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| {{redirect|Smooth curve|the equivalent concept in algebraic geometry|singular point of an algebraic variety}}
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| [[File:nondifferentiable atlas.png|right|frame|A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the middle chart the [[Tropic of Cancer]] is a smooth curve, whereas in the first it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.]]
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| In mathematics, a '''differentiable manifold''' is a type of [[manifold]] that is locally similar enough to a [[linear space]] to allow one to do [[calculus]]. Any manifold can be described by a collection of [[atlas (topology)|charts]], also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is [[differentiable]]), then computations done in one chart are valid in any other differentiable chart.
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| In formal terms, a '''differentiable manifold''' is a [[topological manifold]] with a globally defined [[differential structure]]. Any topological manifold can be given a differential structure ''locally'' by using the [[homeomorphism]]s in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their [[Function composition|composition]] on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called ''transition maps.''
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| Differentiability means different things in different contexts including: [[smooth function|continuously differentiable]], ''k'' times differentiable, and [[holomorphic function|holomorphic]]. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable [[tangent space]], differentiable functions, and differentiable [[Tensor field|tensor]] and [[Vector field|vector]] fields. Differentiable manifolds are very important in [[physics]]. Special kinds of differentiable manifolds form the basis for physical theories such as [[classical mechanics]], [[general relativity]], and [[Gauge theory|Yang–Mills theory.]] It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the [[exterior derivative|exterior calculus.]] The study of calculus on differentiable manifolds is known as [[differential geometry and topology|differential geometry.]]
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| == History ==
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| {{main|History of manifolds and varieties}}
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| The emergence of differential geometry as a distinct discipline is generally credited to [[Carl Friedrich Gauss]] and [[Bernhard Riemann]]. Riemann first described manifolds in his famous habilitation lecture<ref>B. Riemann (1867).</ref> before the faculty at Göttingen. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments:
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| : ''Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ...''– B. Riemann
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| The works of physicists such as [[James Clerk Maxwell]]{{Citation needed|date=February 2007}}, and mathematicians [[Gregorio Ricci-Curbastro]] and [[Tullio Levi-Civita]]<ref>See G. Ricci (1888), G. Ricci and T. Levi-Civita (1901), T. Levi-Civita (1927).</ref> led to the development of [[tensor analysis]] and the notion of [[general covariance|covariance]], which identifies an intrinsic geometric property as one that is invariant with respect to [[coordinate transformation]]s. These ideas found a key application in [[Einstein]]'s theory of [[general relativity]] and its underlying [[equivalence principle]]. A modern definition of a 2-dimensional manifold was given by [[Hermann Weyl]] in his 1913 book on [[Riemann surface]]s.<ref>See H. Weyl (1955).</ref> The widely accepted general definition of a manifold in terms of an [[atlas (mathematics)|atlas]] is due to [[Hassler Whitney]].<ref>H. Whitney (1936).</ref>
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| == Definition ==
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| A ''presentation'' of a '''topological manifold''' is a [[second countable]] [[Hausdorff space|Hausdorff]] [[topological space|space]] that is locally homeomorphic to a linear space, by a collection (called an ''atlas'') of [[homeomorphism]]s called ''charts''. The composition of one chart with the [[inverse function|inverse]] of another chart is a function called a ''[[transition map]]'', and defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced also contains the data of how it has been patched together. However, different atlases (patchings) may produce "the same" manifold; a manifold does not come with a preferred atlas. And, thus, one defines a '''topological manifold''' to be a space as above with an ''[[equivalence class]]'' of atlases, where one defines equivalence of atlases below.
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| There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following.
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| * A '''differentiable manifold''' is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. In broader terms, a [[differentiability class|''C''<sup>k</sub>]]-manifold is a topological manifold with an atlas whose transition maps are all ''k''-times continuously differentiable.
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| * A '''smooth manifold''' or ''C''<sup>∞</sup>-manifold is a differentiable manifold for which all the transition maps are [[smooth function|smooth]]. That is, derivatives of all orders exist; so it is a ''C<sup>k</sup>''-manifold for all ''k''. An equivalence class of such [[atlas]]es is said to be a [[smooth structure]].
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| * An '''analytic manifold''', or ''C''<sup>ω</sup>-manifold is a smooth manifold with the additional condition that each transition map is [[analytic function|analytic]]: the Taylor expansion is absolutely convergent and equals the function on some open ball.
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| * A '''complex manifold''' is a topological space modeled on a Euclidean space over the [[complex field]] and for which all the transition maps are [[holomorphic]].
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| While there is a meaningful notion of a ''C<sup>k</sub>'' ''atlas,'' there is no distinct notion of a ''C<sup>k</sub>'' ''manifold'' other than ''C''<sup>0</sub> (continuous maps: a topological manifold) and ''C''<sup>∞</sub> (smooth maps: a smooth manifold), because every ''C<sup>k</sub>''-structure with ''k'' > 0, there is a unique ''C<sup>k</sub>''-equivalent ''C''<sup>∞</sub>-structure (every ''C<sup>k</sup>''-structure is ''uniquely smoothable'') – a result of [[Hassler Whitney|Whitney]] (and further, two ''C<sup>k</sup>'' atlases that are equivalent to a single ''C''<sup>∞</sup> atlas are equivalent as ''C<sup>k</sup>'' atlases, so two distinct ''C<sup>k</sup>'' atlases do not collide); see [[Differential_structure#Existence_and_uniqueness_theorems|Differential structure: Existence and uniqueness theorems]] for details. Thus one uses the terms "differentiable manifold" and "smooth manifold" interchangeably. This is in stark contrast to ''C<sup>k</sup>'' ''maps,'' where there are meaningful differences for different ''k.'' For example, the [[Nash embedding theorem]] states that any manifold can be ''C<sup>k</sup>'' isometrically embedded in Euclidean space '''R'''<sup>''N''</sup> – for any 1 ≤ ''k'' ≤ ∞ there is a sufficiently large ''N'', but ''N'' depends on ''k''.
