---
abstract: 'We show that there exists no CR-regular embedding of the 5-sphere $S^5$ into $\mathbb{C}^4$, and also obtain analogous results for embeddings of higher dimensional spheres into complex space.'
author:
- 'Ali M. Elgindi'
title: 'On the Non-Existence of CR-Regular Embeddings of $S^5$'
---

 

Introduction
============

 

The h-principle was developed in the early 1970’s by M. Gromov who applied it to the problem of embeddings of real manifolds into complex space to give necessary and sufficient conditions for the existence of totally real embeddings (see \[5\]). In particular, he demonstrated that the only spheres $S^n$ which admit totally real embeddings to $\mathbb{C}^n$ are in the dimensions $n=1,3$ (the case $n=1$ being trivial).

In the 1980’s, F. Forstneric extended the work on Gromov and proved that every compact, orientable 3-manifold admits a totally real embedding into $\mathbb{C}^3$ (see \[4\]).

 

  In our work in \[1\], we demonstrated that every topological type of knot (or link) in $S^3$ can arise as the set of complex tangents to a $\mathcal{C}^n$-embedding $S^3 \hookrightarrow \mathbb{C}^3$ (for any given integer n). In \[2\], we derived a topological invariant for the local removal of complex tangents to an embedding of a closed oriented 3-manifold into $\mathbb{C}^3$, leaving the embedding unchanged outside a small neighborhood of a chosen set of complex tangents. This led to our work in \[3\], where we demonstrated that any embedding $S^3 \hookrightarrow \mathbb{C}^3$ can be approximated $\mathcal{C}^0$-close by a totally real embedding.

 

In their paper \[6\], N. Kasuya and M. Takase generalized our work in \[1\] to demonstrate that every knot or link in $S^3$ can be exactly assumed as the set of complex tangents to a smooth embedding of $S^3$. In fact, they generalize further to demonstrate that any 1-dimensional submanifold of a closed orientable 3-manifold that is homologically trivial may arise as the set of complex tangents to an embedding $M \hookrightarrow \mathbb{C}^3$. Their method of proof is based on the theory of stable maps in topology and Saeki’s Theorem.

 

In more general situations, in particular for embeddings $f: M^n \hookrightarrow \mathbb{C}^q$ ($n \neq q$) the h-principle still holds, although there cannot be in the literal sense totally real embeddings if $n>q$. In the situation $n>q$, every point $x \in M$ must be complex tangent, and the $\textit{complex dimension}$ of $x$ is defined to be:

$dim(x) = dim_{\mathbb{C}} (f_*(T_x M) \bigcap J f_* (T_x M))$ where $J$ is the complex structure. Note that $dim(x) \geq n-q$, by elementary linear algebra.

If $dim(x) = n-q$, we say that $x$ is a $\textit{CR-regular}$ point of the embedding. If $dim(x) > n-q$, we say $x$ is called $\textit{CR-singular}$.

 

In his paper \[8\], M. Slapar considered the situation of a closed oriented 4-manifold embedded into $\mathbb{C}^3$. In this situation, complex tangents generically are discrete (and finite), and can be classified as either elliptic or hyperbolic by comparing the orientation of the tangent space of the embedded manifold at the complex tangent (which is necessarily complex) with the induced orientation of the tangent space as a complex subspace of the tangent space of the complex ambient manifold. Slapar also demonstrated (in \[8\]) that a 4-manifold admits a CR-regular embedding into $\mathbb{C}^3$ if and only if the manifold is parallelizable.

 

In this paper, we will focus on embeddings $S^5 \hookrightarrow \mathbb{C}^4$. In this situation, a generic embedding will assume CR-singular points along a knot in $S^5$, all other points being CR-regular. We will show that there exists no CR-regular embedding of $S^5 \hookrightarrow \mathbb{C}^4$, and we will also make further generalizations for embeddings of spheres of higher dimensions.

The Result and Proof
====================

 

An embedding of $S^5 \hookrightarrow \mathbb{C}^4$ must have a complex line in the tangent space to each point $x \in S^5$, because of dimensionality reasons. There are two classes of points in such an embedding of $S^5$. In particular a point may have a complex plane in its tangent space, we say such a point is CR-singular. If the tangent plane at the point does not contain a complex plane in its tangent space (only a complex line), we say the point is CR-regular. For a generic embedding $S^5 \hookrightarrow \mathbb{C}^4$, we will have a link of CR-singular points, and all other points will be CR-regular (this readily follows from the dimension of the relevant spaces and using an application of Sard’s theorem).

