S̃_{1} S̃_{2}, ẽ⟩_{M} ∘ π_{M} = ⟨π_{M}_{∗}∇_{S_{1}}S_{2}, π_{M}_{∗}ρe⟩_{M} = ρ^{−1}⟨∇_{S_{1}}S_{2}, e⟩_{N} \\ = ρ^{−1}φ⟨S_{1}, S_{2}⟩_{N} = ρφ⟨S̃_{1}, S̃_{2}⟩_{M} ∘ π_{M}. \end{matrix}

The preceding equality implies that there exists φ̃ ∈ C^{∞}(M^{n}) such that ρφ = φ̃ ∘ π_{M} and that the distribution {ẽ}^{⊥} is totally umbilical with mean curvature normal φ̃ẽ. In particular, if b̃, or equivalently, δ̃_{0}, vanishes on an open subset L_{0}^{p} × M_{0}^{n} ⊂ N^{p+n}, then {ẽ}^{⊥} is a totally geodesic distribution in M_{0}^{n}. In the general case, in order to show that {ẽ}^{⊥} is spherical, it remains to prove that S̃(φ̃) = 0 or, equivalently, that S(φ) = 0. First, using that V is umbilical and invariant by Aξ̃, and that λ ≠ 0, we obtain from (Aξ̃, e, S, X) that ∇_{S}^{⊥}ξ̃ = 0. Now, choosing linearly independent sections S_{1}, S_{2} ∈ Γ({e}^{⊥}) we obtain from (Aξ̃, S_{1}, S_{2}, S_{1}) that S(b̃) = 0. We point out that the assumption that n ≥ 3 is only used here. In particular, if δ̃_{0} is everywhere vanishing then it is enough to assume that n ≥ 2. Using (51) we obtain from (Aξ̃, e, S, e) that S(γ̃_{0}) = 0. Since ∇_{e}S ∈ {e}^{⊥} and ∇_{S}e ∈ {e}^{⊥}, as follows from (51) and the fact that V is totally umbilical, then

Se(b̃) = eS(b̃) + ∇_{e}S(b̃) − ∇_{S}e(b̃) = 0,

and hence S(φ) = 0. Therefore, locally N^{p+n} splits as

N^{p+n} = L^{p−1} ×_{ρ_{1}} M^{n+1} with M^{n+1} := M^{2} ×ρ̄ M^{n−1} and M^{2} := I ×_{ρ_{3}} J,

where J ⊂ ℝ is an open interval and ρ̄ = (ρ_{3} ∘ π_{I})(ρ_{2} ∘ π_{J}). Notice that f is of type A with respect to this decomposition of N^{p+n}. We claim that there exist a warped product representation Ψ_{1} : V^{p−1}×_{σ_{1}}ℚ n+3 ; c̃ → ℚ_{c}^{p+n+2}, an isometric immersion G̃: M^{n+1} → ℚ n+3 ; c̃ and a local isometry i_{1} : L^{p−1} → V^{p−1} such that ρ_{1} = σ_{1} ∘ i_{1} and f = Ψ_{1} ∘ (i_{1} × G̃).

Fix ȳ ∈ L^{p−1} with ρ_{1}(ȳ) = 1 and let iȳ: M^{n+1} → N^{n+p} be the (isometric) inclusion of M^{n+1} into N^{n+p} as a leaf of the vertical subbundle V̄ according to the latter decomposition of N^{n+p}. Define G: M^{n+1} → ℚ_{c}^{p+n+2} by G = f ∘ iȳ. By Corollary 13-(ii), in order to show that the spherical hull of G has dimension n + 3, it suffices to prove that for no point z ∈ N^{n+p} there exists a unit vector ξ̄ ∈ T_{z}^{⊥}N such that Aξ̄|V̄_{z}: V̄_{z} → V̄_{z} is a multiple of the identity tensor. Write ξ̄ = cos θξ +sin θξ̃. Then ⟨Aξ̄X, e⟩ = 0 and ⟨Aξ̄X, X⟩ = ⟨Aξ̄e, e⟩ = ⟨Aξ̄S, S⟩ for any unit vector S ∈ ({e}_{z}^{⊥} ⊂ V_{z}) if and only if

λ cos θ = 0 and β̃_{0} sin θ + β cos θ = γ cos θ + γ̃_{0} sin θ = b̃ sin θ.

