Genuine deformations of submanifolds

Marcos Dajczer & Luis A. Florit

Abstract

We introduce the concept of genuine isometric deformation of an Euclidean submanifold and describe the geometric structure of the submanifolds that admit deformations of this kind. That an isometric deformation is genuine means that the submanifold is not included into a submanifold of larger dimension such that the deformation of the former is given by a deformation of the latter. Our main result says that an Euclidean submanifold together with a genuine deformation in low (but not necessarily equal) codimensions must be mutually ruled, and gives a sharp estimate for the dimension of the rulings. This has several strong local and global consequences. Moreover, the unifying character and geometric nature, as opposed to a purely algebraic one, of our result suggest that it should be the starting point for a deformation theory extending the classical Sbrana - Cartan theory for hypersurfaces to higher codimensions.

The isometric deformation problem for a given isometric immersion f: M^{n} → ℝ^{n+p} of a Riemannian manifold into flat Euclidean space with codimension p and a positive integer q is to describe all possible isometric immersions f̂: M^{n} → ℝ^{n+q}. A satisfactory answer to the local version of the problem for every hypersurface (p=1) and q = 1 going back almost a century is due to Sbrana [19] and Cartan [4]. However, basic questions, like the existence of Sbrana-Cartan hypersurfaces of the discrete type or the possibility of smoothly attaching different types of these deformable hypersurfaces, were answered positively only recently; see [10] and also [3] for a special case. The global version of the problem for hypersurfaces was solved in [11] and [18].

Nothing similar to the Sbrana-Cartan theory for codimensions q = p higher than one has yet been obtained. Nevertheless, the classical Beez-Killing rigidity theorem for hypersurfaces, the starting point for the theory, has several generalizations; see [1], [2], [5] and [20]. All these results provide generic algebraic conditions on the second fundamental form of the isometric immersion that imply isometric rigidity, that is, any other isometric immersion must differ by an isometry (rigid motion) of the ambient space.

There is a large set of isometric deformations once we allow codimension q > p since one can always compose f: M^{n} → ℝ^{n+p} with (local) isometric immersions of ℝ^{n+p} into ℝ^{n+q}. In [7] we found generic algebraic conditions on the second fundamental form that assure that compositions of this kind are the only possible deformations. Such a composition for q = p is just an isometric congruence, and thus the result in [7] reduces to the rigidity one in [5].

Since a submanifold of a deformable one is also deformable, to go deeper into the deformation problem one has to discard those deformations that arise this way, in particular, through compositions as above. Our goal in this paper is twofold. First to introduce the concept of genuine deformation, and then to give the geometric structure of the submanifolds that admit deformations of this kind. As a consequence, we have several applications for a new rigidity concept that extends the ones already discussed.

We say that a pair f: M^{n} → ℝ^{n+p} and f̂: M^{n} → ℝ^{n+q} of isometric immersions extends isometrically when there are an isometric embedding j: M^{n} ↪ N^{m} into a Riemannian manifold N^{m} with m > n and isometric immersions F: N^{m} → ℝ^{n+p} and F̂: N^{m} → ℝ^{n+q} such that f = F ∘ j and f̂ = F̂ ∘ j. In other terms, the following diagram commutes:

[figure: M^{n}; N^{m}; (1); ℝ^{n+p}; ℝ^{n+q}; f; f̂; F; F̂; j]

We called f̂ in [7] a composition if m = n + p and p ≤ q because N^{n+p} is flat and, if f is an embedding, there is an isometric immersion h into ℝ^{n+q} of an open neighborhood of f(M) in ℝ^{n+p} such that f̂ = h ∘ f. Clearly, the concept of composition for p = q reduces to the standard one of congruence. Definition. An isometric immersion f̂: M^{n} → ℝ^{n+q} is a genuine deformation of a given isometric immersion f: M^{n} → ℝ^{n+p} if there is no open subset U ⊂ M^{n} along which the restrictions f|_{U} and f̂|_{U} extend isometrically.

In this paper, we prove that any pair of submanifolds in low codimensions determined by a genuine deformation is mutually ruled (with the same rulings) and give a sharp estimate for the dimension of the rulings. In addition, we show that the relation discussed in the sequel between the normal bundles and second fundamental forms that exists for any pair of mutually ruled submanifolds must satisfy strong additional conditions.

Recall that an isometric immersion f: M^{n} → ℝ^{n+p} is d-ruled, or more specifically, D^{d}-ruled, if M^{n} has a (not necessarily maximal dimensionwise) nontrivial integrable d-dimensional distribution D^{d} ⊂ T M whose leaves are mapped diffeomorphically by f to (open subsets of ) affine subspaces of ℝ^{n+p}. At each point x ∈ M there is an associated orthogonal splitting of the normal bundle T_{f}^{⊥}M = L_{D} ⊕ L_{D}^{⊥} of f, where

L_{D}(x) = span{α(Z, X) : Z ∈ D^{d}(x) and X ∈ T_{x}M}

is given in terms of its second fundamental form α: T M × T M → T_{f}^{⊥}M. Assume that ℓ_{D} = dim L_{D} is constant, and let f̂: M^{n} → ℝ^{n+q} be another D^{d}-ruled isometric immersion. Then, by the Gauss equation for f and f̂, there is a unique vector bundle isometry T_{D} : L_{D} → L̂_{D} satisfying

α̂|_{D×T M} = T_{D} ∘ α|_{D×T M}.

Here and elsewhere we mark with a hat any object that refers to f̂. Then T_{D} may neither preserve second fundamental forms (fully) nor the induced connections on L_{D} and L̂_{D}. In other words, it may not hold that α̂L̂_{D} = T_{D} ∘ α_{L_{D}} or that T_{D} is parallel, where α_{S} = π_{S} ∘ α and π_{S}: T_{f}^{⊥}M → S stands for the orthogonal projection onto a subbundle S ⊂ T_{f}^{⊥}M. The following is an immediate consequence of our main result.

Theorem 1. Let f̂: M^{n} → ℝ^{n+q} be a genuine deformation of f: M^{n} → ℝ^{n+p} with p + q < n and min {p, q} ≤ 5. Then, along each connected component of an open dense subset of M^{n}, the immersions f and f̂ are mutually D^{d}-ruled with

d ≥ n − p − q + 3 ℓ_{D},

and T_{D} is a parallel bundle isometry that preserves second fundamental forms.

Theorem 1 generalizes the one on compositions in [7] (see Theorem 24 below) which extended, beside the result on isometric rigidity in [5], the one on compositions in [13]. We recall that [5] implies Allendoerfer’s rigidity theorem [1] until codimension p ≤ 5 and, in particular, the classical Beez-Killing rigidity theorem for hypersurfaces, which is also immediate from ours.

Theorem 1 also generalizes known results on isometric extensions. This is the case of Theorem 5 in [12] and Theorem 2 in [14]. The former deals with deformable submanifolds in the special case p = q = 2. In this situation, we would have from Theorem 1 that ℓ_{D} ≤ 1 and, even for ℓ_{D} = 1, it follows that the index of relative nullity (i.e., the dimension of the nullity of the second fundamental form α) of both immersions satisfies ν ≥ n − 4. Therefore, if as in [12] one of the immersions has index ν ≤ n − 5 everywhere, we conclude that they extend isometrically to either flat or Sbrana-Cartan hypersurfaces.

The following example shows that the estimate in Theorem 1 is sharp even for p ≠ q.

