vmanifold

A vmanifold is a mathematical object that generalizes the concept of a manifold. While a manifold is locally Euclidean, meaning any point has a neighborhood that looks like an open set in Euclidean space, a vmanifold is locally a quotient of an open set in Euclidean space by a discrete group of symmetries. This "orbifold" structure allows vmanifolds to have singularities, which are points where the local structure is not entirely Euclidean.

The definition of a vmanifold typically involves a topological space and a sheaf of groups. The sheaf assigns a discrete group to each open set, and these groups describe the local symmetries. When considering a point in a vmanifold, its neighborhood can be represented as an open set U in Euclidean space, with a discrete group G acting on U. The local structure is then the quotient space U/G.

Vmanifolds find applications in various areas of mathematics and physics, including algebraic geometry, differential geometry, and string theory. They provide a framework for studying geometric objects with singularities and are particularly useful in situations where one needs to consider spaces with non-trivial isotropy. The study of vmanifolds often involves developing tools and techniques to handle the singular strata present in these spaces.