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| On the other hand, complex manifolds are significantly more restrictive. As an example, [[Algebraic geometry and analytic geometry#Chow.27s theorem|Chow's theorem]] states that any projective complex manifold is in fact a [[projective variety]] – it has an algebraic structure.
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| === Atlases ===
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| {{ Annotated image | caption=Charts on a manifold
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| | image=Two coordinate charts on a manifold.svg
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| | image-width = 250
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| | annotations =
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| {{Annotation|45|70|<math>X</math>}}
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| {{Annotation|70|54|<math>U_\alpha</math>}}
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| {{Annotation|187|66|<math>U_\beta</math>}}
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| {{Annotation|42|100|<math>\varphi_\alpha</math>}}
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| {{Annotation|183|117|<math>\varphi_\beta</math>}}
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| {{Annotation|87|112|<math>\varphi_{\alpha\beta}</math>}}
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| {{Annotation|90|145|<math>\varphi_{\beta\alpha}</math>}}
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| {{Annotation|55|183|<math>\mathbf R^n</math>}}
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| {{Annotation|145|183|<math>\mathbf R^n</math>}}
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| }}
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| An [[atlas (topology)|atlas]] on a topological space ''X'' is a collection of pairs {(''U''<sub>α</sub>,φ<sub>α</sub>)} called ''charts'', where the ''U''<sub>α</sub> are open sets that [[open cover|cover]] ''X'', and for each index α
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| :<math>\varphi_\alpha \colon U_\alpha \to {\mathbf R}^n</math> | |
| is a [[homeomorphism]] of ''U''<sub>α</sub> onto an open subset of ''n''-dimensional real space. The '''transition maps''' of the atlas are the functions
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| :<math>\varphi_{\alpha\beta} = \varphi_\beta\circ\varphi_\alpha^{-1}|_{\varphi_\alpha(U_\alpha\cap U_\beta)} \colon \varphi_\alpha(U_\alpha\cap U_\beta) \to \varphi_\beta(U_\alpha\cap U_\beta).</math>
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| Every topological manifold has an atlas. A ''C<sup>k</sup>''-atlas is an atlas whose transition maps are ''C<sup>k</sup>''. A topological manifold has a ''C''<sup>0</sup>-atlas and in general a ''C<sup>k</sup>''-manifold has a ''C<sup>k</sup>''-atlas. A continuous atlas is a ''C''<sup>0</sup> atlas, a smooth atlas is a ''C''<sup>∞</sup> atlas and an analytic atlas is a ''C''<sup>ω</sup> atlas. If the atlas is at least ''C''<sup>1</sup>, it is also called a ''differential structure'' or ''differentiable structure''. A ''holomorphic atlas'' is an atlas whose underlying Euclidean space is defined on the [[complex field]] and whose transition maps are [[biholomorphic]].
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| === Compatible atlases ===
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| Different atlases can give rise to, in essence, the same manifold. The circle can be mapped by two coordinate charts, but if the domains of these charts are changed slightly a different atlas for the same manifold is obtained. These different atlases can be combined into a bigger atlas. It can happen that the transition maps of such a combined atlas are not as smooth as those of the constituent atlases. If ''C<sup>k</sup>'' atlases can be combined to form a ''C<sup>k</sup>'' atlas, then they are called compatible. Compatibility of atlases is an [[equivalence relation]]; by combining all the atlases in an [[equivalence class]], a '''maximal atlas''' can be constructed. Each ''C<sup>k</sup>'' atlas belongs to a unique maximal ''C<sup>k</sup>'' atlas.
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| ==Alternative definitions==
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| === Pseudogroups ===
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| The notion of a [[pseudogroup]]<ref>Kobayashi and Nomizu (1963), Volume 1.</ref> provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. A ''pseudogroup'' consists of a topological space ''S'' and a collection Γ consisting of homeomorphisms from open subsets of ''S'' to other open subsets of ''S'' such that
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| # If ''f'' ∈ Γ, and ''U'' is an open subset of the domain of ''f'', then the restriction ''f''|<sub>''U''</sub> is also in Γ.
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| # If ''f'' is a homeomorphism from a union of open subsets of ''S'', <math>\cup_i \, U_i </math>, to an open subset of ''S'', then ''f'' ∈ Γ provided <math> f|_{U_i} \in \Gamma </math> for every ''i''.
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| # For every open ''U'' ⊂ ''S'', the identity transformation of ''U'' is in Γ.
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| # If ''f'' ∈ Γ, then ''f''<sup>−1</sup> ∈ Γ.
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| # The composition of two elements of Γ is in Γ.