Example 2. There exist local isometric immersions of the round sphere 𝕊^{n} into ℝ^{2n−1} that cannot be obtained as a composition of the inclusion i_{n}: 𝕊^{n} ↪ ℝ^{n+1} with a local isometric immersions of ℝ^{n+1} into ℝ^{2n−1}; see [15]. These immersions cannot be compositions since they have no umbilical direction. Hence, given isometric immersions g_{n_{j}}: 𝕊^{n_{j}} → ℝ^{2n_{j}−1} as such, the product immersion

g = g_{n_{1}} × ⋯ × g_{n_{p}} × id: 𝕊^{n_{1}} × ⋯ × 𝕊^{n_{p}} × ℝ^{k} → ℝ^{n+q},

where n = k + ∑_{i=1}^{p} n_{i}, is a genuine deformation of the product i_{n_{1}} × ⋯ × i_{n_{p}} × id of inclusions, and satisfies that k = n − p − q is the maximal dimension of the rulings of g.

The need for assumptions on rulings or nullity bounds in several of the aforementioned results was not quite well understood at the time but has now been completely clarified by our constructions; see Theorem 24 below. This shows that a deeper understanding in the study of basic rigidity questions can be reached in the broader framework of this paper.

Theorem 1 has several local and global rigidity consequences dealing with the rigidity concept that arises from the notion of genuine deformation.

Definition. An isometric immersion f: M^{n} → ℝ^{n+p} is genuinely rigid in ℝ^{n+q} for a fixed integer q if for any given isometric immersion f̂: M^{n} → ℝ^{n+q} there is an open dense subset U ⊂ M^{n} such that f|_{U} and f̂|_{U} extend isometrically.

The first and immediate consequence is a sharp general rigidity result.

Corollary 3. Let f: M^{n} → ℝ^{n+p} be an isometric immersion and q a positive integer with p + q < n. If min {p, q} ≤ 5 and f is not (n−p−q)–ruled on any open subset of M^{n}, then f is genuinely rigid in ℝ^{n+q}.

Rather simple arguments give the following three corollaries.

Corollary 4. Let f: M^{n} → ℝ^{n+p} be an isometric immersion of a compact manifold and q a positive integer with p + q < n. If min {p, q} ≤ 5, then there is an open subset U ⊂ M^{n} such that f|_{U} is genuinely rigid in ℝ^{n+q}.

In a more general setting where isometric extensions are allowed to be singular, it was shown in [12] for p = q = 2 in Corollary 4 that f itself is genuinely rigid. Counterexamples to this result if only regular extensions are allowed can be constructed; see Example 29. We take this opportunity to observe that the unproved last assertion of Theorem 1 in [12] is not clear.

Corollary 5. Let f: M^{n} → ℝ^{n+p} be an isometric immersion and q a positive integer such that p + q < n. If min {p, q} ≤ 5 and the Ricci curvature of M^{n} is positive then f is genuinely rigid in ℝ^{n+q}.

Example 2 for p = 1 and k = 0 shows that in the preceding result the first bound for the codimensions is sharp. The following immediate consequence of Theorem 1 gives Theorem 3 in [13] and Theorem 1 in [14]. Both results deal with the characterization of Riemannian manifolds that admit isometric immersions in two space forms of different sectional curvatures.

Corollary 6. Let f: M^{n} → 𝕊^{n+p} ⊂ ℝ^{n+p+1} be an isometric immersion and q a positive integer such that p + q + 1 < n. If min {p + 1, q} ≤ 5, then f is genuinely rigid in ℝ^{n+q}.

We conclude with the following topological criteria for genuine rigidity that follows easily from Theorem 25 below, and relates to the result for p = q = 2 given in [12].

Theorem 7. Let M^{n} be a compact manifold whose first Pontrjagin class satisfies that [p_{1}]^{2} ≠ 0. If n > p + q and p + q ≤ 6, then any analytic immersion f: M^{n} → ℝ^{n+p} is (with the induced metric) genuinely rigid in ℝ^{n+q}.

It seems quite possible that the assumption in Theorem 1 that min {p, q} ≤ 5 cannot be dropped. In that sense, we observe that the key Lemma 16 does not hold for dimensions higher than five; see [8]. Nevertheless, we have a result that includes the case min {p, q} = 6; see Theorem 14 below. To conclude, we observe that it is straightforward to extend our results to ambient spaces of arbitrary constant sectional curvature.

The authors would like to acknowledge valuable comments by D. Gromoll. The first author is grateful to the people of the Stony Brook Mathematical Department for their kindness and hospitality.

A class of ruled extensions.

In this section we provide conditions that allow to extend to mutually ruled isometric submanifolds a pair of isometric immersions of a given Riemannian manifold.

Let f: M^{n} → ℝ^{n+p} and f̂: M^{n} → ℝ^{n+q} be a pair of isometric immersions, and let

T : L ⊂ T_{f}^{⊥}M → L̂ ⊂ T ⊥ ; f̂M

be a vector bundle isometry. Assume that the vector subspaces

D = N (α_{L^{⊥}}) ∩ N (α̂L̂^{⊥}) ⊂ T M

have constant dimension on M^{n}, and that the pair (T , D) satisfies the conditions:

\{ \begin{matrix}(C_{1}) The isometry T is parallel and preserves second fundamental forms; \\ (C_{2}) The subbundles L and L̂ are parallel along D in the normal connections. \end{matrix} (2)

Throughout this paper, given a bilinear form β: V^{n} × U^{m} → W between finite dimensional real vector spaces we denote by S(β) ⊂ W the subspace generated by the image of β, that is,

S(β) = span{β(X, Y) : X ∈ V^{n} and Y ∈ U^{m}},

and by N (β) ⊂ V^{n} the (left) nullity space of β defined as

N (β) = {X ∈ V^{n} : β(X, Y) = 0 for all Y ∈ U^{m}}.

Let ϕ: (T M ⊕ L) × T M → L^{⊥} × L̂^{⊥} be the bilinear form given by

ϕ(Y + ξ, X) = (( ˜∇_{X}(Y + ξ))_{L^{⊥}}, ( ˜∇_{X}(Y + T ξ))L̂^{⊥}) ,  (3)

and assume further that the vector subspaces

Δ = N (ϕ) ⊂ T M ⊕ L

have constant dimension on M^{n}. By condition (C_{1}) the vector bundle isometry defined as T_{0} = I ⊕ T : f_{∗}T M ⊕ L → f̂_{∗}T M ⊕ L̂ is parallel in the connections induced by the Euclidean ambient spaces. It follows that T_{0} |_{Δ}: Δ → Δ̂ is a parallel vector bundle isometry, and hence, we may identify Δ̂ with Δ.

Lemma 8. The distribution D ⊂ Δ is integrable and Δ ∩ T M = D holds.

Proof: By assumption ˜∇_{Z}μ ∈ L^{⊥} for all μ ∈ L^{⊥} and Z ∈ D. The integrability of D now follows easily from R̃(Y, Z)μ = 0 for any Y, Z ∈ D and μ ∈ L^{⊥} or L̂^{⊥}, where R̃ stands for the flat curvature tensor of the Euclidean ambient spaces. If Y ∈ Δ ∩ T M and X ∈ T M, then 0 = ( ˜∇_{X}Y)_{L^{⊥}} = α_{L^{⊥}}(Y, X), and similarly αL̂^{⊥}(Y, X) = 0. Thus Y ∈ D, and therefore Δ ∩ T M = D.

Consider the vector bundle π: Λ = Λ̂ → M^{n} determined by the orthogonal splitting Δ = D ⊕ Λ, and define F: N → ℝ^{n+p} as the restriction of the map

λ ∈ Λ ↦ f(π(λ)) + λ

to a tubular neighborhood of the 0-section j: M^{n} ↪ N ⊂ Λ of Λ along which F is an immersion. Similarly, define F̂: N → ℝ^{n+q}. Henceforth, L^{⊥} and Δ will be understood as vector bundles over N ⊂ Λ by means of L^{⊥}(λ) = L^{⊥}(π(λ)) and Δ(λ) = Δ(π(λ)).

