ON TOTALLY REAL SPHERES IN COMPLEX SPACE

XIANGHONG GONG

Abstract. We shall prove that there are totally real and real analytic embeddings of S^{k} in C^{n} which are not biholomorphically equivalent if k ≥ 5 and n = k+2[\frac{k−1}{4}]. We also show that a smooth manifold M admits a totally real immersion in C^{n} with a trivial complex normal bundle if and only if the complexified tangent bundle of M is trivial. The latter is proved by applying Gromov’s weak homotopy equivalence principle for totally real immersions to Hirsch’s transversal fields theory.

1. Introduction

Let M, N be two totally real and real analytic submanifolds in C^{n}. We say that M and N are biholomorphically equivalent if there is a biholomorphic mapping F defined in a neighborhood of M such that F (M) = N. As a standard fact of complexification, one knows that all totally real and real analytic embeddings of M in C^{n} are biholomorphically equivalent if M is of maximal dimension n. However, the topology of the manifold plays a major role in the existence of totally real immersions or embeddings. For instance, R.O. Wells [19] proved that if an n-dimensional compact and orientable manifold M admits a totally real embedding in C^{n}, then its Euler number must vanish. It was also observed by Wells that if M is a manifold of dimension n and it admits a totally real immersion in C^{n}, then its complexified tangent bundle T^{c}M = T M ⊗ C is trivial. Conversely, the triviality of T^{c}M also implies the existence of totally real immersions of M in C^{n}. This was obtained by M.L. Gromov in [8] through the method of convex integration.

The sphere S^{k}: x_{1}^{2} +…+x_{k+1}^{2} = 1 in R^{k+1} gives us a trivial totally real embedding of S^{k} in C^{k+1}. On the other hand, the works of Gromov [8], Ahern-Rudin [1] and Stout-Zame [17] tell us that S^{k} admits a totally real and real analytic embedding in C^{k} if and only if k = 1, 3. Our main result is the following.

Theorem 1.1. If k ≤ 4, all totally real and real analytic embeddings of S^{k} in C^{n} are biholomorphic equivalent. If k ≥ 5 and n_{k} = k + 2[\frac{k−1}{4}], there exist totally real and real analytic embeddings of S^{k} in C^{n_{k}} which are not biholomorphically equivalent, while all totally real and real analytic embeddings of S^{k} in C^{n} are biholomorphically equivalent if n > n_{k}.

1991 Mathematics Subject Classification. Primary 32F25.

Key words and phrases. totally real embedding, complexification, regular homotopy classification.

Research at MSRI is supported in part by NSF grant DMS-9022140.

In fact, we shall prove a slightly stronger result that all totally real and real analytic embeddings of S^{k} in C^{n} are unimodularly equivalent if k ≤ 4 and n > k, i.e. they are biholomorphic equivalent through a mapping F which preserves the holomorphic n-form dz_{1} ∧ … ∧ dz_{n}. We should mention that using a transversality argument [5], F. Forstnerič and J.-P. Rosay showed that for a compact manifold M of dimension k, all its totally real and real analytic embeddings in C^{n} are biholomorphically equivalent if n ≥ 3k/2.

The proof of Theorem 1.1 is not constructive. It depends on the weak homotopy equivalence (w.h.e.) principle for totally real immersions established by Gromov in [8]. To show Theorem 1.1, we also need to understand the role which the normal bundle of totally real immersions plays. Recall that a C^{1}-smooth mapping f: M → C^{n} is a totally real immersion if f_{∗}T_{x}M spans a k-dimensional complex linear subspace of T_{f(x)}C^{n} for each x ∈ M. We define the complex normal bundle of the immersion f, denoted by ν_{f}, to be the complex vector bundle whose fiber over x ∈ M is the quotient of T_{f(x)}C^{n} by the complex linear span of f_{∗}T_{x}M. We shall see that two totally real and real analytic embeddings of a manifold are biholomorphically equivalent if and only if their complex normal bundles are topologically equivalent. The normal bundle of immersions plays quite important role in the works of S. Smale [15] and M.W. Hirsch [11]. Analogous to the results of Hirsch [11] about transversal fields of smooth immersions, we obtain the following.

Theorem 1.2. Let f: M → C^{n} be a C^{1} totally real immersion. Assume that the complex normal bundle ν_{f} has a topologically trivial subbundle of rank r. Then there is a regular homotopy f_{t}: M → C^{n} of C^{1} totally real immersions such that f_{0} = f and f_{1} : M → C^{n−r}.

We now draw some conclusions from Theorem 1.2.

Corollary 1.3. Let M be a smooth manifold of dimension k. Then M admits a totally real immersion in C^{n} with a trivial complex normal bundle if and only if there exists a totally real immersion of M in C^{k}, i.e. the complexified tangent bundle T^{c}M is trivial.

A smooth manifold M is said to be stably parallelizable if the tangent bundle of M × R is trivial. For instance, the boundary of a smooth domain in euclidean space is always stably parallelizable. By a theorem of Hirsch [11], M is stably parallelizable if and only if it is orientable and admits an immersion in R^{n+1}. From Theorem 1.2, we have the following.