Since λ ≠ 0, we obtain that γ̃_{0} = b̃, a contradiction to the fact that βγ−λ^{2} ≠ 0. Our claim then follows from Theorem 8 by letting ℚ n+3 ; c̃ be the spherical hull of G and defining G̃: M^{n+1} → ℚ n+3 ; c̃ by G = j ∘ G̃, where j is the inclusion

of ℚ n+3 ; c̃ into ℚ_{c}^{p+n+2}.

We now study the isometric immersion G̃: M^{n+1} → ℚ n+3 ; c̃. First observe that the second fundamental form of G = j ∘ G̃ is given by

α_{G}(V, W) = α_{f}(iȳ_{∗}V, iȳ_{∗}W) + ⟨V, W ⟩f_{∗}(η̄ ∘ iȳ) for all V, W ∈ Γ(T M^{n+1}), where η̄ denotes the mean curvature normal of V̄. Let V̂ denote the vertical subbundle of T M^{n+1} according to the decomposition M^{n+1} = M^{2} ×ρ̄ M^{n−1}. Using that A_{ξ}^{f}|_{iȳ_{∗}V̂} = 0 and that Af ; ξ̃|_{iȳ_{∗}V̂} = b̃ id, where id denotes the identity tensor, it follows that

α_{G}(V, V̄) = ⟨V, V̄ ⟩((b̃∘iȳ)(ξ̃∘iȳ)+f_{∗}(η̄∘iȳ)) for all V ∈ Γ(T M^{n+1}), V̄ ∈ V̂.

Therefore η̂ = (b̃ ∘ iȳ)(ξ̃ ∘ iȳ) + f_{∗}(η̄ ∘ iȳ) is a principal curvature normal of G and V̂ is contained in the corresponding eigendistribution. Since j is umbilical, it follows that η̂_{T ℚ n+3 ; c̃} is a principal curvature normal of G̃ with the same eigendistribution as η̂. By means of Corollary 13-(i) and Theorem 8, we conclude that there exist a warped product representation Ψ_{2} : W^{4} ×_{σ_{2}} ℚ n−1 ; c̄ → ℚ n+3 ; c̃, an isometric immersion g: M^{2} → W^{4} and a local isometry i_{2} : M^{n−1} → ℚ n−1 ; c̄ such that ρ̄ = σ_{2} ∘ g and G̃ = Ψ_{2} ∘ (g × i_{2}). Moreover, since the second fundamental form of g is determined by the restriction of α_{G} to the horizontal subbundle Ĥ of T M^{n+1} according to the decomposition M^{n+1} = M^{2} ×ρ̄ M^{n−1}, and hence by the restriction of α_{f} to span{X, e} (see formula (15)), it follows that g is of type B_{2}.

Finally, since {X}^{⊥} ⊂ H is contained in Δ, we obtain that the curvaturelike tensor C defined in Proposition 2 satisfies

C(Y_{1}, Y_{2}, Y_{3}, Y_{4}) = 0 for all Y_{1}, Y_{2}, Y_{3}, Y_{4} ∈ Γ(H).

The last assertion then follows from the fact that, for a fixed x̄ ∈ M^{n}, the inclusion ix̄: L^{p} → N^{p+n} given by ix̄(y) = (y, x̄) is a totally geodesic isometric immersion.

Corollary 32. Let N^{p+n} = L^{p} × M^{n} be a Riemannian product of dimension p + n ≥ 3. Then there exists no isometric immersion f: N^{p+n} → ℚ_{c}^{p+n+2} of type B_{2} if c ≠ 0.

Proof: We may assume p ≥ 2. It follows from Proposition 31 that locally L^{p} splits as a Riemannian product L^{p} = L^{p−1} × I. The statement now follows from the fact that L^{p} has constant sectional curvature c.

In order to complete the classification of isometric immersions f: L^{p} × M^{n} → ℚ_{c}^{p+n+2} of type B of Riemannian products of dimension p + n ≥ 3, it remains to determine those that are of type B_{2} for c = 0. Observe that for such isometric immersions equation (34) in Proposition 25 holds with δ̃_{0} = 0 (see Corollary 26). In the following result we solve the more general problem of classifying isometric immersions of type B_{2} of warped products satisfying that condition.