Proposition 9. The immersions F and F̂ are isometric Δ-ruled extensions of f and f̂. Moreover, there are smooth orthogonal splittings

T_{F}^{⊥}N = L ⊕ L^{⊥} and T ⊥ ; F̂N = L̂ ⊕ L̂^{⊥}  (4)

and a vector bundle isometry T: L → L̂ such that

Δ = N (α_{L^{⊥}}^{F}) ∩ N (α̂F̂ ; L̂^{⊥}),  (5)

and the pair (T, Δ) satisfies conditions (C_{1}) and (C_{2}) in (2).

Proof: It follows from F̂_{∗} = T_{0} ∘ F_{∗} that F and F̂ are isometric. To see that both immersions are Δ-ruled it suffices to check that Δ is constant along the leaves of D.

This follows easily from R̃(Y, X)δ = 0 for any Y ∈ D, X ∈ T M and δ ∈ Δ. The proof of (4) is straightforward. Moreover, taking T = T_{0}|_{L} we obtain (C_{1}) for F and F̂.

Condition (C_{2}) and the inclusion “⊂” in (5) are immediate. For the opposite inclusion observe that α_{L^{⊥}}^{F}|_{T_{j}M×T_{j}M} = α_{L^{⊥}}, and similarly for F̂ and f̂. We easily obtain from T_{j}N = T M ⊕ Λ that equality is satisfied along j(M). To conclude the proof observe that the dimension of the right hand side of the inclusion can only decrease along N from its value on j(M) if N is taken small enough.

Observe that the ruled extensions F and F̂ are trivial (i.e., dim N = n) if f and f̂ are already D-ruled.

Remark 10. The assumptions that the subspaces D and Δ have constant dimension are not essential for local purposes. In fact, without them the same result holds for f and f̂ restricted to connected components of an open dense subset of M^{n}.

The structure.

In this section, we study the structure of the tangent and normal bundles of a pair of isometric submanifolds of Euclidean spaces. Our goal is to give conditions that allow the construction of isometric ruled extensions.

In the sequel, we define pointwise several vector subspaces as either images or kernels of certain tensor fields on a submanifold. To avoid cumbersome repetition, for the remaining of the paper we agree that we are always working restricted to a connected component of an open dense subset of the submanifold where all these subspaces have constant dimensions, and hence form smooth vector subbundles.

Given a pair of isometric immersions f: M^{n} → ℝ^{n+p} and f̂: M^{n} → ℝ^{n+q} with second fundamental forms α and α̂, respectively, we endow the vector bundle T_{f}^{⊥}M ⊕ T ⊥ ; f̂M with the indefinite metric of type (p, q) given by

⟨⟨ , ⟩⟩_{T_{f}^{⊥}M⊕T ⊥ ; f̂M} = ⟨ , ⟩_{T_{f}^{⊥}M} − ⟨ , ⟩_{T ⊥ ; f̂M}.

Set α ⊕ α̂: T M × T M → S(α) ⊕ S(α̂) ⊂ T_{f}^{⊥}M ⊕ T ⊥ ; f̂M, and let Ω ⊂ S(α) ⊕ S(α̂)

be the vector bundle with null fibers Ω = S(α ⊕ α̂) ∩ S(α ⊕ α̂)^{⊥}. Accordingly, there are orthogonal splittings

S(α) = Γ ⊕ Γ^{⊥} and S(α̂) = Γ̂ ⊕ Γ̂^{⊥},

where Γ = S(α) ∩ Ω^{⊥} and Γ̂ = S(α̂) ∩ Ω^{⊥}, and an isometry J: Γ^{⊥} → Γ̂^{⊥} such that

Ω = {(η, J η) : η ∈ Γ^{⊥}} ⊂ Γ^{⊥} ⊕ Γ̂^{⊥}

and α̂Γ̂^{⊥} = J ∘ α_{Γ^{⊥}}. From now on we identify Γ^{⊥} with Γ̂^{⊥} by means of J, and hence

α̂Γ̂^{⊥} = α_{Γ^{⊥}}.  (6)

Define β: T M × T M → Γ ⊕ Γ̂ as β = α_{Γ} ⊕ α̂Γ̂ and a vector subbundle Θ ⊂ T M by

Θ = N (β).

The vector subbundle S ⊂ Γ^{⊥}(= Γ̂^{⊥}) defined by

S = S(α|_{Θ×T M})

satisfies Θ = N (α_{S^{⊥}}) ∩ N (α̂Ŝ^{⊥}). Now define a vector subbundle S_{0} ⊂ S by

S_{0} = ⋂_{X∈T M} ker K(X),

where K(X) ∈ Λ^{2}(S) for any X ∈ T M denotes the skew-symmetric tensor given by

K(X)η = (∇_{X}^{⊥}η)_{S} − ( ˆ∇_{X}^{⊥}η)_{S}.

Then define vector subbundles L^{ℓ} ⊂ S_{0} and D^{d} ⊂ Θ as

L^{ℓ} = {δ ∈ S_{0} : ∇_{Y}^{⊥}δ ∈ S and ˆ∇_{Y}^{⊥}δ ∈ Ŝ for all Y ∈ Θ}

and

D^{d} = N (α_{L^{⊥}}) ∩ N (α̂L̂^{⊥}),

and let T : L^{ℓ} → L^{ℓ} be the induced vector bundle isometry given by

T = J |_{L}: L^{ℓ} ⊂ T_{f}^{⊥}M → L^{ℓ} ⊂ T ⊥ ; f̂M.

Theorem 11. Let f: M^{n} → ℝ^{n+p} and f̂: M^{n} → ℝ^{n+q} be isometric immersions. Then, along each connected component of an open dense subset of M^{n} the pair (T , D^{d}) satisfies (C_{1}) and (C_{2}) in (2). In particular, f and f̂ have (possibly trivial) isometric ruled extensions F: N → ℝ^{n+p} and F̂: N → ℝ^{n+q} satisfying the conclusions of Proposition 9. Therefore, if f is a genuine deformation of f̂, then f and f̂ are mutually D^{d}-ruled.

We will make use of the following result.

Lemma 12. The tensor K: T M → Λ^{2}(S) satisfies:

(i) K(Z) = 0 for all Z ∈ Θ;

(ii) K(X)α(Y, Z) = K(Y)α(X, Z) for all Z ∈ Θ and X, Y ∈ T M.

Proof: Comparing the Codazzi equation for f and f̂ for Z_{1}, Z_{2} ∈ Θ or Z_{3} ∈ Θ yields

K(Z_{1})α(Z_{2}, Z_{3}) = K(Z_{2})α(Z_{1}, Z_{3}).  (7)

Denote ⟨K(X_{1})α(X_{2}, X_{3}), α(X_{4}, X_{5})⟩ = (X_{1}, X_{2}, X_{3}, X_{4}, X_{5}). If Z_{1}, Z_{2}, Z_{3} ∈ Θ, then

(Y, Z_{1}, Z_{2}, Z_{3}, X) = −(Y, Z_{3}, X, Z_{1}, Z_{2}) = −(X, Z_{3}, Y, Z_{1}, Z_{2}) = (X, Z_{1}, Z_{2}, Z_{3}, Y)

= (Z_{2}, Z_{1}, X, Z_{3}, Y) = −(Z_{2}, Z_{3}, Y, Z_{1}, X) = −(Z_{3}, Z_{2}, Y, Z_{1}, X)

= (Z_{3}, Z_{1}, X, Z_{2}, Y) = (Z_{1}, Z_{3}, X, Z_{2}, Y) = −(Z_{1}, Z_{2}, Y, Z_{3}, X)

= −(Y, Z_{1}, Z_{2}, Z_{3}, X).