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| These last three conditions are analogous to the definition of a [[group (mathematics)|group]]. Note that Γ need not be a group, however, since the functions are not globally defined on ''S''. For example, the collection of all local ''C<sup>k</sup>'' [[diffeomorphisms]] on '''R'''<sup>''n''</sup> form a pseudogroup. All [[biholomorphism]]s between open sets in '''C'''<sup>''n''</sup> form a pseudogroup. More examples include: orientation preserving maps of '''R'''<sup>''n''</sup>, [[symplectomorphism]]s, [[Möbius transformation]]s, [[affine transformation]]s, and so on. Thus a wide variety of function classes determine pseudogroups.
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| An atlas (''U<sub>i</sub>'', φ<sub>''i''</sub>) of homeomorphisms φ<sub>''i''</sub> from ''U<sub>i</sub>'' ⊂ ''M'' to open subsets of a topological space ''S'' is said to be ''compatible'' with a pseudogroup Γ provided that the transition functions φ<sub>''j''</sub> o φ<sub>''i''</sub><sup>−1</sup>: φ<sub>''i''</sub>(''U<sub>i</sub>'' ∩ ''U<sub>j</sub>'') → φ<sub>''j''</sub>(''U<sub>i</sub>'' ∩ ''U<sub>j</sub>'') are all in Γ.
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| A differentiable manifold is then an atlas compatible with the pseudogroup of ''C''<sup>''k''</sup> functions on '''R'''<sup>''n''</sup>. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in '''C'''<sup>''n''</sup>. And so forth. Thus pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology.
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| === Structure sheaf ===
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| Sometimes it can be useful to use an alternative approach to endow a manifold with a ''C<sup>k</sup>''-structure. Here ''k'' = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The [[sheaf (mathematics)|structure sheaf]] of ''M'', denoted '''C'''<sup>''k''</sup>, is a sort of [[functor]] that defines, for each open set ''U'' ⊂ ''M'', an algebra '''C'''<sup>''k''</sup>(''U'') of continuous functions ''U'' → '''R'''. A structure sheaf '''C'''<sup>''k''</sup> is said to give ''M'' the structure of a ''C''<sup>''k''</sup> manifold of dimension ''n'' provided that, for any ''p'' ∈ ''M'', there exists a neighborhood ''U'' of ''p'' and ''n'' functions ''x''<sup>1</sup>,...,''x''<sup>''n''</sup> ∈ '''C'''<sup>''k''</sup>(''U'') such that the map ''f'' = (''x''<sup>1</sup>, ..., ''x<sup>n</sup>''): ''U'' → '''R'''<sup>''n''</sup> is a homeomorphism onto an open set in '''R'''<sup>''n''</sup>, and such that '''C'''<sup>''k''</sup>|<sub>''U''</sub> is the [[pullback]] of the sheaf of ''k''-times continuously differentiable functions on '''R'''<sup>''n''</sup>.<ref>This definition can be found in MacLane and Moerdijk (1992). For an equivalent, ''ad hoc'' definition, see Sternberg (1964) Chapter II.</ref>
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| In particular, this latter condition means that any function ''h'' in '''C'''<sup>''k''</sup>(''V''), for ''V'', can be written uniquely as ''h''(''x'') = ''H''(''x''<sup>1</sup>(''x''),...,''x''<sup>''n''</sup>(''x'')), where ''H'' is a ''k''-times differentiable function on ''f''(''V'') (an open set in '''R'''<sup>''n''</sup>). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on '''R'''<sup>''n''</sup>, and [[A fortiori argument|''a fortiori'']] this is sufficient to characterize the differential structure on the manifold.
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| ==== Sheaves of local rings ====
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| A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a [[ringed space]]. This approach is strongly influenced by the theory of [[scheme (mathematics)|schemes]] in [[algebraic geometry]], but uses [[local ring]]s of the [[germ (mathematics)|germs]] of differentiable functions. It is especially popular in the context of ''complex'' manifolds.
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| We begin by describing the basic structure sheaf on '''R'''<sup>''n''</sup>. If ''U'' is an open set in '''R'''<sup>''n''</sup>, let
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| :'''O'''(''U'') = ''C''<sup>''k''</sup>(''U'', '''R''')
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| consist of all real-valued ''k''-times continuously differentiable functions on ''U''. As ''U'' varies, this determines a sheaf of rings on '''R'''<sup>n</sup>. The stalk '''O'''<sub>''p''</sub> for ''p'' ∈ '''R'''<sup>''n''</sup> consists of [[germ (mathematics)|germs]] of functions near ''p'', and is an algebra over '''R'''. In particular, this is a [[local ring]] whose unique [[maximal ideal]] consists of those functions that vanish at ''p''. The pair ('''R'''<sup>''n''</sup>, '''O''') is an example of a [[locally ringed space]]: it is a topological space equipped with a sheaf whose stalks are each local rings.
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| A differentiable manifold (of class ''C<sup>k</sup>'') consists of a pair (''M'', '''O'''<sub>''M''</sub>) where ''M'' is a [[second countable]] [[Hausdorff space|Hausdorff]] [[topological space|space]], and '''O'''<sub>''M''</sub> is a sheaf of local '''R'''-algebras defined on ''M'', such that the locally ringed space (''M'', '''O'''<sub>''M''</sub>) is locally isomorphic to ('''R'''<sup>''n''</sup>, '''O'''). In this way, differentiable manifolds can be thought of as [[scheme (mathematics)|schemes]] modelled on '''R'''<sup>''n''</sup>. This means that,<ref>Hartshorne (1997)</ref> for each point ''p'' ∈ ''M'', there is a neighborhood ''U'' of ''p'', and a pair of functions (''f'', ''f''<sup>#</sup>) where
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| # ''f'': ''U'' → ''f''(''U'') ⊂ '''R'''<sup>n</sup> is a homeomorphism onto an open set in '''R'''<sup>''n''</sup>.