Corollary 33. Let f: N^{p+n} = L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2}, n ≥ 2, be an isometric immersion of type B_{2} for which δ̃_{0} vanishes everywhere. Then c ≤ 0, N^{p+n} has constant sectional curvature c and one of the following holds locally:

(i) If c = 0 then L^{p} and M^{n} split as Riemannian products L^{p} = L^{p−1} ×I and M^{n} = J × M^{n−1}, where I, J ⊂ ℝ are open intervals, and there exist isometries i_{1} : L^{p−1} → U ⊂ ℝ^{p−1} and i_{2} : M^{n−1} → V ⊂ ℝ^{n−1} onto open subsets and an isometric immersion g: I × J → ℝ^{4} such that f = i_{1} × g × i_{2}.

[figure: L^{p−1} × (I × J) × M^{n−1} = N^{p+n}; i_{1}; f = i_{1} × g × i_{2}; g; i_{2}; ∪; U; ∪; V; ℝ^{p−1} × ℝ^{4} × ℝ^{n−1} = ℝ^{p+n+2}]

(ii) If c < 0 then L^{p} splits as a warped product L^{p} = L^{p−1} ×_{ρ_{1}} I, M^{n} splits as a Riemannian product M^{n} = J × M^{n−1}, where I, J ⊂ ℝ are open intervals and ρ = ρ_{1} ∘ π_{L^{p−1}}, and there exist a warped product representation Ψ: V^{p−1} ×_{σ} ℝ^{n+3} → ℚ_{c}^{p+n+2}, isometries i_{1} : L^{p−1} → U ⊂ V ^{p−1} and i_{2} : M^{n−1} → W ⊂ ℝ^{n−1} onto open subsets, and an isometric immersion g: I × J → ℝ^{4} such that f = Ψ ∘ (i_{1} × (g × i_{2})) and ρ_{1} = σ ∘ i_{1}.

[figure: L^{p−1} ×_{ρ_{1}} ((I × J) × M^{n−1}); g; ℚ_{c}^{p+n+2}; f = Ψ ∘ (i_{1} × (g × i_{2})); i_{1}; i_{2}; ∪; U; ∪; W; V^{p−1} ×_{σ} (ℝ^{4} × ℝ^{n−1}); ⟳; Ψ]

Proof: First observe that for the statement of Proposition 31 to hold in this case it is enough to require that n ≥ 2, as follows from the second italicized text in its proof. In order to prove that N^{p+n} has constant sectional curvature c, we must show that the curvature-like tensor C defined in Proposition 2 vanishes identically. Since the relative nullity distribution of f is Δ = (span{X, e})^{⊥}, we have that C(E_{1}, E_{2}, E_{3}, E_{4}) = 0 whenever two of the vectors E_{1}, E_{2}, E_{3}, E_{4} belong to (span{X, e})^{⊥}. Thus it remains to show that C(X, e, e, X) = 0, because C is a curvature-like tensor. But this follows from (31), (46) and the assumption that δ̃_{0} = 0.

We have from Proposition 31 that locally L^{p} and M^{n} split as warped products L^{p} = L^{p−1} ×_{ρ_{1}} I, M^{n} = J ×_{ρ_{2}} M^{n−1}, and N^{p+n} = L^{p−1} ×_{ρ_{1}} ((I ×_{ρ_{3}} J) ×ρ̄ M^{n−1}), where I, J ⊂ ℝ are open intervals and ρ_{1} ∈ C^{∞}(L^{p−1}), ρ_{2} ∈ C^{∞}(J), ρ_{3} ∈ C^{∞}(I) and ρ̄ ∈ C^{∞}(I × J) satisfy

ρ = (ρ_{1} ∘ π_{L^{p−1}})(ρ_{3} ∘ π_{I}) and ρ̄ = (ρ_{3} ∘ π_{I})(ρ_{2} ∘ π_{J}).

But now ⟨X, η⟩ = ⟨∇_{S}S, X⟩ = 0, because Δ is totally geodesic. Hence we may assume that ρ_{3} = 1 (recall the proof of Proposition 31), and therefore ρ = ρ_{1} ∘ π_{L^{p−1}}. On the other hand, by the first italicized text in the proof of Proposition 31, the distribution {ẽ}^{⊥} in M^{n} is now totally geodesic, and hence ρ_{2} = 1, which implies that also ρ̄ = 1. Summing things up, we have

N^{p+n} = L^{p−1} ×_{ρ_{1}} M^{n+1}, with M^{n+1} := M^{2} × M^{n−1} and M^{2} := I × J.

