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A differentiable manifold (of class C k) consists of a pair (M, O M) where M is a second countable Hausdorff space, and O M is a sheaf of local R-algebras defined on M, such that the locally ringed space (M, O M) is locally isomorphic to (R n, O). In this way, differentiable manifolds can be thought of as schemes modeled on R n.
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for >, any smooth -dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean-space, and a (not necessarily one-to-one) immersion in ()-space.
Call an open cover of a manifold a "de Rham cover", if all elements of the cover are de Rham, as well as all of their finite intersections. One shows that convex sets in R n {\displaystyle \mathbb {R} ^{n}} are de Rham, basically by the homotopy invariance of both cohomologies in question.
For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number b 2. For large Betti numbers b 2 > 18 in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many ...
The objects of Man • p are pairs (,), where is a manifold along with a basepoint , and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. : (,) (,), such that () =. [1] The category of pointed manifolds is an example of a comma category - Man • p is exactly ({}), where {} represents an arbitrary singleton ...
The "surgery" on a differentiable manifold M of dimension = + +, could be described as removing an imbedded sphere of dimension p from M. [2] Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., =) vanishes for . In particular, it implies that the de Rham complex yields a resolution of the constant sheaf R M {\displaystyle \mathbb {R} _{M}} on M .
Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a complex subbundle of the complexified tangent bundle = such that [ L , L ] ⊆ L {\displaystyle [L,L]\subseteq L} ( L is formally integrable )