A local and global splitting result for real Kähler Euclidean submanifolds ^{∗}

Luis A. Florit^{†} and Fangyang Zheng^{‡}

Abstract We show that if a real Kähler Euclidean submanifold is as far as possible of being minimal, then it should split locally as a product of hypersurfaces almost everywhere, possibly in lower codimension. In addition, if the manifold is complete, simply connected and has constant nullity, it should split globally as a product of surfaces in ℝ^{3} and an Euclidean factor. Several applications are also given.

The series of papers [8, 9, 11, 12] was devoted to the study of the relative nullity distribution Δ of an isometric immersion f: M^{m} → ℝ^{m+p}, that is, Δ = Ker α is the nullity space of the second fundamental form α of f, when the m–dimensional (connected) Riemannian manifold M^{m} is assumed to have nonpositive sectional curvature. Since, along any open subset where ν = dim Δ is constant, the relative nullity is a smooth integrable distribution with totally geodesic leaves in both M^{m} and ℝ^{m+p}, the positiveness of ν imposes strong restrictions on the submanifold structure.

In [10] we gave analogous results to the main ones in [8] and [11] when f is a real Kähler Euclidean submanifold, that is, f: M^{2n} → ℝ^{2n+p} is a (real) isometric immersion of a Kähler manifold, but with the reversed sign on weaker curvatures, namely, when M^{2n} has either nonnegative Ricci curvature or nonnegative holomorphic sectional curvature. It was shown that, with either curvature assumption, ν ≥ 2n − 2p and, if equality holds everywhere, then f should split locally as a product of p real Kähler hypersurfaces almost everywhere, the splitting being global when M^{2n} is complete. The purpose of this note is to give a general theorem for real Kähler Euclidean submanifolds that, in particular, allows us to further extend these results.

Recall that such a real Kähler Euclidean submanifold f is said to be pluriharmonic when every holomorphic curve in M^{2n} is mapped by f onto a minimal surface in ℝ^{2n+p}. It is easy to see that f is pluriharmonic if and only if the J-invariant subspace Δ_{J}(x) of T_{x}M given by

Δ_{J}(x) = {X ∈ T_{x}M : α(X, JY) = α(JX, Y), ∀ Y ∈ T_{x}M}

∗Mathematics Subject Classification (2000): Primary 53B25; Secondary 53C40.

†IMPA: Estrada Dona Castorina 110, 22460–320, Rio de Janeiro, Brazil; e-mail: luis@impa.br.

Research partially supported by CNPq.

‡The Ohio State University: Columbus, OH 43210, USA; e-mail: zheng@math.ohio-state.edu. Research partially supported by a NSF grant.

satisfies that Δ_{J}(x) = T_{x}M for all x ∈ M^{2n}. Due to this property, we define the index of pluriharmonic nullity of f at x by

ν_{J}(x) = dim_{ℂ} Δ_{J}(x).

It is an interesting fact that f is pluriharmonic, i.e., ν_{J} ≡ n, if and only if it is minimal; cf. [7] and Remark 8 in [10]. On the other side, our aim here is to show that, when f is as far as possible of being pluriharmonic, it should split locally as a product of n − μ/2 hypersurfaces almost everywhere, possibly in lower codimension. Here, μ stands for the index of nullity of M^{2n}, that is, the (even) dimension of the nullity space of its curvature tensor R: μ = dim {X ∈ T M : R(X, · ) = 0}. More precisely, we have:

Theorem 1. Let f: M^{2n} → ℝ^{2n+p} be any real Kähler Euclidean submanifold. Then, ν_{J} ≥ n−p. Assume further that equality holds everywhere and that M^{2n} is nowhere flat. Then, along each connected component W_{λ} of an open dense subset W ⊂ M^{2n}, μ is constant and there are r = n−μ/2 ≥ 1 nowhere flat real Kähler Euclidean hypersurfaces f_{i}: M_{i}^{2n_{i}} → ℝ^{2n_{i}+1}, 1 ≤ i ≤ r, such that W_{λ} is the Riemannian product

W_{λ} = M_{1}^{2n_{1}} × ⋯ × M_{r}^{2n_{r}}.

In addition, there is an open subset V ⊂ ℝ^{2n+r} with (f_{1} × ⋯ × f_{r})(W_{λ}) ⊂ V, and an isometric immersion h: V ⊂ ℝ^{2n+r} → ℝ^{2n+p} such that f|_{W_{λ}} = h ∘ (f_{1} × ⋯ × f_{r}).