Now, for a fixed point ȳ ∈ L^{p} with ρ(ȳ) = 1, let iȳ: M^{n} → N^{p+n} denote the (isometric) inclusion of M^{n} into N^{p+n} as a leaf of V. The second fundamental form of iȳ is α_{iȳ}(V, W) = ⟨V, W ⟩(η ∘ iȳ) for all V, W ∈ Γ(T M^{n}), where η = −grad log(ρ∘π_{L}) is the mean curvature normal of V. Since N^{p+n} has constant sectional curvature c, it follows from the Gauss equation for iȳ that M^{n} has constant sectional curvature c + ∥η ∘ iȳ∥^{2} = c + ∥grad log ρ(ȳ)∥^{2}. We conclude from the fact that M^{n} = J × M^{n−1} is a Riemannian product that it must be flat, hence M^{n−1} must be flat when n ≥ 3 and c + ∥grad log ρ(ȳ)∥^{2} = 0. Now choose any other point y^{∗} ∈ L^{p} and modify the warped product representation of N^{n+p} so that the modified warping function ρ^{∗} satisfies ρ^{∗}(y^{∗}) = 1. By this modification η = −grad log(ρ ∘ π_{L}) does not change. Therefore the preceding argument yields ∥grad log ρ(y^{∗})∥^{2} = −c.

We now distinguish the two possible cases:

Case c = 0. Here η = −grad(log ρ ∘ π_{L^{p}}) vanishes, hence ρ = 1, consequently also ρ_{1} = 1 and therefore N^{p+n} = L^{p−1} × M^{2} × M^{n−1}, with L^{p−1} and M^{n−1} flat and M^{2} = I × J. Using that Δ = (span{X, e})^{⊥} is the relative nullity distribution of f, the main lemma in [14] implies that f splits as

f = i_{1} × g × i_{2} : L^{p−1} × M^{2} × M^{n−1} → ℝ^{p−1} × ℝ^{4} × ℝ^{n−1} = ℝ^{p+n+2},

where i_{1} : L^{p−1} → U ⊂ ℝ^{p−1} and i_{2} : M^{n−1} → V ⊂ ℝ^{n−1} are isometries onto open subsets, and g: M^{2} → ℝ^{4} is an isometric immersion.

Case c < 0. For G = f ∘ iȳ as in the proof of Proposition 31, we have from Corollary 13-(ii) that its spherical hull ℚn+3 ; c̃ has constant sectional curvature c̃ = c+∥grad log ρ(ȳ)∥^{2} = 0. Therefore ℚn+3 ; c̃ is a horosphere ℝ^{n+3} ⊂ ℚ_{c}^{p+n+2}. Let G̃: M^{n+1} → ℝ^{n+3} = ℚ n+3 ; c̃ be such that G = j ∘ G̃, where j denotes the inclusion of ℚ n+3 ; c̃ into ℚ_{c}^{p+n+2}. Using that the vertical subbundle of M^{n+1} correspondent to the splitting M^{n+1} = M^{2}×M^{n−1} is contained in the relative nullity distribution of G̃, the conclusion now follows from the main lemma in [14] applied to G̃: M^{2} × M^{n−1} → ℝ^{n+3}.

Corollary 34. Let N^{p+n} = L^{p} × M^{n} be a Riemannian product of dimension p + n ≥ 3. Then any isometric immersion f: N^{p+n} → ℝ^{p+n+2} of type B_{2} is locally given as in Corollary 33-(i).

Remark 35. By making use of global arguments from [1], a complete description of the possible cases in which an isometric immersion f: L^{p} × M^{n} → ℝ^{p+n+2}, p ≥ 2 and n ≥ 2, of a Riemannian product of complete nonflat Riemannian manifolds may fail locally to be a product of isometric immersions was given in [3]. Namely, it was shown therein that there exists an open dense subset of L^{p} × M^{n} each of whose points lies in an open product neighborhood U_{0} = L_{0}^{p} × M_{0}^{n} restricted to which f is either (i) a product of isometric immersions, (ii) an isometric immersion of type B_{2} given as in Corollary 33 - (i), or (iii) an isometric immersion of type B_{1} of the following special type: either L_{0}^{p} or M_{0}^{n}, say, the latter, splits as M_{0}^{n} = I × ℝ^{n−1}, the manifold L_{0}^{p} is free of flat points and f|_{U_{0}} splits as

f|_{U_{0}} = F × id: (L_{0}^{p} × I) × ℝ^{n−1} → ℝ^{p+3} × ℝ^{n−1} = ℝ^{p+n+2}.

