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  2. Differentiable manifold - Wikipedia

    en.wikipedia.org/wiki/Differentiable_manifold

    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.

  3. Differential structure - Wikipedia

    en.wikipedia.org/wiki/Differential_structure

    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 ...

  4. Diffeology - Wikipedia

    en.wikipedia.org/wiki/Diffeology

    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.

  5. Classification of manifolds - Wikipedia

    en.wikipedia.org/wiki/Classification_of_manifolds

    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.

  6. Immersion (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Immersion_(mathematics)

    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.

  7. Calculus on Manifolds (book) - Wikipedia

    en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)

    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 ...

  8. CR manifold - Wikipedia

    en.wikipedia.org/wiki/CR_manifold

    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 )

  9. Symplectic geometry - Wikipedia

    en.wikipedia.org/wiki/Symplectic_geometry

    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.