In the above situation, we say that f̂ = f_{1} × ⋯ × f_{r} : W_{λ} → ℝ^{2n+r} splits as a product of r hypersurfaces. It is easy to see that any nowhere flat real Kähler Euclidean hypersurface M^{2m} ⊂ ℝ^{2m+1} must have ν ≡ μ ≡ 2m−2. Hence, they are locally classified by means of the Gauss parametrization in terms of pseudoholomorphic surfaces in the sphere; cf. [1, 3, 4, 11, 12]. It turns out that the only complete real Kähler Euclidean hypersurfaces are the products of a complete orientable surface in ℝ^{3} and an Euclidean factor ([13]).

As an easy consequence of Theorem 1, we can now extend the main results in [10]:

Corollary 2. Let f: M^{2n} → ℝ^{2n+p} be an isometric immersion of a Kähler manifold with either nonnegative Ricci curvature or nonnegative holomorphic sectional curvature. Then, the index of relative nullity ν of f satisfies that ν ≥ 2n − 2p. Moreover:

i) If ν ≡ 2n − 2p, then there is an open dense subset W ⊂ M^{2n} such that f|_{W} splits locally as a product of p nowhere flat real Kähler Euclidean hypersurfaces with nonnegative Ricci curvature;

ii) If ν ≡ 2n − 2p + 1, then there is an open dense subset W ⊂ M^{2n} such that f|_{W} = h ∘ f̂, where f̂: M^{2n} → ℝ^{2n+p−1} splits locally as a product of p − 1 nowhere flat real Kähler Euclidean hypersurfaces with nonnegative Ricci curvature, and h is an isometric immersion h: V ⊂ ℝ^{2n+p−1} → ℝ^{2n+p} with f̂(W) ⊂ V.

Part ii) is precisely the positive answer to the question posed in [10], that is, the analogous result to the main one in [12] holds for real Kälher Euclidean submanifolds but with the reversed sign on weaker curvatures. We point out that, since the flat immersion h in part ii) has relative nullity of codimension one, we conclude that in either part the submanifold can be locally described using the Gauss parametrization for each factor, thus given a complete local representation.

As a simple application, we obtain the following for submanifolds of the cylinder over a sphere 𝕊_{c}^{m} of constant sectional curvature c, that strongly generalizes Corollary 5 in [8] for c > 0 since no assumptions on curvatures are needed:

Corollary 3. Let M^{2n} ⊂ 𝕊_{c}^{m} ×ℝ^{k} be a real Kähler submanifold, with 6n ≥ 2m+3k +2. Then, k = 2s is even, m = 3(n−s)−1 and M^{2n} is an open subset of 𝕊_{c_{1}}^{2} ×⋯×𝕊_{c_{n−s}}^{2} ×ℂ^{s}, where 1/c = 1/c_{1}+⋯+1/c_{n−s}. In particular, the only Kälher submanifolds M^{2n} ⊂ 𝕊_{c}^{3n−1} are the (open subsets of) products of umbilical 2-spheres, 𝕊_{c_{1}}^{2}× ⋯ × 𝕊_{c_{n}}^{2} ⊂ 𝕊_{c}^{3n−1} ⊂ ℝ^{3n}.

The presence of a positive definite shape operator clearly implies that ν_{J} = 0. So:

Corollary 4. Let f: M^{2n} → ℝ^{2n+p} be an analytic isometric immersion of a compact Kähler manifold with p ≤ n. Then, p = n and f splits globally as a product of n compact surfaces in ℝ^{3}.

Since the shape operator in the direction of the mean curvature vector of any Euclidean submanifold with positive Ricci curvature is positive definite, we also obtain:

Corollary 5. Let N ⊂ ℝ^{2n+p} be an immersed Euclidean submanifold with positive Ricci curvature, with p ≤ n. Then, N has no immersed Kähler submanifold M^{2n} ⊂ N, unless p = n and M^{2n} ⊂ N ⊂ ℝ^{3n} splits as a product of n surfaces in ℝ^{3}.