Corollary 1.4. Let M be a manifold of dimension n which is immersible in R^{n+1}. Then T^{c}M is trivial if M is orientable, or M is non-orientable with H^{2}(M, Z) = 0.

In fact, Gromov proved a stronger result that M admits an exact Lagrangian immersion in euclidean space when M is stably parallelizable (see [9], p. 61). One notices that all real surface M can be immersed in R^{3} ⊂ C^{3}. On the other hand, Forstnerič [4] proved that a non-orientable compact surface admits a totally real immersion in C^{2} if and only if its genus is even. Therefore, the condition that H^{2}(M, Z) vanishes is essential in Corollary 1.4.

The paper is organized as follows. In section two we shall discuss Gromov’s w.h.e.- principal for totally real immersions. Section three is devoted to the proof of Theorem 1.2 and Corollary 1.4. The proof of Theorem 1.1 will be given in the last section, where we shall also make essential uses of homotopy groups of complex Stiefel manifolds obtained by M.L. Kervaire [13] and F. Sigrist [14].

2. Classification of totally real immersions

In this section, we shall first recall Gromov’s w.h.e.-principal for ample differential relations established in [8]. We shall also discuss the group structure on the regular homotopy classes of totally real immersions of S^{k} in C^{n}.

Let M be a smooth manifold of dimension k. Assume that k ≤ n. By a regular homotopy f_{t} of totally real C^{1}-immersions of M in C^{n}, one means that for each t ∈ [0, 1], f_{t} is a totally real C^{1}-immersion, and df_{t}: T M → T C^{n} depends on t ∈ [0, 1] continuously. Mappings from M to C^{n} can be identified with sections of the trivial bundle X = M ×C^{n} → M. If f: M → C^{n} is a C^{1}-mapping defined in a neighborhood of x ∈ M, we define the 1-jet of f at x to be J_{x}^{1}f = (f(x), df_{x}). We denote by X^{1} the space of 1-jets of C^{1} sections of X → M. Then it is easy to see that X^{1} is a fibration over M whose fiber over x ∈ M consists of R-linear mappings from T_{x}M to T_{z}C^{n} for some z ∈ C^{n}. We shall adapt (compact-open) C^{0}-topology on X^{1}. Here, we should say a few words about the topologies used in the sequel. For a homotopy of sections or immersions, we shall always use the (compact-open) C^{r}-topology, i.e. the weak C^{r} topology. For approximating a mapping or function, we shall always use the fine C^{r}-topology. The reader is referred to [12] for basic properties of these two kinds of topologies.

By Σ_{x}, one denotes the set of 1-jets J_{x}^{1}f such that df_{x}(T_{x}M) spans a complex linear subspace of T_{f(x)}C^{n} of rank k. This is equivalent to say that the complexification df_{x} ⊗ C: T_{x}^{c}M → T_{f(x)}C^{n} is injective. Let Σ be the union of Σ_{x} for all x ∈ M. Then Ω = X^{1} \ Σ is an open subset of X^{1}, which is called a totally real differential relation. Thus, a C^{1}-mapping f: M → C^{n} is totally real if and only if J^{1}f maps M into Ω.

For each x_{0} ∈ M, we choose local coordinates u_{1}, …, u_{k} in a neighborhood U of x_{0}. Fix x ∈ U and 1 ≤ j ≤ n. Let Z be the set of 1-jets J_{x}^{1}f satisfying

f(x) = z, f_{u_{i}}(x) = v_{i}, i ≠ j,

where z and v_{i}(i ≠ j) are fixed k vectors in C^{n}. Then either Z \ Σ is an empty set, or the linear convex hull of each connected component of Z \ Σ is the whole affine space Z. According to the terminology of Gromov [8], Ω is said to be ample in the coordinate directions. To see this, we notice that if v_{1}, …, v̂_{j}, …, v_{k} are not C-linearly independent, then Z ⊂ Σ. Otherwise, Z ∩Σ consists of all 1-jets (z, v_{1}, …, v, …, v_{k}) such that v is a C-linear combination of v_{1}, …, v̂_{j}, …, v_{k}. Therefore, Z ∩Σ is a subspace of the affine space Z with real codimension 2n−2(k −1) ≥ 2. This implies that Z \Σ is connected and it spans the whole space Z. As a consequence of Theorem 1.3.1 in [8], we can state the following result.

Theorem 2.1 (Gromov, [8]). Let M be a smooth manifold of dimension k ≤ n, and let Ω and J^{1} be as above. Then J^{1} is a one-to-one correspondence between the regular homotopy classes of totally real C^{1}-immersion of M in C^{n} and the homotopy classes of continuous sections of Ω → M. In particular, M admits a totally real immersion in C^{n} if and only if Ω → M has a global continuous section.

We now consider the case that M is the sphere S^{k}. Here we need the fact that the complexified tangent bundle T^{c}S^{k} of S^{k} is trivial. This follows from the existence of totally real immersion of S^{k} in C^{k}. An explicit example of Lagrangian (whence totally real) immersions of S^{k} was constructed by A. Weinstein [18]. As we mentioned earlier, Gromov showed that all stably parallelizable manifold, such as a sphere, admits an exact Lagrangian immersion (see [9], p. 61). It seems to us that there are no other proof in the literature about the triviality of T^{c}S^{k}. Throughout the whole paper, we shall fix a topological trivialization of T^{c}S^{k} = S^{k} × C^{k}.