Moreover, F: L_{0}^{p} × I → ℝ^{p+3} is a composition F = H ∘ F̃, where

F̃ = G × i: L_{0}^{p} × I → ℝ^{p+1} × ℝ = ℝ^{p+2}

is a cylinder over a hypersurface G: L_{0}^{p} → ℝ^{p+1}, and H: W → ℝ^{p+3} is an isometric immersion of an open subset W ⊃ F̃(L_{0}^{p} × I) of ℝ^{p+2}.

We take the opportunity to point out that the main theorem in [3] misses the informations that L_{0}^{p} is free of flat points and that F̃ is a cylinder F̃ = G×i, which follow from Corollary 24.

5. Immersions of type C

The aim of this section is to prove the following pointwise result for isometric immersions of type C.

Proposition 36. Let f: L^{p} ×_{ρ} M^{n} → ℚ_{c}^{p+n+2} be an isometric immersion of a warped product. If f is of type C at z ∈ N^{p+n} = L^{p} ×_{ρ} M^{n} and n ≥ 3, then N^{p+n} has constant sectional curvature c at z.

Proof: We must prove that the curvature like tensor C on T_{z}N defined in Proposition 2 vanishes identically. Two possible cases may occur:

Case 1. For every X ∈ H_{z} the linear map B_{X} defined in (22) satisfies rank B_{X} ≤ 1.

Case 2. There exists X ∈ H_{z} such that rank B_{X} = 2.

We first prove the following facts.

(i) In Case 1, for any X ∈ H_{z} with rank B_{X} = 1 we have

(52)  α(E, V) = 0 for all V ∈ D(X) = ker B_{X} and E ∈ T_{z}N,

that is, D(X) is contained in the relative nullity subspace Δ of f at z.

(ii) In Case 2, condition (52) is true for any X ∈ H_{z} such that rank B_{X} = 2.

For (i). Here Lemma 20 applies and we have by (23) that D(Y) = {e}^{⊥} for any Y ∈ H_{z} with rank B_{Y} = 1. In particular, this shows that (52) is satisfied for any E ∈ H_{z}. On the other hand, (24) yields α(V, W) = ⟨V, e⟩⟨W, e⟩α(e, e), and hence (52) also holds for any E ∈ V_{z}.

For (ii). If rank B_{X} = 2, i.e., B_{X}(V_{z}) = T_{z}^{⊥}N, then (52) is equivalent to

⟨α(E, V), α(X, W)⟩ = 0

for all V ∈ D(X), W ∈ V_{z} and E ∈ T_{z}N. But this follows from (13) for E ∈ H_{z} and from (14) for E ∈ V_{z}.

We now prove that

(53)  C(E_{1}, V_{1}, V_{2}, E_{2}) = 0 for all E_{1}, E_{2} ∈ T_{z}N and V_{1}, V_{2} ∈ V_{z}.

We obtain from (14) that (53) holds whenever one of the vectors E_{1}, E_{2} lies in H_{z} and the other in V_{z}. On the other hand, if we are in Case 1 (resp., Case 2) and X ∈ H_{z} satisfies rank B_{X} = 1 (resp., rank B_{X} = 2), then it follows from (52) that (53) is also satisfied if any of the vectors E_{1}, E_{2}, V_{1} or V_{2} belongs to D(X). In particular, C(E_{1}, V, V, E_{2}) = 0 holds for any V ∈ D(X). Notice that D(X) ≠ {0} by our assumption that n ≥ 3. Applying (11) for 0 ≠ W = V ∈ D(X) yields

(54)  ∇_{Y} η − ⟨Y, η⟩η − cY = 0 for any Y ∈ H_{z}.

Notice that if N^{p+n} is a Riemannian product then this implies that c = 0. Moreover, since for c = 0 this equation holds automatically for Riemannian products, in this case it is enough to assume that either n ≥ 2 or p ≥ 2.