The open dense subset W in Theorem 1 is necessary due to the fact that the nullity μ of the manifold does not have to be constant, and because of the “jumping” of the relative nullity between the hypersurface factors, phenomena that indeed occur. When these are controlled, we get global consequences. To state them, we need a slightly weaker notion of compositions than the one considered in Theorem 1 (see [5]):

Definition. Given an isometric immersion g: M^{m} → ℝ^{m+r} we say that an isometric immersion f: M^{m} → ℝ^{m+p}, p ≥ r, is a composition of g when there is an isometric embedding g′: M^{m} ↪ N_{0}^{m+r} into a flat manifold N_{0}^{m+r}, a local isometry j: N_{0}^{m+r} → ℝ^{m+r} satisfying g = j ∘ g′ and an isometric immersion h: N_{0}^{m+r} → ℝ^{m+p} such that f = h ∘ g′.

Observe that, along any open subset U ⊂ M^{m} where g is an embedding, there exists an isometric immersion h: V ⊂ ℝ^{m+r} → ℝ^{m+p} of a tubular neighborhood V of g(U) such that f = h ∘ g. In particular, this holds globally if g itself is an embedding.

Theorem 6. Let f: M^{2n} → ℝ^{2n+p} be a complete simply connected nonflat real Kähler Euclidean submanifold. Assume that ν_{J} ≡ n − p and that μ is constant. Then, there are r = n−μ/2 ≥ 1 complete orientable nowhere flat surfaces f_{i} : M_{i}^{2} → ℝ^{3} such that

M^{2n} = M_{1}^{2} × ⋯ × M_{r}^{2} × ℂ^{n−r},

and f is a composition of the product immersion f_{1}×⋯×f_{r}×I, where I: ℂ^{n−r} → ℝ^{2n−2r} is the identity map.

In particular, if M^{2n} is compact, then each M_{i}^{2} ⊂ ℝ^{3} is a strictly convex surface, hence embedded. So in this case we conclude that r = n = p and f = f_{1} ×⋯×f_{n} splits globally as a product of n surfaces in ℝ^{3}. We can also extend the global results in [10]:

Corollary 7. Let f: M^{2n} → ℝ^{2n+p} be an isometric immersion of a complete simply connected Kähler manifold with either nonnegative Ricci curvature or nonnegative holomorphic sectional curvature. Then ν ≥ 2n − 2p. Moreover:

i) It holds that ν ≡ 2n − 2p if and only if we have the global splittings

M^{2n} = M_{1}^{2} × … × M_{p}^{2} × ℂ^{n−p} and f = f_{1} × ⋯ × f_{p} × I,

where f_{i}: M_{i}^{2} → ℝ^{3}, 1≤i≤p, is a complete oriented surface of positive curvature;

ii) It holds that ν ≡ 2n − 2p + 1 if and only if we have the global splittings

M^{2n} = M_{1}^{2} × … × M_{p−1}^{2} × ℂ^{n−p+1} and f = h ∘ (f_{1} × ⋯ × f_{p−1} × I),

where f_{i}: M_{i}^{2} → ℝ^{3}, 1 ≤ i ≤ p − 1, is a complete oriented surface of positive curvature, and h is an isometric immersion h: V ⊂ ℝ^{2n+p−1} → ℝ^{2n+p}.

Further applications of our main results here will be given in [14].

The proofs

Let us first fix some notations. From now on, f: M^{2n} → ℝ^{2n+p} will be an isometric immersion of the Kähler manifold M^{2n} into the Euclidean space. Fix a point x ∈ M^{2n}. The second fundamental form α = α(x) of f at x is the symmetric bilinear map

α : T_{x}M × T_{x}M → T_{x}^{⊥}M = N,

where T_{x}M is the real tangent space of M^{2n}, and (T_{x}^{⊥}M, ⟨ , ⟩) the normal space of f at x. Extend α bilinearly over ℂ, and still denote it by α : (T_{x}M)⊗ℂ × (T_{x}M)⊗ℂ → N⊗ℂ . Let V be the space of type (1, 0) tangent vectors at x, that is, V is the complex subspace of (T_{x}M)⊗ℂ given by V = \{v − iJv : v ∈ T_{x}M\}. Thus, (T_{x}M)⊗ℂ ≅ V ⊕ \overline{V}. Write

H = α|_{V ×\overline{V}} and S = α|_{V ×V}

for the (1, 1) and (2, 0) parts of α, respectively. Then S : V ×V → N⊗ℂ is a symmetric complex bilinear map, while H : V × \overline{V} → N ⊗ℂ is Hermitian, that is,

H(Y, \overline{X}) = \overline{H(X, \overline{Y})} , ∀ X, Y ∈ V.