Recall that the complex Stiefel manifold V_{n,k} consists of k-frames of C^{n}, i.e. the space of ordered k linear independent vectors in C^{n}. With the fixed trivialization for T^{c}S^{k}, the global sections of Ω → M can be identified with mappings from S^{k} to C^{n} × V_{n,k} as follows. Let e_{1}, …, e_{k} be the set of C-linearly independent continuous sections which defines the trivialization of T^{c}S^{k}. Assume that ϕ: M → Ω is a global section. Then ϕ = (f, φ) where f: M → C^{n} and φ: T M → T C^{n} satisfy the property that for each x ∈ M, φ(x): T_{x}M → T_{f(x)}C^{n} is R-linear and its complexification is injective. Hence,

v(x) = (φ(x)(e_{1}(x)), …, φ(x)(e_{k}(x)))

is a set of linearly independent k vectors of T_{f(x)}C^{n}. Denote by U_{n,k} the space of unitary k-frames of C^{n}, where by a unitary k-frame (v_{1}, …, v_{k}), one means that v_{1}, …, v_{k} satisfy the condition <v_{i}, v_{j}> = δ_{i,j} for the standard hermitian inner product <·, ·> on C^{n}. From the well-known normalization, one knows that V_{n,m} is the product of U_{n,m} with the space T of upper-triangle matrices with positive eigenvalues. Thus, a homotopy ϕ_{t}: S^{k} → Ω induces a homotopy

(f_{t}, A_{t}, φ_{t}): S^{k} → C^{n} × T × U_{n,k}.

Since C^{n} and T are contractible, we see that the set of homotopy classes of sections of Ω → S^{k} is the same as the homotopy classes of mappings from S^{k} to U_{n,k}. It is well-known that the homotopy classes of mappings from S^{k} to U_{n,k} is the homotopy group π_{k}(U_{n,k}) (see p. 88 in [16], p. 211 in [3]). Thus, we obtain a one-to-one mapping

j_{∗}: I(S^{k}, C^{n}) → π_{k}(U_{n,k}),

where I(S^{k}, C^{n}) stands for the set of regular homotopy classes of totally real immersions of S^{k} in C^{n}.

With the fixed trivialization of T^{c}S^{k}, j_{∗} is a canonical mapping in the sense that there is a commutative diagram

\begin{matrix}I(S^{k}, C^{n}) & ⟶^{j_{∗}} & π_{k}(U_{n,k}) \\ ↓i_{∗} & & ↓i′_{∗} \\ I(S^{k}, C^{N}) & ⟶^{j′_{∗}} & π_{k}(U_{N,k}), & N > n, \end{matrix} (2.1)

where the homomorphism i_{∗} is induced by the inclusion of sending a totally real immersion f: S^{k} → C^{n} to a totally real immersion (f, 0): S^{k} → C^{N}, and i′_{∗} is induced by regarding a unitary k-frame of C^{n} as a unitary k-frame of C^{N}. In other words, the adapted group structure on I(S^{k}, C^{m}) is preserved under the inclusion C^{n} ⊂ C^{N}. This will be important for us to prove Theorem 1.1.

3. Proof of Theorem 1.2

In this section we shall apply Gromov’s w.h.e.-principle to prove Theorem 1.2. With necessary modifications, we shall follow very closely the proof of Theorem 6.4 in [11].

Let M be a manifold of dimension k. By F M, one denotes the (complexified) k-frame bundle of M, which consists of ordered k linearly independent vectors in T^{c}M with the same base point. For each e = (v_{1}, …, v_{k}) ∈ F_{x}M and g = (g_{i,j}) ∈ GL(k, C), we define g · e to be the k-frame (v′_{1}, …, v′_{k}) with v′_{i} = ∑_{j=1}^{k} g_{i,j}v_{j}. Then F M is a principal GL(k, C)-bundle over M. For k < m ≤ n, we denote by E_{n,m} be the associated bundle of F M with fiber V_{n,m}. More precisely, we define a GL(k, C)- action on V_{n,m} by

(g, (v_{1}, …, v_{m})) ↦ (g · (v_{1}, …, v_{k}), v_{k+1}, …, v_{m}).

Then E_{n,m} is the set of equivalence classes of the relation ∼ on F M × V_{n,m} with (e, f) ∼ (g · e, g · f) for all g ∈ GL(k, C). The bundle projection p_{n,m}: E_{n,m} → M is induced by the composed projection F M ×V_{n,m} → F M → M. Let p_{n,k}^{n,m}: V_{n,m} → V_{n,k} be the projection of deleting the last m−k vectors from each m-frame. Notice that the GL(k, C)-action on V_{n,m} does not affect the last m − k vectors of an m-frame of C^{n}. Hence, p_{n,k}^{n,m} induces a projection from E_{n,m} to E_{n,k} such that p_{n,m} = p_{n,k} ∘ p_{n,k}^{n,m}. We further remark that p_{n,k}^{n,m}: E_{n,m} → E_{n,k} is a fiber bundle. In particular, p_{n,k}^{n,m}: E_{n,m} → E_{n,k} has the covering homotopy property, i.e. for any finite polyhedron P, a homotopy h_{t}: P → E_{n,k} has a lifting h̃_{t}: P → E_{n,m}, if the initial lifting h̃_{0} exists.