Thus ⟨K(Z_{1})α(Z_{2}, Y), α(Z_{3}, X)⟩ = 0, and this proves (i). The proof of (ii) is (7)

The next result implies that the distribution Θ is integrable if K vanishes. The shape operator A_{ξ}: T M → T M for a normal vector ξ is defined by

⟨A_{ξ}X, Y ⟩ = ⟨α(X, Y), ξ⟩.

Lemma 13. Let Θ_{0} ⊂ Θ be the vector subbundle defined as

Θ_{0} = N (α_{S_{0}^{⊥}}) ∩ N (α̂Ŝ_{0}^{⊥}).

Then [Θ_{0}, Θ] ⊂ Θ.

Proof: Clearly S(β) ∩ S(β)^{⊥} = S(β) ∩ S(α ⊕ α̂)^{⊥} and S(α ⊕ α̂ − β) ⊂ Ω ⊂ S(α ⊕ α̂). In particular, S(β) ⊂ S(α ⊕ α̂), and hence S(β) ∩ S(β)^{⊥} ⊂ Ω ⊂ Γ^{⊥} ⊕ Γ̂^{⊥}. We conclude that the metric induced on S(β) ⊂ Γ ⊕ Γ̂ is nondegenerate.

Taking the inner product of the Codazzi equation (∇_{Z_{1}}^{⊥}α)(Z_{2}, W) = (∇_{Z_{2}}^{⊥}α)(Z_{1}, W) with μ ∈ S^{⊥} ⊂ T_{f}^{⊥}M for Z_{1}, Z_{2} ∈ Θ and any W ∈ T M yields

⟨∇_{Z_{1}}^{⊥}α(Z_{2}, W) − ∇_{Z_{2}}^{⊥}α(Z_{1}, W), μ⟩ = ⟨α_{S^{⊥}}([Z_{1}, Z_{2}], W), μ⟩,  (8)

and an analogous equation holds for f̂. On the other hand, the difference between the Codazzi equations of f and f̂ for δ ∈ S is

A_{∇_{Z}^{⊥}δ}Y − Â_{ˆ∇_{Z}^{⊥}δ}Y = A_{∇_{Y}^{⊥}δ}Z − Â_{ˆ∇_{Y}^{⊥}δ}Z.

In the last equation, choose Z = Z_{1} ∈ Θ and δ = α(Z_{2}, W) ∈ S for Z_{2} ∈ Θ. The inner product with X ∈ T M and Lemma 12–(i) give

⟨∇_{Z_{1}}^{⊥}α(Z_{2}, W), α_{S^{⊥}}(X, Y)⟩ − ⟨ ˆ∇_{Z_{1}}^{⊥}α̂(Z_{2}, W), α̂Ŝ^{⊥}(X, Y)⟩ = ⟨α(Z_{1}, X), K(Y)α(Z_{2}, W)⟩.

The right hand side vanishes if either Z_{1} or Z_{2} belong to Θ_{0}. Then we have using (8) for both immersions that

⟨α_{S^{⊥}}([Z_{1}, Z_{2}], W), α_{S^{⊥}}(X, Y)⟩ − ⟨α̂Ŝ^{⊥}([Z_{1}, Z_{2}], W), α̂Ŝ^{⊥}(X, Y)⟩ = 0.

We obtain that ⟨⟨β([Z_{1}, Z_{2}], W), S(β)⟩⟩ = 0, and the statement follows using that S(β) is nondegenerate.

Proof of Theorem 11: The isometry T preserves the second fundamental forms by (6) and it is parallel since J |_{S_{0}} is parallel. The definition of S yields

S(α_{L^{⊥}∩S}|_{Θ×T M}) = L^{⊥} ∩ S.  (9)

It follows from the Codazzi equation, Lemma 13 and the definition of L^{ℓ} that

(∇_{Y}^{⊥}α_{L^{⊥}∩S_{0}}(Z, X))_{S^{⊥}} = (∇_{Z}^{⊥}α_{L^{⊥}∩S}(Y, X))_{S^{⊥}}  (10)

for all Z ∈ Θ_{0}, Y ∈ Θ and X ∈ T M, and a similar result holds for f̂. Since the left hand side of (10) vanishes if Z ∈ D^{d} and X ∈ T M, we obtain from (9) and the definition of L^{ℓ} that S and Ŝ are both parallel along D^{d} in the normal connections.

We define S_{1} ⊂ S by the orthogonal splitting S = S_{0} ⊕ S_{1}. The skew-symmetry of the K(X) gives

S_{1} = span{K(X)S_{1} : X ∈ T M}.  (11)

By the Ricci equation, ⟨R^{⊥}(X, Z)δ, μ⟩ = ⟨R̂^{⊥}(X, Z)δ, μ⟩ for all δ, μ ∈ S. It follows easily from the definitions of R^{⊥} and S_{0} that

⟨∇_{Z}^{⊥}δ, ∇_{X}^{⊥}μ⟩ − ⟨ ˆ∇_{Z}^{⊥}δ, ˆ∇_{X}^{⊥}μ⟩ = ⟨∇_{X}^{⊥}δ, ∇_{Z}^{⊥}μ⟩ − ⟨ ˆ∇_{X}^{⊥}δ, ˆ∇_{Z}^{⊥}μ⟩

for all δ ∈ S_{0} and μ ∈ S_{1}. We now obtain from Lemma 12–(i) and the parallelism of S and Ŝ along D^{d} that ⟨∇_{Z}^{⊥}δ, K(X)μ⟩ = 0 for all Z ∈ D^{d}, X ∈ T M, δ ∈ L^{ℓ} and μ ∈ S_{1}. It follows from (11) that

∇_{Z}^{⊥}δ ∈ S_{0} for all Z ∈ D^{d} and δ ∈ L^{ℓ}.  (12)

The Ricci equation for Z ∈ D^{d}, Y ∈ Θ, δ ∈ L^{ℓ} and ξ ∈ S^{⊥}, yields

0 = ⟨R^{⊥}(Y, Z)δ, ξ⟩ = ⟨∇_{Y}^{⊥}∇_{Z}^{⊥}δ, ξ⟩,

where for the second equality we have to use (12), Lemma 13 and the parallelism of S along D^{d}. From the definition of L^{ℓ} we conclude that L^{ℓ} and L̂^{ℓ} are parallel along D^{d} in the corresponding normal connections.

The main result.

In this section, we show that for pairs of isometric submanifolds of low codimension the foliation D^{d} is nontrivial by giving an estimate that implies the one in Theorem 1.

With the notation of the last section we now state the main result of this paper.

Theorem 14. Let f̂: M^{n} → ℝ^{n+q} be a genuine deformation of f: M^{n} → ℝ^{n+p} with p+q < n and min {p, q} ≤ 6. Then f and f̂ are mutually D^{d}-ruled along each connected component of an open dense subset of M^{n} with

d ≥ n − p − q + 3ℓ,

unless min {p, q} = 6 and ℓ = 0 in which case d ≥ n−p−q −1, and the bundle isometry T : L^{ℓ} → L^{ℓ} satisfies (C_{1}) and (C_{2}) in (2).

Notice that Theorem 14 provides more information than Theorem 1, in particular, because L^{ℓ} in the former can be larger than L_{D} in the latter for the same rulings. Moreover, already for p = q = 2, we may have that L_{D} = L^{ℓ} but the rulings D^{d} in Theorem 1 that one considers can be smaller than the ones determined by Theorem 14.

The proof of the theorem will be made in several steps. First, we obtain some estimates of the dimension of the nullity space of a vector valued bilinear map.

Given a bilinear form β: V^{n} × U^{m} → W, we call a vector Y ∈ U^{m} a (right) regular element of β if the map B_{Y} = β(Y, ·) satisfies

dim B_{Y} (V^{n}) = max{dim B_{Z}(V^{n}) : Z ∈ U^{m}}.