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| # ''f''<sup>#</sup>: '''O'''|<sub>''f''(''U'')</sub> → ''f''<sub>*</sub> ('''O'''<sub>''M''</sub>|<sub>''U''</sub>) is an isomorphism of sheaves.
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| # The localization of ''f''<sup>#</sup> is an isomorphism of local rings
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| :: ''f''<sup>#</sup><sub>p</sub>: '''O'''<sub>''f''(''p'')</sub> → '''O'''<sub>''M'', ''p''</sub>.
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| There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no ''a priori'' reason that the model space needs to be '''R'''<sup>n</sup>. For example (in particular in [[algebraic geometry]]), one could take this to be the space of complex numbers '''C'''<sup>''n''</sup> equipped with the sheaf of [[holomorphic function]]s (thus arriving at the spaces of [[complex analytic geometry]]), or the sheaf of [[polynomial]]s (thus arriving at the spaces of interest in complex ''algebraic'' geometry). In broad terms, this concept can be adapted for any suitable notion of a scheme (see [[topos theory]]). Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair (''f'', ''f''<sup>#</sup>), but these merely quantify the idea of ''local isomorphism'' rather than being central to the discussion (as in the case of charts and atlases). Third, the sheaf '''O'''<sub>''M''</sub> is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a ''consequence'' of the construction (via the quotients of local rings by their maximal ideals). Hence it is a more primitive definition of the structure (see [[synthetic differential geometry]]).
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| A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.
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| * The [[cotangent space]] at a point is ''I<sub>p</sub>''/''I<sub>p</sub>''<sup>2</sup>, where ''I<sub>p</sub>'' is the maximal ideal of the stalk '''O'''<sub>''M'', ''p''</sub>.
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| * In general, the entire [[cotangent bundle]] can be obtained by a related technique (see [[cotangent bundle]] for details).
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| * [[Taylor series]] (and [[jet (mathematics)|jets]]) can be approached in a coordinate-independent manner using the [[completion (ring theory)|''I''<sub>''p''</sub>-adic filtration]] on '''O'''<sub>''M'', ''p''</sub>.
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| * The [[tangent bundle]] (or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of '''O'''<sub>''M''</sub> into the ring of [[dual numbers]].
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| == Differentiable functions ==
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| A real valued function ''f'' on an ''n''-dimensional differentiable manifold ''M'' is called '''differentiable''' at a point ''p'' ∈ ''M'' if it is differentiable in any coordinate chart defined around ''p''. In more precise terms, if (''U'', φ) is a chart where ''U'' is an open set in ''M'' containing ''p'' and φ: ''U'' → '''R'''<sup>n</sup> is the map defining the chart, then ''f'' is differentiable [[if and only if]]
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| :<math>f\circ \phi^{-1} \colon \phi(U)\subset {\mathbf R}^n \to {\mathbf R}</math>
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| is differentiable at φ(''p''). The definition of differentiability depends on the choice of chart at ''p''; in general there will be many available charts. However, it follows from the [[chain rule]] applied to the transition functions between one chart and another that if ''f'' is differentiable in any particular chart at ''p'', then it is differentiable in all charts at ''p''. Analogous considerations apply to defining ''C<sup>k</sup>'' functions, smooth functions, and analytic functions.
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| ===Differentiation of functions===
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| There are various ways to define the [[derivative]] of a function on a differentiable manifold, the most fundamental of which is the [[directional derivative]]. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable [[affine space|affine]] structure with which to define [[vector (geometric)|vectors]]. The directional derivative therefore looks at curves in the manifold instead of vectors.
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| ====Directional differentiation====
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| Given a real valued function ''f'' on an ''m'' dimensional differentiable manifold ''M'', the directional derivative of ''f'' at a point ''p'' in ''M'' is defined as follows. Suppose that γ(''t'') is a curve in ''M'' with γ(0) = ''p'', which is ''differentiable'' in the sense that its composition with any chart is a [[differentiable curve]] in '''R'''<sup>''m''</sup>. Then the '''directional derivative''' of ''f'' at ''p'' along γ is
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| :<math>\left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}.</math>
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| If γ<sub>1</sub> and γ<sub>2</sub> are two curves such that γ<sub>1</sub>(0) = γ<sub>2</sub>(0) = ''p'', and in any coordinate chart φ,
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| :<math>\left.\frac{d}{dt}\phi\circ\gamma_1(t)\right|_{t=0}=\left.\frac{d}{dt}\phi\circ\gamma_2(t)\right|_{t=0}</math> | |
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| then, by the chain rule, ''f'' has the same directional derivative at ''p'' along γ<sub>1</sub> as along γ<sub>2</sub>. This means that the directional derivative depends only on the [[tangent vector]] of the curve at ''p''. Thus the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space.
