On totally real spheres in complex space

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a K\"ahler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the normal bundle and of constant length of the second fundamental form (or equivalently of constant scalar curvature) of a complex space form N is totally geodesi...

In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into $\mathbb{C}^3$ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient condition for the (local) removal of the isolated complex tangent. The obstruction is a certain homotopy class of the space $\mathbb{Y}$ consisting of totally rea...

We consider a family $M_t^n$, with $n\ge 2$, $t>1$, of real hypersurfaces in a complex affine $n$-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in ${\mathbb C}^n$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of $M_t^n$ in ${\mathbb C}^n$ for $n=3,7$. In our earlier article we showed that $M_t^7$ is not embeddable in ${\...

This article is the continuation of the first named author work "On maximal totally real embeddings". For real analytic compact manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle, a construction of canonical maximal totally real embeddings is known from previous works by Guillemin-Stenzel, Lempert, Lempert-Sz{\"o}ke, Sz{\"o}ke and Bielawski. The construction is based on the use of Jacobi fields, which are far fro...

This paper is devoted to the study of the embeddings of a complex submanifold $S$ inside a larger complex manifold $M$; in particular, we are interested in comparing the embedding of $S$ in $M$ with the embedding of $S$ as the zero section in the total space of the normal bundle $N_S$ of $S$ in $M$. We explicitely describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho-Mova...

We consider two classes of smooth maps M^n\to C ^N. Definition. A map f:M^n\to C^N is called an independent map if df_1(p)\wedge...\wedge df_N (p)\neq 0. We are interested in the optimal value of N for all manifolds of dimension n for independent maps and also for totally real immersions.

Ahern and Rudin have given an explicit construction of a totally real embedding of $S^3$ in $\mathbb{C}^3$. As a generalization of their example, we give an explicit example of a CR regular embedding of $S^{4n-1}$ in $\mathbb{C}^{2n+1}$. Consequently, we show that the odd dimensional sphere $S^{2m-1}$ with $m>1$ admits a CR regular embedding in $\mathbb{C}^{m+1}$ if and only if $m$ is even.

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real $n$-dimensional manifolds into $\mathbb{C}^n$. The generic topological structure of the set complex tangents to such embeddings $M^n \hookrightarrow \mathbb{C}^n$ takes the form of a (stratified) $(n-2)$-dimensional submanifiold of $M^n$. In this paper, we generalize our results from our previous work for the 3-dimensional ...

Let $f\colon M^{2n}\to\mathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $n\geq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component of an open dense subset of $M^{2n}$, either $f$ is holomorphic in $\mathbb{R}^{2n+4}\cong\mathbb{C}^{n+2}$ or it is in a unique way a composition $f=F\circ h$ of isometric immersions. In the latter case, we have that $h\colon M^{2n}\to N^{2...

This article has been withdrawn due to an error in a proof of the main result.