To use the covering homotopy property, we consider the set of sections of the fiber bundle E_{n,m} → M. Given a section s: M → E_{n,m}, we define a mapping φ: F M → V_{n,m} by φ(e) = f if s(x) is the equivalence class of (e, f) ∈ F M × V_{n,m}. Then φ is well-defined. For if (e, f) and (e, f′) are in the same equivalence class s(x).

Then there is g ∈ GL(k, C) such that g · e = e and g · f = f′. Obviously, g · e = e implies that g = id. Hence, f′ = f. Moreover, if (e, f) is in the equivalence class s(x), so is (g · e, g · f). Hence, φ(g · e) = g · φ(e), i.e. φ is a GL(k, C)-equivariant mapping. Conversely, given a GL(k, C)-equivariant mapping φ: F M → V_{n,m}, we set s(x) to be the equivalence class of (e, φ(e)) in E_{n,m} for e ∈ F_{x}M. Thus, there is a one-to-one correspondence between the set of GL(k, C)-equivariant mappings from F M to V_{n,m} and the set of sections of the fiber bundle E_{n,m} → M. Therefore, the covering homotopy property gives us the following.

Theorem 3.1. Let φ_{t} be a homotopy of G(k, C)-equivariant mappings from F M to V_{n,k}. Assume that φ_{0} has a lifting φ̃_{0} : F M → V_{n,k+r} such that φ̃_{0} is GL(k, C)- equivariant. Then there is a homotopy φ̃_{t} of lifting of φ_{t} such that each φ̃_{t}: F M → V_{n,k+r} is GL(k, C)-equivariant.

We now let p′: V_{n,k+r} → V_{n,r} be the mapping of projecting a (k + r)-frame to its last r components. Let ϕ: F M → V_{n,k+r} be a GL(k, C)-equivariant mapping. Then φ = pϕ: F M → V_{n,k} is also a GL(k, C)-equivariant mapping. Notice that GL(k, C) acts on each fiber of F M transitively and that the last r components of a (k+r)-frame of C^{n} is fixed under the GL(k, C)-action. This implies that p′φ is constant along fibers of F M. Therefore, p′φ is the lifting of some mapping ψ: M → V_{n,r}. Following [11], we shall call ψ: M → V_{n,r} a transversal r-field of φ: M → V_{n,k}. Thus, we identify the set of GL(k, C)-equivariant mappings from F M to V_{n,k+r} with the set of GL(k, C)- equivalent mappings from F M to V_{n,k} with transversal r-fields. In general, we say that ψ: M → V_{n,r} is transverse to a totally real immersion f: M → C^{n} if for each x ∈ M and ψ(x) = (v_{1}, …, v_{r}), v_{j} is not contained in f_{∗}(T_{x}^{c}M) (1 ≤ j ≤ r).

Corollary 3.2. Let f_{t} be a homotopy of totally real immersions of M in C^{n}. Assume that ν_{f_{0}} has a topological trivial subbundle of rank r. Then ν_{f_{1}} also contains a topological trivial subbundle of rank r.

Proof. We identify f_{t∗}: T^{c}M → T C^{n} with a homotopy f_{t} of mappings from M to C^{n} and a homotopy φ_{t} of GL(k, C)-equivariant mappings from F M to V_{n,k}. By assumptions, φ_{0} has a transversal r-field. Hence, Theorem 3.1 implies that φ_{1} also admits a transversal r-filed, i.e. ν_{f_{1}} has a trivial subbundle of rank r.

We also need the following.

Lemma 3.3. Let M be a smooth manifold and f: M → C^{n} a C^{∞}-smooth totally real immersion. Assume that ν_{f} has a topologically trivial subbundle of rank r. Then there is a smooth unitary r-field ψ: M → U_{n,r} which is transverse to f.

Proof. By assumptions, there is a continuous r-field ψ_{0} : M → V_{n,r} which is transverse to the immersion f. Using the approximation in fine C^{0}-topology, we can replace ψ_{0} by a smooth r-field ψ_{1} which is still transverse to the immersion f. We now project ψ_{1}(x) to the orthogonal complement of f_{∗}T_{x}^{c}M. By the well-known normalization, we readily obtain the desired unitary r-filed.

Proof of Theorem 1.2. Let f: M → C^{n} be a C^{1}-smooth totally real immersion. Assume that the complex normal bundle ν_{f} has a topologically trivial subbundle of rank r. We shall seek a homotopy f_{t} of totally real immersions of M in C^{n} such that f_{0} = f and f_{1}(M) ⊂ C^{n−1}. We further require that the complex normal bundle of f_{1} : M → C^{n−1} has a trivial subbundle of rank r −1. Thus, Theorem 1.2 follows from the induction.

