In mathematics, a submanifold of a manifold M {\displaystyle M} is a subset S {\displaystyle S} which itself has the structure of a manifold, and for...

immersed) submanifolds whose tangent spaces lie inside the contact field: these are called Legendrian submanifolds. Legendrian submanifolds are analogous...

subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called co-isotropic. Lagrangian submanifolds of a symplectic...

possibly disconnected). The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of M; it is unique (up to isotopy)...

In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every q ∈ M {\displaystyle q\in M}...

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

A Riemannian submanifold N {\displaystyle N} of a Riemannian manifold M {\displaystyle M} is a submanifold N {\displaystyle N} of M {\displaystyle M}...

when the curve γ {\displaystyle \gamma } is restricted to lie on a submanifold M {\displaystyle M} of M ¯ {\displaystyle {\bar {M}}} (e.g. for curves...

In dynamical systems, a spectral submanifold (SSM) is the unique smoothest invariant manifold serving as the nonlinear extension of a spectral subspace...

true for any submanifold of Euclidean space of any dimension. Although John Nash proved that every Riemannian manifold arises as a submanifold of Euclidean...

field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to...

constructed the parallel 4-form. A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if...

on a surface can be broken down the same way. More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point...

compact manifold (without boundary), whereas a closed submanifold N of a manifold M means a submanifold that is a closed subset of M, not necessarily compact...

submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as Lagrangian submanifolds and...

Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics...

the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback...

basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine...

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

idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative...

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...

topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category...

In mathematics, an algebraic manifold is an algebraic variety which is also a manifold. As such, algebraic manifolds are a generalisation of the concept...

More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold. The second fundamental form of a parametric...

Subanalytic sets still have a reasonable local description in terms of submanifolds. A subset V of a given Euclidean space E is semianalytic if each point...

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

that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R2n, whereas it is "rare" for a complex manifold to have a holomorphic...

plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the...

distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same...

Intuitively, one can think of a submanifold as a surface embedded inside of a Calabi–Yau manifold, although submanifolds can also exist in dimensions different...