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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.
A circle is a compact 1-manifold. A torus and a Klein bottle are compact 2-manifolds (or surfaces). The n-dimensional sphere S n is a compact n-manifold. The n-dimensional torus T n (the product of n circles) is a compact n-manifold.
In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. Two important classes of differentiable manifolds are smooth and analytic manifolds ...
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 ...
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 .
As another example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space, although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous, the continuous image of a compact space is compact, and the 2-sphere is compact whereas Euclidean 2-space is not.
Then the result is being extended to manifolds having a basis which is a de Rham cover. This step is more technical. Finally, one easily shows that open subsets of and consequently any manifold has a basis which is a de Rham cover. Thus, invoking the previous step, finishes the proof.
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 ...