centermanifold

A centermanifold is a concept in differential geometry that generalizes the idea of a manifold to a space that is not necessarily orientable or has a boundary. It is a topological space equipped with a chart, which is a local homeomorphism to an open subset of Euclidean space. The collection of charts is required to satisfy certain compatibility conditions, ensuring that the transitions between charts are smooth.

The key distinction of a centermanifold from a standard manifold lies in its definition of smoothness. Instead of requiring the transition maps between charts to be differentiable, centermanifolds only require them to be continuous. This relaxation allows for the inclusion of spaces that might not be "smooth" in the traditional sense, such as certain stratified spaces or spaces with singularities.

Centermanifolds are built upon the notion of a "center," which can be thought of as a distinguished subspace within the centermanifold. The structure of the centermanifold is often defined in relation to this center. For instance, charts might be required to map a neighborhood of a point in the centermanifold to a neighborhood of a corresponding point in Euclidean space, with additional conditions imposed on how the center is mapped.

The study of centermanifolds is motivated by the desire to develop geometric tools that can be applied to a broader range of spaces than those typically handled by classical differential geometry. This includes applications in areas such as geometric measure theory, the study of singularities in algebraic geometry, and certain branches of theoretical physics where non-smooth structures arise.