We first find a smooth mapping g: M → C^{n} which is sufficiently close to f in fine C^{1}-topology such that each f_{t} = (1 − t)f + tg is still a totally real immersion for 0 ≤ t ≤ 1. Rename g by f. Then, Corollary 3.2 implies that ν_{f} still contains a trivial subbundle of rank r. By Lemma 3.3, there exists a smooth unitary r-field ψ = (ξ_{1}, …, ξ_{r}) which is transverse to the smooth totally real immersion f: M → C^{n}.

From the standard embedding S^{2n−1} ⊂ C^{n}, one gets a complex vector bundle T^{(1,0)}S^{2n−1} whose fiber over v ∈ S^{2n−1} consists of vectors in C^{n} which are orthogonal to v with respect to the standard hermitian metric on C^{n}. Let E be the frame bundle over S^{2n−1} whose fiber consists of linear independent vectors v_{1}, …, v_{k+r−1} of T^{(1,0)}S^{2n−1} with the same base point. Consider the mapping

ξ̃_{r}(x): e ↦ (df_{x}(e), ξ_{1}(x), …, ξ_{r−1}(x)), e ∈ F_{x}M. (3.1)

Since ξ_{r} is orthogonal to f_{∗}T^{c}M and ξ_{j} (j < r), then ξ̃_{r}(x) maps F_{x}M into E_{ξ_{r}(x)}. Hence, ξ̃_{r}: F M → E is a GL(k, C)-equivariant mapping which covers the mapping ξ_{r}: M → S^{2n−1}. Since the dimension of M is less than 2n − 1, the smoothness of the mapping ξ_{r} implies that there is y_{0} ∉ ξ_{r}(M). Set Y = S^{2n−1} − {y_{0}}. Since Y is contractible, then there is a homotopy h_{t}: M → Y such that h_{0} = ξ_{r} and h_{1} ≡ y_{1} ∈ Y. Also, there is a trivialization

T^{(1,0)}S^{2n−1}|_{Y} = Y × C̃^{n−1}, C̃^{n−1} = T_{y_{1}}^{(1,0)}S^{2n−1}. (3.2)

Therefore, ξ̃_{r} can be written as (h_{0}, ϕ_{0}) with ϕ_{0} a GL(k, C)-equivariant mapping from F M to E_{y_{1}}. Put h̃_{t} = (h_{t}, ϕ_{0}). Returning to T^{(1,0)}S^{2n−1}|_{Y} from the trivialization (3.2), we obtain a homotopy h̃_{t} of GL(k, C)-equivariant mappings from F M to E|_{Y}. Returning to the ambient space C^{n}, we then have T^{(1,0)}S^{2n−1} ⊂ S^{2n−1}×C^{n}. Thus, we obtain a homotopy h̃_{t} of GL(k, C)-equivariant mappings from F M to C^{n} × V_{n,k+r−1} such that h̃_{1} : F M → E_{y_{1}}.

Let us identify y_{1} with a point in C^{n−1}. Put T_{y_{1}}C^{n−1} = C̃^{n−1}. Then we have E_{y_{1}} = y_{1} × V_{n−1,k+r−1}. We now write h̃_{t} as (h_{t}, φ_{t}, ψ_{t}), where each φ_{t}: F M → V_{n,k} is GL(k, C)-equivariant, and ψ_{t} is a transversal (r − 1)-field of φ_{t}. From (3.1), it is clear that φ_{0} : F M → V_{n,k} is just the GL(k, C)-equivariant mapping induced by f_{∗}: T^{c}M → T C^{n}. This implies that f_{∗} and (h_{1}, φ_{1}) are homotopic as fiberwisely injective C-linear mappings from T^{c}M to T C^{n}. Notice that h_{1} ≡ y_{1} ∈ C^{n−1} and φ_{1} : F M → V_{n−1,k}. Now Theorem 2.1 implies that there is a totally real immersion g: M → C^{n−1} such that g_{∗} and (h_{1}, φ_{1}) are joined by a homotopy k_{t} of fiberwisely injective C-linear mappings from T^{c}M to T C^{n−1}. Thus, we have proved that f_{∗}: T^{c}M → T C^{n} is homotopic to g_{∗}: T^{c}M → T C^{n−1} as fiberwisely injective Clinear mappings from T^{c}M to T C^{n}. Using Theorem 2.1 again, we know that there is a homotopy of totally real immersions joining f and g. To complete the proof of Theorem 1.2, we need to show that the complex normal bundle of g: M → C^{n−1} has a topologically trivial subbundle of rank r − 1. To this end, we notice that φ_{1} has a transversal (r − 1)-field ψ_{1} in C^{n−1} since h̃_{1} : F M → E_{y_{1}} ≡ V_{n−1,k+r−1}. By applying Corollary 3.2 to the homotopy k_{t}, we see that the complex normal bundle of g has a trivial subbundle of rank r − 1. The proof of Theorem 1.2 is complete.