 

Now, let $\mathbb{G}_{5,8}$ be the Grassmannian of 5-planes in $\mathbb{R}^8 = \mathbb{C}^4$, and consider its subspace $\mathbb{Y} = \{P \in \mathbb{G}_{5,8} | P \bigcap JP  \cong \mathbb{C} \}$ of planes that contain only a complex line. We say a 5-plane $P \in \mathbb{Y}$ is $\textit{totally real}$.

 

Consider the Gauss map of the embedding: $\textit{G}: S^5 \rightarrow \mathbb{G}$. Then the CR-regular points of the embedding are exactly the points whose image under $\textit{G}$ is contained in $\mathbb{Y}$. In particular:

$\{$CR-regular points$\} = \textit{G}^{-1} (\mathbb{Y}) \subset S^5$

 

A CR-regular embedding $S^5 \hookrightarrow \mathbb{C}^4$ will have that its Gauss map has image only in $\mathbb{Y}$, that is the tangent plane at every point is totally real.

  We will show that no embedding can satisfy this, in particular:

 

: There exists no CR-regular embedding $S^5 \hookrightarrow \mathbb{C}^4$.

 

 

******:-

  First, let us denote $\mathbb{G} = \mathbb{G}_{3,8}$ to be the Grassmannian of 3-planes in $\mathbb{R}^8$. Note that by taking orthogonal complements: $\mathbb{G} = \mathbb{G}_{3,8} \cong \mathbb{G}_{5,8}$. Now denote by $\mathbb{V} = \mathbb{V}_{3,8}$ the Steifel manifold of real 3-frames on $\mathbb{R}^8$. Note that the Steifel manifold manifold deformation retracts (and hence is homotopy equivalent) to the manifold of orthonormal 3-frames in $\mathbb{R}^8$, which we denote by $\mathbb{V}^o$.

 

Next consider the Stiefel manifold: $V^{tr}_{3,8} \subset \mathbb{V}$ of real 3-frames of $\mathbb{R}^8 \cong \mathbb{C}^4$ for which the 5-plane which is the complement of their span (as real vectors) is totally real in $\mathbb{C}^4$ (contains only a complex line).

 

Let $(v_1,v_2,v_3) \in V^{tr}_{3,8}$ be such a 3-frame. Applying first the Gram-Schmidt process to this 3-frame, we may assume without loss that these vectors are orthonormal (their span will not change). Denote their span by $\mathcal{N}=span\{v_1,v_2,v_3\}$, and its complement by $\mathcal{P} = \mathcal{N}^{\perp} \subset \mathbb{R}^8$, and let $L \subset \mathcal{P}$ be the (unique) complex line contained in $\mathcal{P}$. In particular, if we denote the complex structure as $J:\mathbb{R}^8 \rightarrow \mathbb{R}^8$, we will have that: $L = \mathcal{P} \cap J(\mathcal{P}) \cong \mathbb{C}$.

 

Note the following from linear algebra:

$(\mathcal{N}+J\mathcal{N})^\perp \subset \mathcal{N}^\perp \cap (J\mathcal{N})^\perp = \mathcal{N}^\perp \cap J(\mathcal{N}^\perp) = \mathcal{P} \cap J(\mathcal{P}) = L$.

 

Hence, by taking dimensions: $dim(\mathcal{N}+J\mathcal{N}) \geq 6$.

  But, as each of the subspaces $\mathcal{N}$ and $J\mathcal{N}$ are of dimension 3, it must be that $\mathcal{N} \cap J\mathcal{N} = \{0\}$ and $(\mathcal{N} \oplus J\mathcal{N})^\perp = L$. Note then that $\mathcal{N}$ is a totally real 3-plane.

 

Now consider the (orthonormal) 6-frame of $\mathbb{R}^8$:

$\{v_1,v_2,v_3,Jv_1,Jv_2,Jv_3\}$. This will form a basis for $\mathcal{N} \oplus J\mathcal{N} \subset \mathbb{R}^8$, whose complement is $L$.

 

Applying the Gram-Schmidt process to these six (orthonormal) vectors, we necessarily get a vector $\theta \in L$.

  We can now form an orthonormal basis of $\mathbb{R}^8$ by adding to the collection two vectors: $\{v_1,v_2,v_3,Jv_1,Jv_2,Jv_3, \theta, J\theta\}$.

 

Suppose now that we had a CR-regular embedding $F: S^5 \hookrightarrow \mathbb{C}^4$. As the normal bundle to any embedding $S^5 \hookrightarrow \mathbb{R}^8$ is trivial (see Massey in \[7\]), with a choice of trivialization of the normal bundle we can choose a normal frame to the embedding $F$; denote this normal frame by $\eta:S^5 \rightarrow V^{tr}_{3,8}$. Let $\eta(m) = \{v_1,v_2,v_3\}$ for a general point $m \in S^5$.