Therefore, (53) is satisfied for all E_{1}, E_{2} ∈ H_{z} and V_{1}, V_{2} ∈ V_{z}. This completes the proof of (53) in Case 1 and shows that in Case 2 it remains to prove that (53) is satisfied if E_{1}, E_{2}, V_{1}, V_{2} all belong to the two-dimensional subspace D(X)^{⊥}. Since C is a curvature like tensor, this will follow once we prove the existence of an orthonormal basis {e_{1}, e_{2}} of D(X)^{⊥} such that

(55)  C(e_{1}, e_{2}, e_{2}, e_{1}) = 0.

In order to prove (55) take an orthonormal basis e_{1}, e_{2} of D(X)^{⊥} such that e_{1} is one of the two points on the unit circle S^{1} in D(X)^{⊥} where ϕ: S^{1} → ℝ given by ϕ(V) = ∥B_{X}V ∥^{2} assumes its maximum value. Differentiating ψ(t) = ϕ(cos te_{1} + sin te_{2}) yields

0 = ψ′(0) = 2⟨B_{X}e_{1}, B_{X}e_{2}⟩.

Thus, there exist an orthonormal basis {ξ_{1}, ξ_{2}} of T_{z}^{⊥}N and positive real numbers λ_{1}, λ_{2} such that B_{X}e_{r} = λ_{r}ξ_{r} for r = 1, 2. We have from (14) that

(56)  λ_{1}⟨α(e_{2}, e_{r}), ξ_{1}⟩ = λ_{2}⟨α(e_{1}, e_{r}), ξ_{2}⟩, r = 1, 2.

Using that (53) holds for E_{1} = E_{2} = X, V_{1} = e_{s} and V_{2} = e_{t}, 1 ≤ s, t ≤ 2, we obtain

⟨α(X, X), α(e_{r}, e_{r})⟩ = ⟨α(X, e_{r}), α(X, e_{r})⟩ = λ_{r}^{2},

⟨α(X, X), α(e_{1}, e_{2})⟩ = ⟨α(X, e_{1}), α(X, e_{2})⟩ = 0.

Setting γ_{st}^{r} = ⟨α(e_{s}, e_{t}), ξ_{r}⟩ = γ_{st}^{r}, α_{r} = ⟨α(X, X), ξ_{r}⟩ and D = γ_{11}^{1}γ_{22}^{2} − γ_{11}^{2}γ_{22}^{1}, where 1 ≤ s, t ≤ 2, it follows that

(57)  a_{1}γ_{rr}^{1} + a_{2}γ_{rr}^{2} = λ_{r}^{2} and a_{1}γ_{12}^{1} + a_{2}γ_{12}^{2} = 0.

If we compute Da_{1} and Da_{2} from the first equation in (57) and put the result into the second we obtain an equation which because of (56) is equivalent to

λ_{1}λ_{2}(⟨α(e_{1}, e_{1}), α(e_{2}, e_{2})⟩ − ⟨α(e_{1}, e_{2}), α(e_{1}, e_{2})⟩) = 0,

and this gives (55) and concludes the proof of (53).

Because of Proposition 2 it remains to show that

(58)  C(Y_{1}, Y_{2}, Y_{3}, Y_{4}) = 0 for all Y_{1}, Y_{2}, Y_{3}, Y_{4} ∈ H_{z}.

We divide the proof into the same two cases considered before.

Case 1. By Lemma 20 the linear map H_{z} → T_{z}^{⊥}N, Y ↦ α(Y, e) is surjective. Hence p ≥ 2, and we can take X_{1}, X_{2} ∈ H_{z} such that the vectors ξ_{j} := α(X_{j}, e), j = 1, 2, form an orthonormal normal basis and α(Z, e) = 0 for all Z ∈ {X_{1}, X_{2}}^{⊥}. We have from (12) and (13) that

0 = C(Z, X_{j}, e, E) = ⟨A_{ξ_{j}}Z, E⟩

for all Z ∈ {X_{1}, X_{2}}^{⊥} and any E ∈ T_{z}N. Therefore, A_{ξ_{j}}Z = 0 for all Z ∈ {X_{1}, X_{2}}^{⊥}, that is, {X_{1}, X_{2}}^{⊥} ⊂ Δ(z). Hence (58) holds whenever Y_{i} ∈ {X_{1}, X_{2}}^{⊥} for some 1 ≤ i ≤ 4. Thus, it remains to prove that

(59)  C(X_{1}, X_{2}, X_{2}, X_{1}) = 0.