It is easy to see that the subset RE(β) ⊂ U^{m} of regular elements of β is open and dense and that

S(β|_{ker B_{Y} ×U^{m}}) ⊂ B_{Y} (V^{n}) for any Y ∈ RE(β).  (13)

Therefore,

S(β) = S(β|_{R×U^{m}})  (14)

if the subspace R ⊂ V^{n} satisfies V^{n} = R ⊕ ker B_{Y}; see [6] or [17] for details.

Lemma 15. Let β: V^{n} × U^{m} → W be a nonzero bilinear form and let L^{k} ⊂ U^{m} be a vector subspace of minimal dimension k such that S(β|_{V^{n}×L^{k}}) = S(β). Then there is an integer ρ satisfying 1 ≤ ρ ≤ dim S(β) − k + 1 such that

dim N (β|_{V^{n}×L^{k}}) ≥ n − k(ρ − 1) − 1.

Proof: Take ρ = dim B_{Y} (V^{n}) for Y ∈ RE(β). If L^{k} = span{Y_{1}, …, Y_{k}} for Y_{j} ∈ RE(β), 1 ≤ j ≤ k, then B_{Y_{i}}(V^{n}) ≠ B_{Y_{j}}(V^{n}) if i ≠ j, and an easy argument using (13) gives the proof in this case. To obtain the proof for an arbitrary L^{k} observe that there is a sequence L_{j}^{k} → L^{k} such that each L_{j}^{k} satisfies the assumption and is spanned by vectors in RE(β) as before.

Let W^{p,q} be a (p+q)–dimensional vector space endowed with a possibly indefinite inner product of type (p, q). We call a subspace U ⊂ W^{p,q} degenerate if the restriction of the metric of W^{p,q} to U is degenerate, and denote by rank U the rank of the induced metric. Thus rank U = dim U − dim U ∩ U^{⊥}. We call null the degenerate subspace U if rank U = 0, and thus U = U ∩ U^{⊥}. A bilinear form β: V^{n} × U^{m} → W^{p,q} is said to be flat when

⟨β(X, Y), β(Z, T)⟩ − ⟨β(X, T), β(Z, Y)⟩ = 0

for all X, Y ∈ V^{n} and Z, T ∈ U^{m}. It follows from (13) that

S(β|_{ker B_{Y} ×U^{m}}) ⊂ B_{Y} (V^{n}) ∩ B_{Y} (V^{n})^{⊥} if Y ∈ RE(β).  (15)

The proof of Theorem 3 in [7] on flat symmetric bilinear forms stated below gives the following slightly stronger result.

Lemma 16. Let β: V^{n}×V^{n} → W^{p,q} be a flat symmetric bilinear form. If min{p, q} ≤ 6 and S(β) is nondegenerate, then

dim N (β) ≥ n − dim Im B_{Y} − dim S(β|_{ker B_{Y} ×T M}) − δ_{min {p,q}}^{6}

for any Y ∈ RE(β).

Corollary 17 ([7]). Let β: V^{n} × V^{n} → W^{p,q} be a flat symmetric bilinear form. If min{p, q} ≤ 6 and S(β) is nondegenerate, then dim N (β) ≥ n − dim S(β) − δ_{min {p,q}}^{6}.

To our surprise, we recently constructed in [8] a family of examples that shows not only that the last two results are false for min{p, q} = 6 without the Kronecker symbol in the estimates, but also that there is no linear estimate in p and q. Since Lemma 16 is a key point in the argument, there can be no linear estimate for d in Theorem 14 for higher codimensions as well.

Lemma 18. Under the assumptions of Theorem 14 we have that

dim Θ ≥ n − p − q + 2τ + τ_{0} − δ_{0},  (16)

where τ = dim S, τ_{0} = dim S_{0} and δ_{0} = δ_{min{dim Γ,dim Γ̂}}^{6}.

Proof: It follows from Corollary 17 and that S(β) is nondegenerate (see the proof of Lemma 13) that

dim Θ ≥ n − p − q + 2τ − δ_{0}.  (17)

To prove the better estimate (16) we have to work with the more elaborate flat bilinear form defined next, and make strong use of the fact that the deformation is genuine.

Let ϕ: (T M ⊕ S_{0}) × T M → S^{⊥} × Ŝ^{⊥} be the bilinear form given by

ϕ(X + ξ, Y) = β′(X, Y) + ψ(ξ, Y),

where ψ: S_{0} × T M → S^{⊥} ⊕ Ŝ^{⊥} is defined as

ψ(ξ, Y) = ((∇_{Y}^{⊥}ξ)_{S^{⊥}}, ( ˆ∇_{Y}^{⊥}ξ)Ŝ^{⊥})

and β′: T M × T M → S^{⊥} ⊕ Ŝ^{⊥} by

β′ = α_{S^{⊥}} ⊕ α̂Ŝ^{⊥}.

Notice that S(β′) is degenerate if S ≠ Γ^{⊥}. A long but straightforward computation using the Gauss, Codazzi and Ricci equations for f and f̂ and the definitions of S_{0} and S gives that ϕ is flat. Now a key observation is that there is no local section X_{0} + ξ_{0} ∈ T M ⊕ S_{0} with ξ_{0} ≠ 0 such that the subspace ϕ(X_{0} + ξ_{0}, T M) is null. This is so because this is the precise condition for the maps F (x, t) = f(x) + t(X_{0} + ξ_{0}) and F̂(x, t) = f̂(x) + t(X_{0} + ξ_{0}) to be local isometric extensions of f and f̂ in M^{n} × (−ϵ, ϵ), for some ϵ > 0.

Set C_{Y} = ϕ( · , Y) and B′_{Y} = β′( · , Y) for Y ∈ RE(ϕ). By (15) and the flatness of ϕ the subspace S(ϕ|_{ker C_{Y} ×T M}) is null. It follows using the above that ker C_{Y} = ker B′_{Y}. Therefore,

dim Im C_{Y} = dim Im B′_{Y} + τ_{0}.  (18)

Set K = ker B_{Y} and K′ = ker B′_{Y}, where now Y ∈ RE(ϕ) ∩ RE(β). The subspaces Im C_{Y} and S(β′|_{K′×T M}) are orthogonal by the flatness of ϕ. In particular,

dim Im C_{Y} + dim S(β′|_{K′×T M}) ≤ p + q − 2τ.  (19)

The Gauss equation for f and f̂ and (6) give that β = α_{Γ} ⊕ α̂Γ̂ is flat. By Lemma 16 and since S(β) is nondegenerate, we have

dim Θ ≥ n − dim Im B_{Y} − dim S(β|_{K×T M}) − δ_{0}.  (20)

We obtain from (18), (19), (20) and dim Im B′_{Y} + dim K′ = dim Im B_{Y} + dim K that

dim Θ ≥ n − p − q − δ_{0} + 2τ + τ_{0} + dim S(β′|_{K′×T M}) − dim S(β|_{K×T M}) + dim K − dim K′.

Since K′ ⊂ K, then (16) follows unless K′ ⊊ K and

dim S(β′|_{K′×T M}) − dim S(β|_{K×T M}) + dim K − dim K′ < 0.

In the latter situation the null subspace S(β_{0}) ≠ 0, where β′ = β ⊕β_{0}. Moreover, we can assume that β′|_{K′×T M} ≠ 0 since, otherwise, K′ = N (β′) ⊂ N (β) and we easily obtain (16) from (18). It follows that

min{p, q} − τ_{0} > dim S(β|_{K×T M}) > dim S(β′|_{K′×T M}) + dim K − dim K′ ≥ 2,

where the first inequality follows from the fact that S(β|_{K×T M}) ⊂ Γ⊕Γ̂ is a null subspace, and that τ_{0} ≤ dim S < dim Γ^{⊥} because β_{0} ≠ 0. But min{p, q} ≤ 6 by assumption, and hence τ_{0} ≤ 2. Corollary 17 now yields

dim Θ + δ_{0} ≥ n − dim S(β) ≥ n − p − q + 2τ + 2 dim S(β_{0}) ≥ n − p − q + 2τ + τ_{0},

as we wished.