It then follows from the above that the 5-frame: $\{Jv_1,Jv_2,Jv_3, \theta, J\theta\}$ will span the tangent space to the embedded $S^5$ at the (general) point $m$. As the Gram-Schmidt process is continuous, this will give us as a global trivialization of the tangent bundle of the 5-sphere. But the 5-sphere is not parallelizable!

 

With this contradiction, we conclude that there exists no CR-regular embedding of $S^5 \hookrightarrow \mathbb{C}^4$.

 

$\textbf{\textit{QED}}$

 

We now recall the following result of Kervaire (see Massey in \[7\]):

The normal bundle to an $n$-sphere embedded in $\mathbb{R}^{n+k}$ with $k>\frac{n+1}{2}$ is necessarily trivial.

 

We may now directly generalize our above proof to achieve higher dimensional analogues:

: There exists no CR-regular embedding $S^n \hookrightarrow \mathbb{C}^{n-r}$ for $r < \frac{n-1}{4}$, $n \neq 3,7$.

  ******:-

  Applying the theorem of Kervaire for $k=n-2r$, we find that the normal bundle of an $n$-sphere embedded in $\mathbb{R}^{2n-2r}$ is necessarily trivial for $r < \frac{n-1}{4}$. Note we already achieved the analogous result earlier for the special case $n=5, r=1$ (not in the dimension range of this theorem), and that the $r=0$ cases reduce to Gromov’s work for totally real embeddings $S^k \hookrightarrow \mathbb{C}^k$.

 

Suppose there exists a CR-regular embedding $F:S^n \hookrightarrow \mathbb{C}^{n-r}$, and let $x \in S^n$. We would then get the holomorphic tangent space $T_x \cap JT_x = H$ is a complex plane of dimension $r$. As the normal bundle is trivial, we can specify a normal frame for the normal bundle $\mathcal{N}$ at $x$: $\{v_1,...,v_{n-2r}\}$, which we may assume without loss is orthonormal. It is clear that $\mathcal{N} \cap J\mathcal{N} = \{0\}$ in direct analogy to our work in the proof of Theorem 1, and that: $TS^n = J\mathcal{N} \oplus H$. We may then extend the frame using the complex structure: $\{v_1,...,v_{n-2r}, Jv_1,...,Jv_{n-2r}\}$ to obtain an orthonormal $(2n-4r)$-frame of $\mathbb{R}^{2n-2r}$. Now, applying Gram-Schmidt to extend by one vector, we obtain a vector $\theta_1 \in H$.

 

We then have the following frame tangent to $S^n$: $\{Jv_1,...,Jv_{n-2r}, \theta_1, J \theta_1\}$, to which we apply Gram-Schmidt once more to obtain a vector $\theta_2$. Proceeding inductively, we apply Gram-Schmidt to the extended frame: $\{Jv_1,...,Jv_{n-2r}, \theta_1, J \theta_1, \theta_2, J \theta_2\}$, and we repeat ($r$ times in total) to obtain the frame: $\{Jv_1,...,Jv_{n-2r}, \theta_1, J \theta_1,..., \theta_r, J \theta_r\}$ which will necessarily span the tangent space to $S^n$ at each point. This will then trivialize the tangent bundle globally, but for $n \neq 3,7$, $S^n$ is not parallelizable. Hence, the theorem follows.

 

$\textbf{\textit{QED}}$

 

[14]{}

A.M. Elgindi, “On the Topological Structure of Complex Tangencies to Embeddings of $S^3$ into $\mathbb{C}^3$,” New York J. of Math. Vol. 18 (2012), 295-313.

A.M. Elgindi “A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups,” International Journal of Mathematics Vol. 26 (2015).

A.M. Elgindi, “Totally real perturbations and nondegenerate embeddings of $S^3$,” New York J. of Math. Volume 21 (2015) 1283-1293

F. Forstneric, “On totally real embeddings into $\mathbb{C}^n$,” Expositiones Mathematicae, 4 (1986), pp. 243-255.

M.L. Gromov, “Convex Integration of Differential Relations,” 1973 Math. USSR Izv. 7.

N. Kasuya and M. Takase, “Knots and links of complex tangents.” arXiv preprint 1606.03704 (2016).

W.S. Massey, “On the Normal Bundle of a Sphere Imbedded in Euclidean Space,” Proc. of the Amer. Math. Soc., Vol. 10, No. 6 (Dec., 1959), pp. 959-964.

M. Slapar, “Cancelling complex points in codimension two,” Bull. Aust. Math. Soc. 88 (2013), no. 1, 64-69.
