Existence results of totally real immersions and embeddings into $\mathbb{C}^N$

TL;DR

This paper investigates conditions under which manifolds can be totally real immersed or embedded into complex Euclidean spaces, establishing new existence results and topological obstructions in various dimensions.

Contribution

It proves that totally real immersion existence is preserved under cut-and-paste operations and characterizes embeddings for certain 5- and 6-manifolds, highlighting differences from lower dimensions.

Findings

Totally real immersion existence is closed under cut-and-paste operations.

Fundamental group is not an obstruction for high-dimensional embeddings.

Classifies diffeomorphism and homotopy types of specific manifolds.

Abstract

We prove that the existence of totally real immersions of manifolds is a closed property under cut-and-paste constructions along submanifolds including connected sums. We study the existence of totally real embeddings for simply connected 5-manifolds and orientable 6-manifolds and determine the diffeomorphism and homotopy types. We show that the fundamental group is not an obstruction for the existence of a totally real embedding for high-dimensional manifolds in contrast with the situation in dimension four.