Explore: Local Submanifolds

|_{S}} is a symplectic form on S {\displaystyle S} . Isotropic submanifolds are submanifolds where the symplectic form restricts to zero, i.e. each tangent

called a closed embedded submanifold of M {\displaystyle M} . Closed embedded submanifolds form the nicest class of submanifolds. Smooth manifolds are sometimes

locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Violations of local flatness describe

thing as symplectic submanifolds. Another important generalisation of Poisson submanifolds is given by coisotropic submanifolds, introduced by Weinstein

topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold. To define this more precisely, first

More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely

{\displaystyle x} such that the image f ( U ) {\displaystyle f(U)} is an embedded submanifold, and f | U : U → f ( U ) {\displaystyle f|_{U}:U\to f(U)} is a diffeomorphism

The submanifolds are called the leaves of the foliation. The 3-sphere has a famous codimension-1 foliation called the Reeb foliation. The submanifolds are

intrinsically the property of being a hypersurface (or certain real submanifolds of higher codimension) in complex space by studying the properties of

positive there). Subanalytic sets still have a reasonable local description in terms of submanifolds. A subset V of a given Euclidean space E is semianalytic