CR manifold

In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a special kind of geometric space. It is built on a differentiable manifold, which is a smooth surface that can be bent and stretched without tearing. This CR structure copies the shape of a real surface found inside a complex vector space, or something called the edge of a wedge.

A CR manifold has a special set of directions called a complex subbundle within its tangent bundle. This subbundle follows certain rules that make it integrable, meaning the directions fit together smoothly. From this structure, mathematicians can create a special operator called the ∂̅_b-operator. This operator helps them study functions on the manifold.

The term CR stands for Cauchy–Riemann or Complex-Real. Functions that work well with this structure, called CR-functions, are like more general versions of the smooth functions studied in complex analysis. These ideas help mathematicians understand the geometry and analysis of complex spaces.

Introduction and motivation

A CR structure is a way to describe special surfaces inside complex space using the properties of holomorphic vector fields. Think of it like this: imagine a curved shape floating inside a space filled with two kinds of directions — like up-down and left-right, but in a more complex way. The CR structure looks at which of these directions just touch the surface without cutting through it.

For example, take a surface defined by the equation |z|² + |w|² = 1, where z and w are complex numbers. The CR structure picks out special directions that stay on this surface. These directions help us understand the surface in a way that connects to complex numbers and their rules. This idea can be extended to more complicated surfaces and even to abstract spaces, where we define the rules without starting from a complex space.

Embedded CR manifolds

Embedded CR manifolds are submanifolds of $\mathbb{C}^n$. They are defined using complex vector fields that annihilate certain functions. These vectors help organize the structure of the manifold.

When a real submanifold is embedded in $\mathbb{C}^n$, it follows specific rules that ensure its compatibility with complex structures. This compatibility is crucial for understanding the geometry and analysis on these manifolds. The study of these structures involves advanced mathematical tools and concepts.

Examples

A common example of a CR manifold is a special kind of sphere. This sphere lives inside a bigger space with complex numbers, making it a great example to study. Another example is the Heisenberg group, which is not compact but still follows the rules of a CR manifold. These spaces help mathematicians understand curves and shapes in more complex geometries, similar to how they study curves in regular space.