For the sake of simplicity, we write S_{XY} for S(X, Y) and H_{X\overline{Y}} for H(X, \overline{Y}). Let us also extend to N ⊗ℂ the inner product ⟨ , ⟩ on N bilinearly over ℂ, and still denote it by ⟨ , ⟩. We have the following key result from [10]:

Lemma 8. Let f: M^{2n} → ℝ^{2n+p} be a real Kähler Euclidean submanifold, p ≥ 1. Take x ∈ M^{2n}, and let V, N, H, S be as above. Then, ν_{J}(x) ≥ n − p. Moreover, if ν_{J}(x) = n − p, there exists a basis \{e_{1}, …, e_{n}\} of V such that H_{e_{i}\overline{e_{j}}} = S_{e_{i}e_{j}} = 0, if either i ≠ j or i = j > p. Moreover, for 1 ≤ i ≤ p, H_{e_{i}\overline{e_{i}}} ≠ 0, S_{e_{i}e_{i}} is collinear to w_{i} = H_{e_{i}\overline{e_{i}}}/|H_{e_{i}\overline{e_{i}}}|, and \{w_{1}, …, w_{p}\} is an orthonormal basis of the normal space N at x.

Assume from now on that ν_{J}(x) = n − p, for all x ∈ M^{2n}. Our purpose first is to realize M^{2n} as an Euclidean submanifold in its optimal codimension by means of the Fundamental Theorem for submanifolds.

By Lemma 8, ν ≥ 2n−2p and there exists a tangent diagonalizing frame {e_{1}, …, e_{p}} of type (1, 0) vectors in Δ_{1,0}^{⊥} = V ∩ (Δ^{⊥} ⊗ ℂ) at each x. Note that this frame is unique up to permutations and scalings. In other words, the set {[e_{1}], …, [e_{p}]} is unique in the symmetric power S^{n}(ℙ(Δ_{1,0}^{⊥})) of the projectified holomorphic Δ_{1,0}^{⊥} bundle. Hence, since H is smooth, in a sufficiently small neighborhood U of x, we can take a smooth frame {e_{1}, …, e_{n}} such that it has the diagonalization property of Lemma 8 at each point in U. In particular, we also obtain from Lemma 8 the smooth orthonormal normal frame {w_{1}, …, w_{p}}.

For each 1 ≤ i ≤ p, consider the shape tensor A_{w_{i}} on M^{2n} defined by ⟨A_{w_{i}}X, Y ⟩ = ⟨α_{f}(X, Y), w_{i}⟩, and set

V_{i} = Im A_{w_{i}}.  (1)

By Lemma 8, 1 ≤ dim V_{i} ≤ 2 and

V_{1} ⊕ ⋯ ⊕ V_{p} = Δ^{⊥}.  (2)

Let Γ be the nullity space of the curvature tensor R of M^{2n}, i.e., Γ = {X ∈ T M : R(X, ·) = 0} and μ = dim Γ the index of nullity of M^{2n}. We have that 2n − 2r = μ ≥ ν ≥ 2n − 2p, where 2r = rank R. Changing the order of the base {e_{1}, …, e_{p}} if necessary, we thus can assume that

dim V_{j} = 2, ∀ 1 ≤ j ≤ r, and dim V_{s} = 1, ∀ r+1 ≤ s ≤ p.  (3)

Hence, ν = 2n − p − r and

V_{1} ⊕ ⋯ ⊕ V_{r} = Γ^{⊥}.  (4)

If μ is constant, which is always the case on each connected component of an open dense subset of M^{2n}, the decompositions (2) and (4) are smooth. Let L ⊂ T^{⊥}M be the normal subbundle of rank r spanned by {w_{1}, …, w_{r}} and π: T^{⊥}M → L its orthogonal projection. Set

ˆ∇^{⊥} = π ∘ ∇^{⊥}|_{T M×L} : T M × L → L and α̂ = π ∘ α : T M × T M → L.

Proposition 9. If M^{2n} is simply connected and μ is constant, there is an isometric immersion f̂: M^{2n} → ℝ^{2n+r} whose second fundamental form is α̂ = τ ∘ α, for some parallel bundle isometry τ: L → T ⊥ ; f̂M. In particular, the nullities of f̂ satisfy that μ̂ ≡ ν̂ ≡ 2ν̂_{J} ≡ 2n − 2r.