Proof of Corollary 1.4. Assume that M is immersed in R^{n+1} ⊂ C^{n+1}. When M is orientable, the real normal line bundle of the immersion of M in R^{n+1} is trivial. Let ν be a normal vector field of M. It is obvious that ν is still transverse to T^{c}M. Therefore, Theorem 1.2 implies that M admits a totally real immersion in C^{n}, i.e. T^{c}M is trivial. We now consider the case that M is non-orientable and H^{2}(M, Z) = 0. By a theorem of Whitney, a smooth manifold has a compatible analytic structure. Using the analytic approximations of smooth mappings in fine C^{r}-topology [7], we can replace the original immersion of M in R^{n+1} by an analytic immersion f: M → R^{n+1}. Now the complexification of f gives us a holomorphic mapping from M^{c} to C^{n+1}, where M^{c} is a complexification of M depending on f. By Grauert’s Tube theorem [7], there is a Stein neighborhood X of M in M^{c} such that M is a strong deformation retraction of X. Since X is Stein, then all holomorphic line bundle over X is determined by its first Chern class. On the other hand, H^{2}(X, Z) = H^{2}(M, Z) = 0, since M is a deformation retraction of X. Hence, the normal bundle of X in C^{n+1} is trivial. Now, Theorem 1.1 implies that M admits a totally real immersion in C^{n}. The proof of Corollary 1.4 is complete.

We notice that a real surface can be immersed in R^{3}. Thus, Corollary 1.4 gives another proof that all orientable surfaces admit totally real immersions in C^{2}, which is due to Forstnerič [4] when the surfaces are compact. Corollary 1.4 is inconclusive when M is a non-orientable compact surface, since H^{2}(M, Z) = Z_{2}. It was proved by Forstnerič [4] that a non-orientable compact surface admits a totally real immersion in C^{2} if and only if its genus is even. In view of Corollary 1.3, we see that a totally real immersion of a non-orientable compact surface in C^{3} has a trivial complex normal bundle if and only if the genus of the surface is even. This also indicates that the hypothesis that H^{2}(M, Z) = 0 is needed in Corollary 1.4, although it is not a necessary condition for the triviality of T^{c}M.

4. Proof of Theorem 1.1

In this section we shall prove Theorem 1.1 by using Theorem 1.2. We shall see that the proof of Theorem 1.1 is eventually related to homotopy groups of complex Stiefel manifolds.

Let us first consider the case k ≤ 4. We need the following lemma.

Lemma 4.1. Let M and N are two totally real and real analytic submanifolds of C^{n}. Let ν_{M} and ν_{N} be the complex normal bundles of M and N respectively. Then M and N are biholomorphically equivalent if and only if there is an analytic diffeomorphism f: M → N such that the pull-back f^{∗}ν_{N} is topologically isomorphic to ν_{M}.

Proof. If M, N are biholomorphically equivalent by F, then it is clear that dF maps T^{c}M into T^{c}N and df also induces an isomorphism from ν_{M} to ν_{N}. We now assume that there is an analytic diffeomorphism f: M → N such that ν_{M} is topologically isomorphic to f^{∗}ν_{N}. Let f^{c} be an biholomorphic extension of f which sends a Stein neighborhood M^{c} of M onto N^{c} = f^{c}(M^{c}). We may assume that M is a strong deformation retraction of M^{c}, i.e. there is a continuous mapping r: M^{c} → M^{c} with r|_{M} = id and a homotopy r_{t}: M^{c} → M with r_{0} = r and r_{1} = id. This implies that any complex vector bundle V over M^{c} is isomorphic to r^{∗}(V |_{M}). Therefore, ν_{M^{c}} is topologically isomorphic to f^{∗}ν_{N^{c}}. By a theorem of Grauert [6], ν_{M^{c}} and ν_{N^{c}} are also holomorphically isomorphic. Let s be the zero section of ν_{M^{c}}. By a theorem of Docquier and Grauert (see [10], p. 257), the zero-section mapping can be extends to a biholomorphic mapping from a neighborhood of M^{c} in C^{n} to a neighborhood of zero section of ν_{M^{c}}. Therefore, f^{c} extends to biholomorphic mapping defined in a neighborhood of M in C^{n}.

To start the proof of Theorem 1.1, we first notice that Forstnerič and Rosay [5] proved that any two totally real immersions of a smooth manifolds M in C^{n} are regularly homotopic through totally real immersions, if dimM ≤ 2n/3. Consequently, their argument also showed that such two totally real and real analytic embeddings are also biholomorphically equivalent if M is compact. In the special case of spheres, we notice that the standard embedding of S^{k} in C^{k+1} has a trivial complex normal bundle, then Lemma 4.1 and Corollary 3.2 also imply that any two totally real and real analytic embeddings of S^{k} in C^{n} are biholomorphically equivalent when n ≥ 3k/2. In particular, all totally real and real analytic embeddings of S^{k} in C^{n} are biholomorphically equivalent when k = 1, 2. We also notice that complex line bundles on S^{k} is classified by the group π_{k−1}(U_{1}) (see [3]). Hence, complex line bundles on S^{k} are trivial if k ≠ 2. Now it is easy to see that all totally real and real analytic embeddings of S^{k} in C^{n} are biholomorphically equivalent if k ≤ 4.