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| ====Tangent vectors and the differential====
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| A '''tangent vector''' at ''p'' ∈ ''M'' is an [[equivalence class]] of differentiable curves γ with γ(0) = ''p'', modulo the equivalence relation of first-order [[contact (mathematics)|contact]] between the curves. Therefore,
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| :<math> \gamma_1\equiv \gamma_2 \iff \left.\frac{d}{dt}\phi\circ\gamma_1(t)\right|_{t=0} = \left.\frac{d}{dt}\phi\circ\gamma_2(t)\right|_{t=0}</math>
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| in every coordinate chart ''φ''. Therefore, the equivalence classes are curves through ''p'' with a prescribed [[velocity vector]] at ''p''. The collection of all tangent vectors at ''p'' forms a [[vector space]]: the [[tangent space]] to ''M'' at ''p'', denoted ''T''<sub>''p''</sub>''M''.
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| If ''X'' is a tangent vector at ''p'' and ''f'' a differentiable function defined near ''p'', then differentiating ''f'' along any curve in the equivalence class defining ''X'' gives a well-defined directional derivative along ''X'':
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| :<math>Xf(p) := \left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}.</math>
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| Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative.
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| If the function ''f'' is fixed, then the mapping
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| :<math>X\mapsto Xf(p)</math>
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| is a [[linear functional]] on the tangent space. This linear functional is often denoted by ''df''(''p'') and is called the '''differential''' of ''f'' at ''p'':
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| :<math>df(p) \colon T_pM \to {\mathbf R}.</math>
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| ===Partitions of unity===
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| One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits [[partition of unity|partitions of unity]]. This distinguishes the differential structure on a manifold from stronger structures (such as analytic and holomorphic structures) that in general fail to have partitions of unity.
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| Suppose that ''M'' is a manifold of class ''C<sup>k</sup>'', where 0 ≤ ''k'' ≤ ∞. Let {''U''<sub>α</sub>} be an open covering of ''M''. Then a '''partition of unity''' subordinate to the cover {''U''<sub>α</sub>} is a collection of real-valued ''C<sup>k</sup>'' functions φ<sub>''i''</sub> on ''M'' satisfying the following conditions:
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| * The [[support (mathematics)|supports]] of the φ<sub>''i''</sub> are [[compact space|compact]] and [[locally finite collection|locally finite]];
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| * The support of φ<sub>''i''</sub> is completely contained in ''U''<sub>α</sub> for some α;
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| * The φ<sub>''i''</sub> sum to one at each point of ''M'':
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| ::<math>\sum_i \phi_i(x) = 1.\,</math>
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| (Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φ<sub>''i''</sub>.)
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| Every open covering of a ''C<sup>k</sup>'' manifold ''M'' has a ''C<sup>k</sup>'' partition of unity. This allows for certain constructions from the topology of ''C<sup>k</sup>'' functions on '''R'''<sup>''n''</sup> to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of '''R'''<sup>''n''</sup>. Partitions of unity therefore allow for certain other kinds of [[function space]]s to be considered: for instance [[Lp space|L<sup>p</sup> spaces]], [[Sobolev spaces]], and other kinds of spaces that require integration.
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| ===Differentiability of mappings between manifolds===
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| Suppose ''M'' and ''N'' are two differentiable manifolds with dimensions ''m'' and ''n'', respectively, and ''f'' is a function from ''M'' to ''N''. Since differentiable manifolds are topological spaces we know what it means for ''f'' to be continuous. But what does "''f'' is ''C<sup>k</sup>''(''M'', ''N'')" mean for ''k'' ≥ 1? We know what that means when ''f'' is a function between Euclidean spaces, so if we compose ''f'' with a chart of ''M'' and a chart of ''N'' such that we get a map that goes from Euclidean space to ''M'' to ''N'' to Euclidean space we know what it means for that map to be ''C<sup>k</sup>''('''R'''<sup>''m''</sup>, '''R'''<sup>''n''</sup>). We define "''f'' is ''C<sup>k</sup>''(''M'', ''N'')" to mean that all such compositions of ''f'' with charts are ''C<sup>k</sup>''('''R'''<sup>''m''</sup>, '''R'''<sup>''n''</sup>). Once again the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on ''M'' and ''N'' are selected. However, defining the derivative itself is more subtle. If ''M'' or ''N'' is itself already a Euclidean space, then we don't need a chart to map it to one.
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| === Algebra of scalars ===
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| For a ''C<sup>k</sup>'' manifold ''M'', the [[Set (mathematics)|set]] of real-valued ''C<sup>k</sup>'' functions on the manifold forms an [[algebra over a field|algebra]] under pointwise addition and multiplication, called the ''algebra of scalar fields'' or simply the ''algebra of scalars''. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of [[regular function]]s in algebraic geometry.
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| It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the [[Banach–Stone theorem]], and is more formally known as the [[spectrum of a C*-algebra]]. First, there is a one-to-one correspondence between the points of ''M'' and the algebra homomorphisms φ: ''C<sup>k</sup>''(''M'') → '''R''', as such a homomorphism ''φ'' corresponds a codimension one ideal in ''C<sup>k</sup>''(''M'') (namely the kernel of ''φ''), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec (the Max Spec) of ''C<sup>k</sup>''(''M'') recovers ''M'' as a point set, though in fact it recovers ''M'' as a topological space.
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| One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry (interpreting rings geometrically) and [[operator theory]] (interpreting Banach spaces geometrically). For example, the tangent bundle to ''M'' can be defined as the derivations of the algebra of smooth functions on ''M''.