Lemma 19. Let f: M^{n} → ℝ^{n+p} and f̂: M^{n} → ℝ^{n+q} be isometric immersions satisfying min {p, q} ≤ 6. Then,

dim Θ_{0} ≥ dim Θ − τ_{1} − δ_{τ_{1}}^{6},  (21)

where τ_{1} = dim S_{1}.

Proof: We only have to argue for τ_{1} ≥ 1. Set V_{Z} = γ(T M, Z) ⊂ S_{1} and m = dim V_{Z}, where γ = α_{S_{1}}|_{T M⊕Θ} and Z ∈ RE(γ) ⊂ Θ. We claim that 1 ≤ m ≤ [τ_{1}/2], where [ · ] denotes the entire part function. Observe that S_{1} = S(γ), and take the minimal number of elements Z_{1}, …, Z_{k_{0}} ∈ RE(γ) such that

S_{1} = S(γ|_{T M×span{Z_{1},...,Z_{k_{0}}}}) = ∑_{j=1}^{k_{0}} V_{Z_{j}}.  (22)

Suppose that m > [τ_{1}/2]. Then k_{0} ≤ τ_{1} − m + 1 ≤ [(τ_{1} + 1)/2]. Since τ_{1} ≤ 6, we easily see that L_{0} = ⋂_{j=1}^{k_{0}} V_{Z_{j}} ⊂ S_{1} satisfies L_{0} ≠ 0. On the other hand, Lemma 12–(ii) easily gives ⟨K(X)α(Y, Z), α(T, Z)⟩ = 0 for any Z ∈ Θ. Equivalently, K(X)V_{Z} ⊂ V_{Z}^{⊥} ⊂ S_{1} for any Z ∈ Θ and X ∈ T M. It follows using (22) that K(X)L_{0} ⊂ ⋂_{j=1}^{k_{0}} V_{Z_{j}}^{⊥} = 0 for any X ∈ T M. Hence, L_{0} ⊂ S_{0} ∩ S_{1} = 0. This is a contradiction and proves the claim.

Notice that Θ_{0} = N (α_{S_{1}}|_{Θ×T M}). For L(X) = γ(X, ·): Θ → S_{1} this is equivalent to

Θ_{0} = ⋂_{X∈T M} ker L(X).  (23)

Fix Z ∈ RE(γ). Then (14) gives S_{1} = ∑_{i=1}^{m} Im L(X_{i}), where X_{1}…. , X_{m} ∈ T M are such that V_{Z} = span{γ(X_{i}, Z), 1 ≤ i ≤ m}. Let {Y_{1}, …, Y_{m_{0}}} ⊂ {X_{1}, …, X_{m}} be a subset with the minimum number of elements satisfying

S_{1} = ∑_{j=1}^{m_{0}} Im L(Y_{j}).  (24)

We show that (23) can be replaced by

Θ_{0} = ⋂_{1≤j≤m_{0}} ker L(Y_{j}).  (25)

In fact, from (11) and (24) we easily obtain S_{1} = ∑_{j=1}^{m_{0}} Im K(Y_{j}). Equivalently, we have

⋂_{1≤j≤m_{0}} ker K(Y_{j}) = 0.  (26)

Since Z ∈ ⋂_{j=1}^{m_{0}} ker L(Y_{j}) if and only if K(X)γ(Y_{j}, Z) = K(Y_{j})γ(X, Z) = 0 for all X ∈ T M and 1 ≤ j ≤ m_{0}, then (25) follows from (23) and (26).

We show that (21) holds without the Kronecker symbol when a nonsingular K(Y) exists. By (26) this is the case when m_{0} = 1. We have,

K(Y)γ(X, ker L(Y)) = K(X)γ(Y, ker L(Y)) = 0.

Thus, ker L(Y) ⊂ Θ_{0}, and dim Θ_{0} ≥ dim ker L(Y) ≥ dim Θ − τ_{1} as wished. Since [τ_{1}/2] ≥ m ≥ m_{0}, it suffices to argue for τ_{1} ≥ 4 and the K(Y) are all singular. Assume m_{0} = 2. After taking linear combinations, if necessary, we obtain from (26) that there are Y_{1}, Y_{2} such that rank K(Y_{j}) = 4. Thus τ_{1} ≥ 5, and dim ker L(Y_{1}) ≥ dim Θ − τ_{1} + 1 since Im L(Y_{1}) ≠ S_{1} by (24). On the other hand,

K(Y_{1})γ(Y_{2}, ker L(Y_{1})) = K(Y_{2})γ(Y_{1}, ker L(Y_{1})) = 0,

and (21) follows from (25) and (26). If m_{0} = 3, then τ_{1} = 6 and dim Im L(Y_{j}) ≤ 4. Moreover, dim ker K(Y_{1}) ∩ ker K(Y_{2}) ≤ 1, and now (21) follows similarly.

Proof of Theorem 14: If L = S then (9) gives D^{d} = Θ_{0} = Θ, and the estimate for d follows from (16). Thus, we may assume that α_{L^{⊥}∩S}|_{Θ×T M} does not vanish and apply Lemma 15. By (9) there is a subspace V_{0}^{k_{0}} ⊂ T M of minimal dimension such that

S(α_{L^{⊥}∩S}|_{Θ×V_{0}}) = L^{⊥} ∩ S,  (27)

and such that Θ′ = N (α_{L^{⊥}∩S}|_{Θ×V_{0}}) ⊂ Θ satisfies

dim Θ′ ≥ dim Θ − k_{0}(ρ − 1) − 1  (28)

with 1 ≤ k_{0} ≤ τ_{0} + τ_{1} − ℓ − ρ + 1 and 1 ≤ ρ ≤ τ_{0} + τ_{1} − ℓ ≤ 6.

We first show that D_{0} = Θ′ ∩ Θ_{0} satisfies

dim D_{0} ≥ n − p − q + 2 ℓ + τ_{0}.  (29)

We have by (28) that

dim D_{0} ≥ dim Θ_{0} − k_{0}(ρ − 1) − 1.  (30)

It follows from the definition of Θ_{0} that D_{0} = N (α_{L^{⊥}∩S_{0}}|_{Θ_{0}×V_{0}}). In particular,

dim D_{0} ≥ dim Θ_{0} − k_{0}(τ_{0} − ℓ).  (31)

By obtain form (30) and (31) that

dim D_{0} ≥ dim Θ_{0} − min{k_{0}(ρ − 1) + 1, k_{0}(τ_{0} − ℓ)}.

Now we use that the deformation is genuine. We have from (16) and (21) that

dim D_{0} ≥ n − p − q + τ_{1} + 3τ_{0} − δ_{0} − δ_{τ_{1}}^{6} − min{k_{0}(ρ − 1) + 1, k_{0}(τ_{0} − ℓ)}.

To prove (29) we have to verify that

min{k_{0}(ρ − 1) + 1, k_{0}(τ_{0} − ℓ)} ≤ 2(τ_{0} − ℓ) + τ_{1} − δ_{0} − δ_{τ_{1}}^{6},  (32)

where 1 ≤ k_{0} ≤ τ_{0} + τ_{1} − ℓ − ρ + 1 and 1 ≤ ρ ≤ τ_{0} + τ_{1} − ℓ ≤ 6. First observe that δ_{0} = 0 since we are assuming τ_{1} ≥ 1. Moreover, δ_{τ_{1}}^{6} = 1 only if τ_{1} = 6 and τ_{0} = ℓ = 0, and then (32) holds. The remaining of the argument is a straightforward verification that can be done, for instance, with a simple computer script.

