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.

A CR manifold has a special set of directions called a complex subbundle within its tangent bundle. These directions fit together smoothly. From this structure, mathematicians can create a special 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 are like more general versions of smooth functions. 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. Think of it like this: imagine a curved shape floating inside a space 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 better. This idea can be used for more complicated surfaces and even abstract spaces.

Embedded CR manifolds

Embedded CR manifolds are special shapes inside $\mathbb{C}^n$. They are described using complex vectors that zero out some functions. These vectors help give the shape its structure.

When a real shape is placed inside $\mathbb{C}^n$, it must follow certain rules. These rules make sure the shape works well with complex structures. This is important for studying the shape's geometry and properties. Learning about these shapes uses advanced math tools and ideas.

Examples

A common example of a CR manifold is a special kind of sphere. This sphere is found inside a bigger space with complex numbers, which makes it useful for study. Another example is the Heisenberg group, which is not compact but still follows the rules of a CR manifold. These spaces help mathematicians learn about curves and shapes in more complex geometries, just like how they study curves in regular space.