Because of (53) we have

(60)  ⟨α(X_{i}, X_{j}), α(e, e)⟩ = ⟨ξ_{i}, ξ_{j}⟩.

We now may assume that α(e, e) = a ξ_{2} with a ≠ 0. We obtain from (60) that

(61)  ⟨A_{ξ_{2}}X_{1}, X_{2}⟩ = 0

and

a⟨A_{ξ_{2}}X_{j}, X_{j}⟩ = 1, j = 1, 2.

Since a ≠ 0, it follows from the last equation that

(62)  ⟨A_{ξ_{2}}X_{1}, X_{1}⟩ = ⟨A_{ξ_{2}}X_{2}, X_{2}⟩.

In addition, (12) yields

(63)  0 = C(X_{i}, X_{j}, e, X_{i}) = ⟨A_{ξ_{j}}X_{i}, X_{i}⟩ − ⟨A_{ξ_{i}}X_{i}, X_{j}⟩, i ≠ j.

Then (59) follows from (61), (62) and (63), and the proof is completed in this case.

Case 2. Since (55) holds, it follows from a result of E. Cartan ([5]; cf. Theorem 1 in [15]), that we may choose nonzero vectors v_{1}, v_{2} in D(X)^{⊥} (not necessarily orthogonal) such that ⟨α(v_{1}, v_{1}), α(v_{2}, v_{2})⟩ = 0 and α(v_{1}, v_{2}) = 0. From (53) we have

0 = C(Z, v_{1}, v_{2}, Y) = −⟨α(Z, v_{2}), α(Y, v_{1})⟩.

Therefore, the subspaces α(H_{z}, v_{1}), α(H_{z}, v_{2}) are orthogonal lines spanned by η_{1} = α(X, v_{1}) and η_{2} = α(X, v_{2}), respectively, which we may assume to have unit length by rescaling v_{1} and v_{2} if necessary. In particular, the kernel H_{j} of the linear map F_{j}: H_{z} → T_{z}^{⊥}N given by F_{j}(Y) = α(Y, v_{j}), j = 1, 2, has codimension one in H_{z}. On the other hand, for Y ∈ H_{j} and any Z ∈ H_{z} we obtain using (12) that

⟨A_{η_{j}}Y, Z⟩=⟨α(Y, Z), α(X, v_{j})⟩ = ⟨α(Y, v_{j}), α(X, Z)⟩ = ⟨F_{j}(Y), α(X, Z)⟩ = 0.

Let Z_{j} be a unit vector in H_{z} orthogonal to H_{j}, j = 1, 2. Then

C(Y_{1}, Y_{2}, Y_{3}, Y_{4}) = ⟨α(Y_{1}, Y_{4}), α(Y_{2}, Y_{3})⟩ − ⟨α(Y_{1}, Y_{3}), α(Y_{2}, Y_{4})⟩

= ∑_{j=1}^{2}(⟨A_{η_{j}}Y_{1}, Y_{4}⟩⟨A_{η_{j}}Y_{2}, Y_{3}⟩ − ⟨A_{η_{j}}Y_{1}, Y_{3}⟩⟨A_{η_{j}}Y_{2}, Y_{4}⟩)

= ∑_{j=1}^{2}⟨Y_{1}, Z_{j}⟩⟨Y_{2}, Z_{j}⟩(⟨A_{η_{j}}Z_{j}, Y_{4}⟩⟨A_{η_{j}}Z_{j}, Y_{3}⟩−
⟨A_{η_{j}}Z_{j}, Y_{3}⟩⟨A_{η_{j}}Z_{j}, Y_{4}⟩) = 0.

Proposition 36 and the italicized text in its proof yield the following for the case of Riemannian products.

Corollary 37. Let f: L^{p} × M^{n} → ℚ_{c}^{p+n+2} be an isometric immersion of a Riemannian product. Assume that f is of type C at z ∈ N^{p+n} = L^{p} × M^{n}. Then we have

(i) If either p ≥ 3 or n ≥ 3 then c = 0.

(ii) If p + n ≥ 3 and c = 0 then N^{p+n} is flat at z.

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IMPA – Estrada Dona Castorina 110, 22460-320-Rio de Janeiro, Brazil

E-mail address: marcos@impa.br

Universidade Federal de São Carlos, 13565-905-São Carlos, Brazil

E-mail address: tojeiro@dm.ufscar.br