We claim that D_{0} = D^{d}, and then the estimate follows since τ_{0} ≥ ℓ . Clearly, D^{d} ⊂ D_{0}. On the other hand, since the left hand side of (10) vanishes if Z ∈ D_{0} and X ∈ V_{0}, we have from (27) and the definition of L^{ℓ} that S and Ŝ are parallel along D_{0} in the normal connections. Now (10) yields ∇_{Y}^{⊥}α_{L^{⊥}∩S_{0}}(Z, X) ∈ S for all Y ∈ Θ, Z ∈ D_{0} and X ∈ T M for both immersions. From the definition of L^{ℓ} we have that α_{L^{⊥}∩S_{0}}(Z, X) = 0 for all Z ∈ D_{0} and X ∈ T M, that is, D_{0} ⊂ D^{d}, and the claim follows.

Proof of Theorem 1: The result follows from L_{D} ⊂ L by definition of L_{D}.

Remarks 20. (1) Two important facts can be added to the conclusions of Theorem 1. It can easily be proved using L_{D} ⊂ L^{ℓ} that

D^{d} = N (α_{L_{D}^{⊥}}) ∩ N (α̂L̂_{D}^{⊥}).

Moreover, we saw in Theorem 11 that L^{ℓ} is parallel in the normal connections along the leaves of D^{d}. This also holds in Theorem 1 for L_{D} since, by the Codazzi equation, any D^{d}-ruled submanifold has that property.

(2) That T_{D} in Theorem 1 is parallel can be proved directly by comparing the Codazzi equations of both immersions.

(3) Without the assumption that the deformation is genuine in Theorem 14 we still have the estimate d ≥ n − p − q + 2ℓ. To see this we use (17) instead of (16) in its proof.

(4) Theorem 14 does not assume that the second fundamental form of one of the immersions spans the full normal space as it is usually asked for rigidity results. On the other hand, we see from the proof that we can replace p and q in the estimate of d by dim S(α) and dim S(α̂). Of course, everything just said holds for Theorem 1.

Further applications.

In this section, we give several applications of our main result beside the ones already stated in the introduction that are now proved. The corresponding statements for the case min {p, q} = 6 are straightforward.

Our first an immediate application extends Corollary 3 in the introduction. We define the (local) index ν_{R}(f) for f: M^{n} → ℝ^{n+p} by

ν_{R}(f) = max{d − 3ℓ_{D} : f|_{U} is D^{d}-ruled for some open subset U ⊂ M^{n}}.

Corollary 21. Let f: M^{n} → ℝ^{n+p} be an isometric immersion and let q be a positive integer such that p + q < n and min {p, q} ≤ 5. If ν_{R}(f) ≤ n − p − q − 1, then f is genuinely rigid in ℝ^{n+q}.

Even for submanifolds with very degenerate second fundamental form one can have genuine rigidity. In fact, we show in [9] that there is a large class of n-dimensional submanifolds in codimension 2 with constant index of relative nullity ν = n − 2 that are genuinely rigid in ℝ^{n+2}.

Proof of Corollary 4: If the function h(x) = ∥f(x)∥^{2} reaches a maximum at x_{0} ∈ M^{n}, then ξ_{0} = f(x_{0}) ∈ T_{x_{0}}^{⊥}M and A_{ξ_{0}} is definite. Hence, A_{ξ} is definite in a neighborhood U of x_{0} for a given smooth extension ξ of ξ_{0}. Therefore, f|_{U} must be genuinely rigid in ℝ^{n+q} because, otherwise, f would be D^{d}–ruled with d ≥ n − p − q > 0 on an open dense subset, and thus A_{ξ}|_{D×D} = 0.

The following result extends Corollary 5 in the introduction. Example 2 shows that the bound given for μ is sharp.

Corollary 22. Let f: M^{n} → ℝ^{n+p} be an isometric immersion and let q be a positive integer such that p + q < n and min {p, q} ≤ 5. If M^{n} has nonnegative Ricci curvature and the dimension of nullity of the curvature tensor μ satisfies μ < n − p − q, then f is genuinely rigid in ℝ^{n+q}.

Proof: If otherwise, then f is (n−p−q)–ruled on an open subset. By the Gauss equation and the assumption on the Ricci curvature the rulings belong to the relative nullity of the immersion, and hence to the nullity of the curvature tensor.

Corollary 23. Let f̂: M^{n} → ℝ^{n+q} be a genuine deformation of an isometric immersion f: M^{n} → ℝ^{n+p} such that p + q < n and min {p, q} ≤ 5. If M^{n} has nonpositive sectional curvature, then f and f̂ have common relative nullity of dimension ν ≥ n − p − q + 2ℓ.

Proof: It was shown in Proposition 8 in [16] that if α has nonpositive sectional curvature and an asymptotic subspace D^{d}, i.e., α|_{D×D} = 0, then ν ≥ d−s, where s = dim γ(D, Y) with Y ∈ RE(γ) and γ = α|_{D×D^{⊥}}. By Theorem 14 we have S(γ) ⊂ L^{ℓ}, and we obtain the estimate for ν from the one for d. Observe that the relative nullity contained in D^{d} must be shared by f̂ since it coincides with the set of vectors in D^{d} that belong to the nullity of the curvature tensor of M^{n}.

The s-nullity ν_{s}^{f}(x) of f: M^{n} → ℝ^{n+p} at x ∈ M^{n} for 1 ≤ s ≤ p is defined by

ν_{s}^{f}(x) = max{dim N (α_{V^{s}}(x)) : V^{s} ⊂ T_{f(x)}^{⊥}M}.

The following result was proved in [7] by a different argument.

Theorem 24 ([7]). Let f: M^{n} → ℝ^{n+p} be an isometric immersion and q ≥ p a positive integer. Suppose p ≤ 5, and assume that f satisfies everywhere

ν_{s}^{f} ≤ n + p − q − 2s − 1 for all 1 ≤ s ≤ p.

For q ≥ p + 5 assume further that ν_{1}^{f} ≤ n − 2(q − p) + 1 everywhere. Then any isometric immersion f̂: M^{n} → ℝ^{n+q} is a composition on connected components of an open dense subset of M^{n}.

Proof: Theorem 1 applies and gives (possibly trivial) local isometric Δ-ruled extensions F and F̂ of maximal dimension defined on a manifold N^{n+r} with 0 ≤ r ≤ p. It remains to show that r = p. It is easy to see from the hypothesis on the s-nullities that L^{⊥} = 0, that is, α̂F̂ = α^{F} ⊕ γ, and that dim Δ ≥ n + 2p − q. If r < p, then take a normal vector field η ≠ 0 to F and conclude that f satisfies ⟨A_{η} D, D⟩ = 0 along points in M^{n}, where D = Δ ∩ T M. Since dim D ≥ dim Δ − r ≥ n − (q − p) + 1, then ν_{1}^{f} ≥ n − 2(q − p) + 2, and this is a contradiction with our assumptions on ν_{1}^{f}.

We conclude this section with the following intrinsic criteria for genuine rigidity.

Theorem 25. Let f: M^{n} → ℝ^{n+p} be an isometric immersion and q a positive integer such that p+q < n and min{p, q} ≤ 6. If there is a genuine deformation of f into ℝ^{n+q}, then the k-th Pontrjagin form p_{k} of M^{n} vanishes for

k ≥ \{ \begin{matrix}1  if  p + q ≤ 3, \\ 2  if  4 ≤ p + q ≤ 6, \\ (p + q − 2)/2  if  7 ≤ p + q ≤ 13, \\ (3(p + q) − 17)/4 if  p + q ≥ 14. \end{matrix}

In particular, when M^{n} is compact and the k-th Pontrjagin class [p_{k}] does not vanish, then any analytic isometric immersion of M^{n} in ℝ^{n+p} is genuinely rigid in ℝ^{n+q} in the C^{∞}-category if p, q and k are related as above.