Proof: The last assertion follows from the construction of α̂ and (4). For the first one, we apply the Fundamental Theorem of submanifolds to the pair (α̂, ˆ∇^{⊥}). We have to show that it satisfies Gauss, Codazzi and Ricci equations for flat ambient space. It is clear that α̂ satisfies the Gauss equation since we took from α precisely the normal directions of rank one that does not contribute to the curvature.

Fix the following index conventions:

1 ≤ j, k ≤ r, r+1 ≤ s ≤ p, 1 ≤ i ≤ p.

Recall that the Codazzi equation for the shape operator A_{w_{i}} of f is

∇_{X}(A_{w_{i}}Y) − A_{w_{i}}∇_{X}Y − A_{∇_{X}^{⊥}w_{i}}Y = ∇_{Y} (A_{w_{i}}X) − A_{w_{i}}∇_{Y} X − A_{∇_{Y}^{⊥}w_{i}}X.  (5)

Taking in (5) X, Y ∈ V_{i}^{⊥} = Ker A_{w_{i}} we easily obtain using (2) that

A_{w_{j}}(ψ_{ij}(X)Y − ψ_{ij}(Y)X) = 0, ∀ X, Y ∈ V_{i}^{⊥},

where ψ_{ij} is defined by ψ_{ij}(X) = ⟨∇_{X}^{⊥}w_{i}, w_{j}⟩. Suppose that ψ_{ij}|_{V_{i}^{⊥}} ≠ 0. Then, the above implies that V_{i}^{⊥} ∩ Ker ψ_{ij} ⊂ Ker A_{w_{j}} = V_{j}^{⊥}. This is a contradiction to (2) since dim V_{j} = 2. Therefore,

V_{i}^{⊥} = Ker A_{w_{i}} ⊂ Ker ψ_{ij}.

Since ψ_{ij} = −ψ_{ji} if i = k ≤ r, we have that T M = (V_{j} ∩ V_{k})^{⊥} = V_{j}^{⊥} + V_{k}^{⊥} ⊂ Ker ψ_{jk}. Hence,

ψ_{jk} = 0,  (6)

i.e., w_{j} is parallel in L, and, by dimension reasons,

ψ_{js} = 0,  or Ker ψ_{js} = Ker A_{w_{s}}.  (7)

In particular, A_{w_{i}}(ψ_{ij}(X)Y − ψ_{ij}(Y)X) = 0, or, equivalently, A_{∇_{X}^{⊥}w_{j}}Y = A_{∇_{Y}^{⊥}w_{j}}X, for all X, Y ∈ T M. From (6) and (5) for i = j we conclude that (α̂, ˆ∇^{⊥}) verifies Codazzi equation.

The Ricci equation for f says that [A_{w_{j}}, A_{w_{k}}] = ∑_{s} ψ_{js} ∧ ψ_{ks} = 0, where the last equality follows from (7). Therefore, by (6) we conclude that (α̂, ˆ∇^{⊥}) satisfies Ricci equation.

Proof of Theorem 1: The estimate on ν_{J} follows from the first part of Lemma 8. Assume that ν_{J} ≡ n−p everywhere. Consider any simply connected open subset U ⊂ M^{2n} where μ is constant, and let f̂ be the isometric immersion given by Proposition 9 when applied to f|_{U}. Proposition 10 part ii) in [10] says that f̂ splits locally as a product of r nowhere flat real Kähler Euclidean hypersurfaces almost everywhere in U. Therefore, all we have to show is that f|_{U} is a composition of f̂.

With the notations of Lemma 8 and Proposition 9, let

T = span{( ˜∇_{X}ξ)_{T M⊕L} : X ∈ T M, ξ ∈ L^{⊥}},

where ˜∇ stands for the usual connection in ℝ^{2n+p} and (v)_{T M⊕L} denotes the orthogonal projection of v ∈ ℝ^{2n+p} onto T M ⊕ L. By Proposition 8 in [5], all we have to show is that T ∩ L = 0, since it is clear from Lemma 8, (2) and (4) that α decomposes regularly.