We now consider the case of k > 4. Assume first that for some n > k, all totally real and real analytic embeddings of S^{k} in C^{n} are biholomorphically equivalent. This implies that all totally real and real analytic embeddings of S^{k} in C^{n} have trivial complex normal bundle. Notice that when n > k, a C^{1}-smooth totally real immersions of S^{k} in C^{n} can be connected to a totally real and real analytic embeddings of S^{k} in C^{n} by C^{1} totally real immersions. Thus, Corollary 3.2 implies that for any totally real immersion of S^{k} in C^{n}, its complex normal bundle is trivial. Therefore, the commutative diagram (2.1) and Theorem 1.2 imply that

i_{∗}: π_{k}(U_{k}) → π_{k}(U_{n,k}) (4.1)

is an epimorphism, where i_{∗} is induced by the inclusion U_{k} ⊂ U_{n,k}. Therefore, the proof of Theorem 1.1 will be complete if we can show that (4.2) is not an epimorphism for n = n_{k}, and also that

i_{∗}: π_{k}(U_{n_{k},k}) → π_{k}(U_{n,k}) (4.2)

is an epimorphism for all n > n_{k}. Notice that π_{k}(U_{n,k}) = 0 for n ≥ 3k/2. This also follows from the fact that all totally real immersions of S^{k} in C^{n} are regularly homotopic. Hence, it suffices to show that (4.2) holds for n_{k} < n < 3k/2.

It is well-known from the Bott periodicity theorem that π_{2l}(U_{n}) = 0 and π_{2l+1}(U_{n}) = π_{2l+1}(U_{l+1}) = Z for n > l. We shall discuss in two cases.

Case 1. k = 2l (l ≥ 3). In this case, π_{2l}(U_{2l}) = 0 implies that n_{k} is the largest integer n such that π_{2l}(U_{n,2l}) is non-trivial. From [14, p. 127], one sees that

π_{2l}(U_{3l−1,2l}) = \{ \begin{matrix}0, & if l is even and l ≥ 2, \\ Z_{2}, & if l is odd and l ≥ 3. \end{matrix}

Hence, n_{2l} = 3l − 1 when l is odd and l ≥ 3. We now assume that l is even. Then, one has

π_{2l}(U_{3l−2,2l}) = \{ \begin{matrix}Z_{2}, & l = 4, \\ Z_{48/U(l+1,3)}, & l ≥ 6, \end{matrix}

where U(·, 3) is a James number which divides 24 [14, p. 128]. In particular, π_{2l}(U_{3l−2,2l}) ≠ 0, if l even and l ≥ 4. This showed that if l is even and l ≥ 4, n_{2l} = 3l − 2 satisfies the property stated in Theorem 1.1.

Case 2. k = 2l + 1 (l ≥ 2). In this case, one has π_{2l+1}(U_{2l+1}) = Z. From [14], we find

π_{2l+1}(U_{3l+1,2l+1}) = Z, (4.3)

(4.4) π_{2l+1}(U_{3l,2l+1}) = \{ \begin{matrix}Z, & if l is even and l ≥ 2, \\ Z + Z_{2}, & if l is odd and l ≥ 3. \end{matrix}

We now consider the homomorphism

i_{∗}: π_{2l+1}(U_{3l,2l+1}) → π_{2l+1}(U_{3l+1,2l+1}), (4.5)

where i_{∗} is induced by the inclusion U_{3l,2l+1} ⊂ U_{3l+1,2l+1}. We need the following.

Lemma 4.2. Let i_{∗} be defined by (4.5). Then i_{∗} is an epimorphism if and only if l is odd.

Let us postpone the proof of Lemma 4.2 for a while and finish our proof of Theorem 1.1. Assume first that l is odd. It is clear that there is no epimorphism from Z to Z + Z_{2}. Hence, (4.1) is not onto for k = 2l + 1 and n = 3l. On the other hand, Lemma 4.2 and the vanishing of π_{k}(U_{n,k}) for n > 3l + 1 imply that (4.2) is onto for all n > 3l. Therefore, n_{2l} = 3l satisfies the property stated in Theorem 1.1. Next, we assume that l is even. Then Lemma 4.2 implies that (4.1) is not onto for n = 3l + 1. On the other hand, (4.2) is onto for n = 3l+2 and k = 2l+1 because of the vanishing of π_{2l+1}(U_{3l+2,2l+1}). Therefore, we conclude that for even l, n_{2l+1} = 3l + 1 satisfies the property stated in Theorem 1.1.