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| This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of a [[C*-algebra]] – a commutative C*-algebra being precisely the ring of scalars of a manifold, by Banach–Stone, and allows one to consider ''non''commutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of [[noncommutative geometry]].
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| {{Expand section|date=June 2008}}
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| ==Bundles==
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| ===Tangent bundle===
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| {{details|tangent bundle}}
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| The [[tangent space]] of a point consists of the possible directional derivatives at that point, and has the same [[dimension]] ''n'' as does the manifold. For a set of (non-singular) coordinates ''x<sub>k</sub>'' local to the point, the coordinate derivatives <math>\partial_k=\frac{\partial}{\partial x_k}</math> typically define a basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the [[tangent bundle]], whose dimension is 2''n''. The tangent bundle is where [[vector field|tangent vector]]s lie, and is itself a differentiable manifold. The [[Lagrangian]] is a function on the tangent bundle. One can also define the tangent bundle as the bundle of 1-[[jet (mathematics)|jets]] from '''R''' (the [[real line]]) to ''M''.
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| One may construct an atlas for the tangent bundle consisting of charts based on ''U''<sub>α</sub> × '''R'''<sup>''n''</sup>, where ''U''<sub>α</sub> denotes one of the charts in the atlas for ''M''. Each of these new charts is the tangent bundle for the charts ''U''<sub>α</sub>. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class.
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| ===Cotangent bundle===
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| {{details|cotangent bundle}}
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| The [[dual space]] of a vector space is the set of real valued linear functions on the vector space. The [[cotangent space]] at a point is the dual of the tangent space at that point, and the [[cotangent bundle]] is the collection of all cotangent spaces.
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| Like the tangent bundle the cotangent bundle is again a differentiable manifold. The [[Hamiltonian mechanics|Hamiltonian]] is a scalar on the cotangent bundle. The [[total space]] of a cotangent bundle has the structure of a [[symplectic manifold]]. Cotangent vectors are sometimes called ''[[covector]]s''. One can also define the cotangent bundle as the bundle of 1-[[jet (mathematics)|jets]] of functions from ''M'' to '''R'''.
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| Elements of the cotangent space can be thought of as [[infinitesimal]] displacements: if ''f'' is a differentiable function we can define at each point ''p'' a cotangent vector ''df<sub>p</sub>'', which sends a tangent vector ''X<sub>p</sub>'' to the derivative of ''f'' associated with ''X<sub>p</sub>''. However, not every covector field can be expressed this way. Those that can are referred to as [[exact differential]]s. For a given set of local coordinates ''x<sup>k</sup>'' the differentials ''dx''{{su|p=''k''|b=p}} form a basis of the cotangent space at ''p''.
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| ===Tensor bundle===
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| {{details|tensor bundle}}
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| The tensor bundle is the [[direct sum of vector bundles|direct sum]] of all [[tensor product]]s of the tangent bundle and the cotangent bundle. Each element of the bundle is a [[tensor field]], which can act as a [[multilinear operator]] on vector fields, or on other tensor fields.
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| The tensor bundle cannot be a differentiable manifold, since it is infinite dimensional. It is however an [[algebra (ring theory)|algebra]] over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent factors it has. Sometimes these ranks are referred to as ''[[Covariance|covariant]]'' and ''[[Covariance and contravariance of vectors|contravariant]]'' ranks, signifying tangent and cotangent ranks, respectively.
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| ===Frame bundle===
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| {{details|frame bundle}}
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| A frame (or, in more precise terms, a tangent frame) is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of '''R'''<sup>''n''</sup> to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F(''M''), a [[general linear group|GL(''n'', '''R''')]] [[principal bundle]] made up of the set of all frames over ''M''. The frame bundle is useful because tensor fields on ''M'' can be regarded as [[equivariant]] vector-valued functions on F(''M'').
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| ===Jet bundles===
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| {{details|jet bundle}}
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| On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order [[contact (mathematics)|contact]]. By analogy, the ''k''-th order tangent bundle is the collection of curves modulo the relation of ''k''-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the ''k''-jet bundle is the bundle of their ''k''-jets. These and other examples of the general idea of jet bundles play a significant role in the study of [[differential operator]]s on manifolds.
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| The notion of a frame also generalizes to the case of higher-order jets. Define a ''k''-th order frame to be the ''k''-jet of a [[diffeomorphism]] from '''R'''<sup>''n''</sup> to ''M''.<ref>See S. Kobayashi (1972).</ref> The collection of all ''k''-th order frames, ''F<sup>k</sup>''(''M''), is a principle ''G<sup>k</sup>'' bundle over ''M'', where ''G<sup>k</sup>'' is the [[jet group|group of ''k''-jets]]; i.e., the group made up of [[jet (mathematics)|''k''-jets]] of diffeomorphisms of '''R'''<sup>''n''</sup> that fix the origin. Note that GL(''n'', '''R''') is naturally isomorphic to ''G''<sup>1</sup>, and a subgroup of every ''G<sup>k</sup>'', ''k'' ≥ 2. In particular, a section of ''F''<sup>2</sup>(''M'') gives the frame components of a [[Connection (mathematics)|connection]] on ''M''. Thus, the quotient bundle ''F''<sup>2</sup>(''M'')/ GL(''n'', '''R''') is the bundle of linear connections over ''M''.