Indeed, if η ∈ T, there are a_{l} ∈ ℝ such that η = ∑_{l} a_{l}( ˜∇_{X_{l}}ξ_{l})_{T M⊕L}. Writing ξ_{l} = ∑_{s=r+1}^{p} b_{sl}w_{s}, we have that

η = ( ∑_{s=r+1}^{p} ˜∇_{Y_{s}}w_{s})_{T M⊕L}

where Y_{s} = ∑_{l} a_{l}b_{ls}X_{l}. If, in addition, η ∈ L, from (2) we get that Y_{s} ∈ Ker A_{w_{s}}. But then we conclude from (7) that η = 0.

Proof of Corollary 2: It is easy to check that ν ≥ 2ν_{J} under either curvature assumption (see the proof of Theorem 1 in [10]). Hence, ν_{J} = n−p everywhere, and r = p or r = p−1 in i) or ii), respectively. The result then follows from Theorem 1.

Proof of Corollary 3: Considering 𝕊_{c}^{m} × ℝ^{k} ⊂ ℝ^{m+k+1}, that has a positive semidefinite second fundamental form A, we have that ν_{J} ≥ n − (m + k + 1 − 2n) ≥ k/2. If ν_{J} > k/2, then there is X ∈ Δ_{J} such that ⟨AX, X⟩ > 0. But then ⟨AJX, JX⟩ < 0, a contradiction. So, 2ν_{J} = k = 2s, m = 3(n − s) − 1, and Δ_{J} is always parallel to the ℝ^{k} factor. Thus, M^{2n} ⊂ N^{2n−2s} × ℂ^{s}, for some N^{2(n−s)} ⊂ 𝕊^{3(n−s)−1} that is locally a product of n − s surfaces in ℝ^{3}. This can only occurs if each surface is itself (an open subset of) a two dimensional sphere.

Proof of Theorem 6: We have that ν = 2n−p−r is also constant. Hence, by Proposition 9 we have the isometric immersion f̂: M^{2n} → ℝ^{2n+r} with ν̂_{J} = n − r and ν̂ = 2n − 2r everywhere. We claim that the relative nullity Δ̂ of f̂ is, in fact, parallel in ℝ^{2n+r}.

Apply Theorem 1 to f̂, and consider a connected component W_{λ} where f̂ splits as a product of nowhere flat hypersurfaces. Take one of these hypersurfaces, say, f̂_{1}, and assume it is not locally a product of a surface and an Euclidean factor. So, there is a vector T in the relative nullity Δ̂_{1} ⊂ Δ̂ of f̂_{1} such that C_{T} ≠ 0, where C_{T} : Δ̂_{1}^{⊥} → Δ̂_{1}^{⊥} is the splitting tensor of Δ̂_{1}, i.e., C_{T} X = −(∇_{X}T)Δ̂_{1}^{⊥}. Observe that Δ̂_{1}^{⊥} has dimension two and is J-invariant, since Δ̂_{1} coincides with the nullity of the curvature tensor of f̂_{1}. By the product structure of f̂ in W_{λ}, the splitting tensor Ĉ_{T} : Δ̂^{⊥} → Δ̂^{⊥} of f̂ in the direction of T is zero except on Δ̂_{1}^{⊥}, where it equals C_{T}. On the other hand, since ν̂ is constant and M^{2n} is complete, the leaves of Δ̂ are complete. Since Ĉ_{JT} = JĈ_{T}, we easily check that there are a, b ∈ R such that Ĉ_{aT +bJT} = (aI + bJ)Ĉ_{T} has a nonzero real eigenvalue. But this is in contradiction with Lemma 1.8 in [6]. We conclude that each hypersurface factor in W_{λ} is locally a product of a surface and an Euclidean factor. Therefore, Ĉ_{T} = 0 for any T ∈ Δ̂, that is, M^{2n} = N^{2r} × ℂ^{n−r} and f̂ = f̂′ × I split for some complete real Kähler submanifold f̂′ : N^{2r} → ℝ^{3r} and the claim is proved.

We end the proof applying Proposition 10 part iii) in [10] to f̂′, since we then get that f̂′ splits globally as a product of nowhere flat orientable surfaces in ℝ^{3}.

Proof of Corollary 7: Again, we have everywhere that ν_{J} = n − p, and r = p or r = p − 1 in i) or ii), respectively. Theorem 6 then gives the result, since f̂ is an embedding because the surfaces are strictly convex.

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