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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.
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 ...
Any differentiable manifold is a diffeological space by considering its maximal atlas (i.e., the plots are all smooth maps from open subsets of to the manifold); its D-topology recovers the original manifold topology. With this diffeology, a map between two smooth manifolds is smooth if and only if it is differentiable in the diffeological sense.
These categories are related by forgetful functors: for instance, a differentiable manifold is also a topological manifold, and a differentiable map is also continuous, so there is a functor . These functors are in general neither one-to-one nor onto on objects; these failures are generally referred to in terms of "structure", as follows.
A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H : M × [0,1] → N such that for all t in [0, 1] the function H t : M → N defined by H t (x) = H(x, t) for all x ∈ M is an immersion, with H 0 = f, H 1 = g. A regular homotopy is thus a homotopy through immersions.
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...
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 )
Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry.. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.