We now turn to the proof of Lemma 4.2. Let U_{3l} → U_{3l,2l+1} be the standard fibration with fiber U_{l−1}, and U_{3l+1} → U_{3l+1,2l+1} the fibration with fiber U_{l}. Then the inclusion U_{3l,2l+1} ⊂ U_{3l+1,2l+1} induces the following commutative diagram of exact sequences

π_{2l+1}(U_{3l}) ⟶^{j_{∗}} π_{2l+1}(U_{3l,2l+1}) ⟶ π_{2l}(U_{l−1}) → π_{2l}(U_{3l})
∥  ↓ i_{∗}  ↓ i′_{∗}  ∥
π_{2l+1}(U_{3l+1}) ⟶^{j′_{∗}} π_{2l+1}(U_{3l+1,2l+1}) ⟶^{δ} π_{2l}(U_{l}) → π_{2l}(U_{3l+1}).
∥  ↓ p_{∗}  ∥
Z  π_{2l}(S^{2l−1})  0

It is clear that if i_{∗} is epimorphic, so is i′_{∗}. However, one knows that π_{2l}(U_{l}) = Z_{l!} (see [2]), and π_{2l}(U_{l−1}) = ℤ_{l!/2} for l even (see [13]). Thus i′_{∗} is not epimorphic when l is even. We now assume that l is odd. In this case, Kervaire showed that p_{∗} = 0 (see [13], Lemma I.1). Hence, i′_{∗} is an epimorphism, i.e. δi_{∗} is an epimorphism. Using (4.3) and (4.4), we can write more explicitly the following diagram

Z ⟶^{j_{∗}} Z + Z_{2}
∥  ↓ i_{∗}
Z ⟶^{j′_{∗}} Z ⟶^{δ} Z_{l!} → 0.

Write j_{∗}(1) = g + ϵ, where g ∈ Z and ϵ ∈ Z_{2}. Clearly, i_{∗}(ϵ) = 0. Hence, we get

g · i_{∗}(e) = i_{∗}j_{∗}(1) = j′_{∗}(1) = ±l!

for e = 1 ∈ Z ⊂ Z + Z_{2}. Thus, i^{∗}(e) divides l!. On the other hand, δ ∘ i_{∗} is an epimorphism. Hence, i_{∗}(e)δ(1) must be a generator of Z_{l!}. Therefore i_{∗}(e) = ±1, i.e. i_{∗} is an epimorphism. The proof of Lemma 4.2 is complete.

Next, we want to show that all totally real and real analytic embeddings of S^{k} in C^{n} are unimodularly equivalent if n > k and k ≤ 4, i.e. they are equivalent through a biholomorphic mapping F satisfying F^{∗}Ω = Ω. To this end we shall use the Cauchy-Kowalewski theorem to prove a slightly general result.

Proposition 4.3. Let M, N be two totally real and real analytic submanifolds of C^{n} which are biholomorphically equivalent. Assume that for some complexification M^{c}, the holomorphic normal bundle of M^{c} contains a holomorphic subbundle of rank one. Then M and N are unimodularly equivalent.

Proof. Assume that M and N are equivalent by a biholomorphic mapping φ defined near M. It suffices to show that there is a biholomorphic mapping ψ defined near M such that ψ^{∗}Ω = φ^{∗}Ω and ψ(M) = M. By shrinking M^{c} if necessary, we may assume that M^{c} ⊂ C^{n} is a Stein manifold. Now we have the decomposition ν_{M^{c}} = ν′ ⊕ L, where L is a line bundle. Moreover, we may assume that ν is a subbundle of M^{c}×C^{n}. Thus, a neighborhood U of the zero section of ν_{M^{c}} is identified with a neighborhood of M^{c}. We now want to show that there is a holomorphic mapping

ψ: (u, v) ↦ (u, λ(u, v)v), u ∈ ν′, v ∈ L

such that ψ^{∗}Ω = φ^{∗}Ω. Let w = (w_{1}, ⋯, w_{k}) be local coordinates on M^{c}, and let ξ = (ξ_{1}, …, ξ_{n−k−1}) and t be local trivializations of ν′ and L respectively. Then in local coordinates, ψ must be in the form (w, ξ, t) → (w, ξ, t′(w, ξ, t)) with t′|_{t=0} = 0. Put

φ^{∗}Ω = f(w, ξ, t)dw ∧ dξ ∧ dt, Ω = a(w, ξ, t)dw ∧ dξ ∧ dt,

where dw = dw_{1} ∧ … ∧ w_{k} , dξ = dξ_{1} ∧ … ∧ dξ_{n−k−1}. Then ψ^{∗}Ω = φ^{∗}Ω is equivalent to the equation a(w, ξ, t′)∂t′/∂t = f(w, ξ, t). By the Cauchy-Kowalewski theorem, we know that for small |t| the solution t′ exists uniquely. This means that the required mapping ψ is uniquely determined in local coordinates. Therefore, there is a holomorphic mapping defined in a neighborhood of M^{c} such that ψ^{∗}Ω = φ^{∗}Ω. Since the restriction of ψ to M^{c} is the identity mapping, it is easy to see that ψ is one-to-one in some neighborhood of M^{c}.

Notice that all totally real and real analytic embeddings of a compact surface M in C^{n} (n ≥ 3) are biholomorphically equivalent. Also, M admits a totally real and real analytic embedding in C^{3}. Hence, the complex normal bundle of a totally real embeddings of M in C^{n} (n ≥ 3) is the direct sum of a line bundle and a trivial bundle. From Proposition 4.3, we have the following.

Corollary 4.4. All totally real and real analytic embeddings of a compact surface in C^{n} (n ≥ 3) are unimodularly equivalent.

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