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| ==Calculus on manifolds==
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| Many of the techniques from [[multivariate calculus]] also apply, ''[[mutatis mutandis]]'', to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the [[total derivative]] of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least [[local property|locally]]. For example, there are versions of the [[implicit function|implicit]] and [[inverse function theorem]]s for such functions.
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| There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:
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| * The [[Lie derivative]], which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
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| * An [[affine connection]], which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.
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| Ideas from [[integral calculus]] also carry over to differential manifolds. These are naturally expressed in the language of [[exterior calculus]] and [[differential form]]s. The fundamental theorems of integral calculus in several variables — namely [[Green's theorem]], the [[divergence theorem]], and [[Stokes' theorem]] — generalize to a theorem (also called [[Stokes' theorem]]) relating the [[exterior derivative]] and integration over [[submanifold]]s.
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| ===Differential calculus of functions===
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| Differentiable functions between two manifolds are needed in order to formulate suitable notions of [[submanifold]]s, and other related concepts. If ''f'': ''M'' → ''N'' is a differentiable function from a differentiable manifold ''M'' of dimension ''m'' to another differentiable manifold ''N'' of dimension ''n'', then the [[pushforward (differential)|differential]] of ''f'' is a mapping ''df'': T''M'' → T''N''. It also denoted by ''Tf'' and called the '''tangent map'''. At each point of ''M'', this is a linear transformation from one tangent space to another:
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| :<math>df(p)\colon T_p M \to T_{f(p)} N.</math>
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| The '''rank''' of ''f'' at ''p'' is the [[rank of a matrix|rank]] of this linear transformation.
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| Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given by [[Sard's theorem]]. Functions of maximal rank at a point are called [[immersion (mathematics)|immersions]] and [[submersion (mathematics)|submersions]]:
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| * If ''m'' ≤ ''n'', and ''f'' : ''M'' → ''N'' has rank ''m'' at ''p'' ∈ ''M'', then ''f'' is called an '''immersion''' at ''p''. If ''f'' is an immersion at all points of ''M'' and is a [[homeomorphism]] onto its image, then ''f'' is an '''[[embedding]]'''. Embeddings formalize the notion of ''M'' being a [[submanifold]] of ''N''. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.
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| * If ''m'' ≥ ''n'', and ''f'': ''M'' → ''N'' has rank ''n'' at ''p'' ∈ ''M'', then ''f'' is called a '''submersion''' at ''p''. The implicit function theorem states that if ''f'' is a submersion at ''p'', then ''M'' is locally a product of ''N'' and '''R'''<sup>''m''−''n''</sup> near ''p''. In formal terms, there exist coordinates (''y''<sub>1</sub>, ..., ''y<sub>n</sub>'') in a neighborhood of ''f''(''p'') in ''N'', and ''m''−''n'' functions ''x''<sub>1</sub>,...,''x''<sub>''m''−''n''</sub> defined in a neighborhood of ''p'' in ''M'' such that
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| ::<math>(y_1\circ f,\dotsc,y_n\circ f, x_1, \dotsc, x_{m-n})</math>
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| :is a system of local coordinates of ''M'' in a neighborhood of ''p''. Submersions form the foundation of the theory of [[fibration]]s and [[fibre bundle]]s.
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| ===Lie derivative===
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| A [[Lie derivative]], named after [[Sophus Lie]], is a [[derivation (abstract algebra)|derivation]] on the [[Algebra over a field|algebra]] of [[tensor field]]s over a [[manifold]] ''M''. The [[vector space]] of all Lie derivatives on ''M'' forms an infinite dimensional [[Lie algebra]] with respect to the [[Lie bracket of vector fields|Lie bracket]] defined by
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| :<math> [A,B] := \mathcal{L}_A B = - \mathcal{L}_B A.</math>
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| The Lie derivatives are represented by [[vector field]]s, as [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generator]]s of flows ([[active transformation|active]] [[diffeomorphism]]s) on ''M''. Looking at it the other way round, the [[group (mathematics)|group]] of diffeomorphisms of ''M'' has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the [[Lie group]] theory.
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| ===Exterior calculus===
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| {{details|differential form}}
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| The exterior calculus allows for a generalization of the [[gradient]], [[divergence]] and [[Curl (mathematics)|curl]] operators.
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| The bundle of [[differential form]]s, at each point, consists of all totally [[antisymmetric]] [[multilinear]] maps on the tangent space at that point. It is naturally divided into ''n''-forms for each ''n'' at most equal to the dimension of the manifold; an ''n''-form is an ''n''-variable form, also called a form of degree ''n''. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, an ''n''-form is a tensor with cotangent rank ''n'' and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric.
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| ====Exterior derivative====
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| There is a map from scalars to covectors called the [[exterior derivative]]
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| :<math>\mathrm{d} \colon \mathcal{C}(M) \to \mathrm{T}^*(M) : f \mapsto \mathrm{d}f</math>
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| such that
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| :<math>\mathrm{d}f \colon \mathrm{T}(M) \to \mathcal{C}(M) : V \mapsto V(f).</math>
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| This map is the one that relates covectors to infinitesimal displacements, mentioned above; some covectors are the exterior derivatives of scalar functions. It can be generalized into a map from the ''n''-forms onto the (''n''+1)-forms. Applying this derivative twice will produce a zero form. Forms with zero derivative are called closed forms, while forms that are themselves exterior derivatives are known as exact forms.