For the proof of the last result we need the following fact.

Proposition 26. If the immersion f: M^{n} → ℝ^{n+p} is d-ruled, then the k-th Pontrjagin form p_{k} of M^{n} vanishes for any k such that 4k > 3(n − d).

Proof: Let {e_{1}, …, e_{n}} be an orthonormal tangent frame of M^{n} and {w_{1}, …, w_{n}} its dual frame. For 1 ≤ a, b ≤ n we have the curvature 2-forms Ω_{ab} = ∑_{r,s=1}^{n} R_{abrs}w_{r} ∧ w_{s}, where R_{abrs} are the components of the curvature tensor in the given frame. Then, the Pontrjagin form p_{k} has the local expression

p_{k} = ∑_{(i)} Θ_{(i)}^{2k} ∧ Θ_{(i)}^{2k},

where Θ_{(i)}^{2k} = ∑_{(j)} δ_{(i)}^{(j)}Ω_{j_{1}j_{2}} ∧ Ω_{j_{2k−1}j_{2k}}. Here, (i) = (i_{1}, ⋯, i_{2k}) and (j) = (j_{1}, ⋯, j_{2k})

run over all the 2k-uples of distinct elements in {1, …, n}, and δ_{(i)}^{(j)} is +1 (resp., −1) if (j_{1}, ⋯, j_{2k}) is an even (resp., odd) permutation of (i_{1}, ⋯, i_{2k}) and zero otherwise. Therefore, p_{k} is a sum (up to a constant factor) of terms of the form

Ω_{j_{1}j_{2}} ∧ ⋯ ∧ Ω_{j_{2k−1}j_{2k}} ∧ Ω_{j′_{1}j′_{2}} ∧ ⋯ ∧ Ω_{j′_{2k−1}j′_{2k}},  (33)

with {j_{1}, ⋯, j_{2k}} = {j′_{1}, ⋯, j′_{2k}}. By the Gauss equation

Ω_{ab} = \frac{1}{2} ∑_{r,s=1}^{n}⟨α(e_{a}, e_{r}), α(e_{b}, e_{s})⟩ w_{r} ∧ w_{s}.

We assume that e_{n−d+1}, …, e_{n} belong to the rulings. If a ≥ n−d+1 (resp., b ≥ n−d+1), then r (resp., s) in the last equation runs only up to n − d. Thus, if 2k > n − d, then (33) is a linear combination of terms of the form w_{r_{1}} ∧ ⋯ ∧ w_{r_{2k}} ∧ w_{r′_{1}} ∧ ⋯ ∧ w_{r′_{2k}}, where at least 2k − n + d of the r_{i}’s and also of the r′_{i}’s are less or equal than n − d. We conclude that all terms must vanish if 2(2k − n + d) > n − d.

Proof of Theorem 25: If f has a genuine deformation, then the immersions must have the structure given by Theorem 14. It follows that either f is (n − p − q + 3ℓ)-ruled if min{p, q} ≤ 5 or ℓ ≥ 1, or its index of relative nullity satisfies ν ≥ n − p − q − 1 if min{p, q} = 6 and ℓ = 0. In the first case, it is easy to see that ν ≥ n−(ℓ+1)(p+q−3ℓ). The result is then a consequence of Proposition 26 and that p_{k} = 0 if 4k > n − ν, since the Pontrjagin forms are curvature forms and the relative nullity spaces are contained in the nullity of the curvature tensor.

Proof of Theorem 7: We use Theorem 25 and that p_{k} = p_{1}^{k} since at least one of the immersions has codimension at most three.

Final comments.

First we argue that the estimate d ≥ n − p − q + 3ℓ in our main result does not come as a complete surprise. The bilinear form ϕ defined by (3) for (T , D^{d}) given by Theorem 11 is flat. In fact, nothing changes if ϕ is replaced with

ϕ′ = ϕ|_{(D^{⊥}⊕L)×T M}: (D^{⊥} ⊕ L) × T M → L^{⊥} × L̂^{⊥}

since ϕ(D, T M) = 0. Take a vector field X ∈ RE(ϕ′) and set B_{X} = ϕ′( · , X). If a section Y_{0} + ξ_{0} ∈ ker B_{X} satisfies ξ_{0} ≠ 0 everywhere, then the maps F (x, t) = f(x) + t(Y_{0} + ξ_{0}) and F̂(x, t) = f̂(x) + t(Y_{0} + ξ_{0}) are local isometric extensions of f and f̂ in M^{n}×(−ϵ, ϵ), for some ϵ > 0. But this is not possible if f̃ is a genuine deformation of f. Observe that

dim ker B_{X} ≥ n − d − p − q + 3ℓ,

and hence, we cannot conclude that ker B_{X} ≠ 0 if d satisfies our estimate.

Next we show that isometric extensions is indeed a matter for pairs of submanifolds even for codimensions as low as p = q = 2. In fact, in (1) we may have different extensions F for different isometric deformations f̂ of f. Moreover, it may happen that the given pair f and f̂ does extend isometrically but there exists an isometric deformation of f that is genuine. To see that all this is possible, consider the following examples.

Example 27. Let i: M^{n} = N_{1}^{n+1} ∩ N_{2}^{n+1} ↪ ℝ^{n+2} be the transversal intersection of two distinct Sbrana-Cartan hypersurfaces. Assume that the index of relative nullity of i takes its generic value n − 4 everywhere. Now consider two additional isometric immersions g_{j}: M^{n} → ℝ^{n+2} determined by isometric deformations N̂_{j}^{n+1} of N_{j}^{n+1}. One can prove that the isometric extension of the pair of immersions i and g_{j} of M^{n} recreates N_{j}^{n+1} and N̂_{j}^{n+1}, and thus the extension of i also depends on g_{j}. Moreover, g_{2} must be a genuine deformation of g_{1} since, otherwise, their second fundamental forms would have to coincide on a normal subbundle, and that is not possible.

Example 28. We showed in [9] that a nonholomorphic isometric immersion of rank two and codimension two of a simply connected and nowhere flat Kaehler manifold is the intersection of a one-parameter family of (non isometric) deformable hypersurfaces.

Our ruled extensions are always nonsingular by definition and trivial if the pair of isometric immersions we start with are mutually ruled. Nevertheless, there are situations where it is convenient to consider more general ruled extensions. In fact, this is the case of the extensions in Examples 28 and the ones in [12]. In the latter case, it was shown that the subset of singular points of a so called Generalized Sbrana-Cartan hypersurface N^{n+1} in ℝ^{n+2} is a hypersurface N_{0}^{n} ⊂ N^{n+1} with index of relative nullity ν = n − 2 as a submanifold of ℝ^{n+2}. Moreover, deformations of N^{n+1} yield deformations of N_{0}^{n} that preserve the relative nullity distribution. Given such an isometric deformation of N_{0}^{n}, we thus have an example of a pair of submanifolds in codimension two that satisfies the conclusion of Theorem 1 whose “isometric extensions” are singular. In fact, one can show that their unique (possibly singular) isometric extensions are immersions of N^{n+1} in ℝ^{n+2}, and hence the condition Δ ∩ T M = D is no longer satisfied. In particular, we obtain the following example that relates to Corollary 4.

Example 29. A deformation of a compact hypersurface M^{n} ↪ N^{n+1} that contains an open subset of N_{0}^{n} as above given by a deformation of N^{n+1} can not be extended isometrically and regularly along connected components of an open dense subset.

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IMPA – Estrada Dona Castorina, 110 22460-320 — Rio de Janeiro — Brazil marcos@impa.br —— luis@impa.br