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...

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

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

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}...

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

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

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...

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

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...

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...

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

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...

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

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

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...

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

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

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...

Codimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold. Connected sum Connection Cotangent...

conditions and reasons for PDRs to reduce to homotopy theory. Immersed submanifold Isometric immersion Submersion This definition is given by Bishop & Crittenden...

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...

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...

hypersurface to the unit sphere Sn − 1  ⊆  Rn. For a general oriented k-submanifold of Rn the Gauss map can also be defined, and its target space is the...

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...

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

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...

to be determined in the calculus of variations Lagrangian submanifold, a class of submanifolds in symplectic geometry Lagrangian system, a pair consisting...

it is (locally) a submanifold. One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to...

intersection may fail to be a submanifold, having some sort of singular point. In particular, this means that transverse submanifolds of complementary dimension...