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| The space of differential forms at a point is the archetypal example of an [[exterior algebra]]; thus it possesses a wedge product, mapping a ''k''-form and ''l''-form to a (''k''+''l'')-form. The exterior derivative extends to this algebra, and satisfies a version of the [[product rule]]:
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| ::<math>\mathrm{d}(\omega \wedge \eta) = \mathrm{d} \omega \wedge \eta+(-1)^{{\rm deg\,}\omega}(\omega \wedge \mathrm{d} \eta).</math>
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| From the differential forms and the exterior derivative, one can define the [[de Rham cohomology]] of the manifold. The rank ''n'' cohomology group is the [[quotient group]] of the closed forms by the exact forms.
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| ==Topology of differentiable manifolds==
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| ===Relationship with topological manifolds===
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| Every topological manifold in dimension 1, 2, or 3 has a unique differential structure (up to diffeomorphism); thus the concepts of topological and differentiable manifold are distinct only in higher dimensions. It is known that in each higher dimension, there are some topological manifolds with no smooth structure, and some with multiple non-diffeomorphic structures.
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| The existence of non-smoothable manifolds was proven by {{harvtxt|Kervaire|1960}}, and later explained in the context of [[Donaldson's theorem]] (compare [[Hilbert's fifth problem]]);<ref>S. Donaldson (1983).</ref> a good example of a non-smoothable manifold is the [[E8 manifold|E<sub>8</sub> manifold]].
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| The classic example of manifolds with multiple incompatible structures are the [[exotic sphere|exotic 7-spheres]] of [[John Milnor]].<ref>J. Milnor (1956). These are the first examples of exotic spheres.</ref>
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| ===Classification===
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| Every second-countable 1-manifold without boundary is homeomorphic to a disjoint union of countably many copies of '''R''' (the [[real line]]) and '''S''' (the [[circle]]); the only connected examples are '''R''' and '''S''', and of these only '''S''' is compact. In higher dimensions, classification theory normally focuses only on compact connected manifolds.
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| For a classification of 2-manifolds, see [[surface]]: in particular compact connected oriented 2-manifolds are classified by their genus, which is a nonnegative integer.
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| A classification of [[3-manifold]]s follows ''in principle'' from the [[Thurston's geometrization conjecture|geometrization of 3-manifolds]] and various recognition results for geometrizable 3-manifolds, such as [[Mostow rigidity]] and Sela's algorithm for the isomorphism problem for hyperbolic groups.<ref>Z. Sela (1995). However, 3-manifolds are only classified in the sense that there is an (impractical) algorithm for generating a non-redundant list of all compact 3-manifolds.</ref>
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| The classification of ''n''-manifolds for ''n'' greater than three is known to be impossible, even up to [[homotopy equivalence]]. Given any finitely [[presentation of a group|presented]] group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to [[decision problem|decide]] the isomorphism problem for finitely presented groups, there is no algorithm to decide if two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is [[decision problem|undecidable]]<!-- this is A.A. Markov's result -->. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide if a manifold has trivial fundamental group, i.e. is [[simply connected]].
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| Simply connected [[4-manifold]]s have been classified up to homeomorphism by [[Michael Freedman|Freedman]] using the [[intersection theory|intersection form]] and [[Kirby–Siebenmann invariant]]. Smooth 4-manifold theory is known to be much more complicated, as the [[exotic R4|exotic smooth structure]]s on '''R'''<sup>4</sup> demonstrate.
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| However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the [[h-cobordism theorem]] can be used to reduce the classification to a classification up to homotopy equivalence, and [[surgery theory]] can be applied.<ref>See A. Ranicki (2002).</ref> This has been carried out to provide an explicit classification of simply connected [[5-manifold]]s by Dennis Barden.
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| ==Structures on manifolds==
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| === (Pseudo-)Riemannian manifolds ===
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| A [[Riemannian manifold]] is a differentiable manifold on which the tangent spaces are equipped with [[inner product space|inner products]] in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the [[Riemannian metric]]. This metric can be used to interconvert vectors and covectors, and to define a rank 4 [[Riemann curvature tensor]]. On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian structure.
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| A [[pseudo-Riemannian manifold]] is a variant of [[Riemannian manifold]] where the [[metric tensor]] is allowed to have an [[metric signature|indefinite signature]] (as opposed to a [[definite bilinear form|positive-definite]] one). Pseudo-Riemannian manifolds of signature (3, 1) are important in [[general relativity]]. Not every differentiable manifold can be given a pseudo-Riemannian structure; there are topological restrictions on doing so.
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| A [[Finsler manifold]] is a generalization of a Riemannian manifold, in which the inner product is replaced with a [[vector norm]]; this allows the definition of length, but not angle.
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| ===Symplectic manifolds===
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| {{details|symplectic manifold}}
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| A [[symplectic manifold]] is a manifold equipped with a [[closed form (calculus)|closed]], [[nondegenerate form|nondegenerate]] [[2-form]]. This condition forces symplectic manifolds to be even-dimensional. Cotangent bundles, which arise as phase spaces in [[Hamiltonian mechanics]], are the motivating example, but many compact manifolds also have symplectic structure. All [[orientable]] surfaces [[embedding|embedded]] in [[Euclidean space]] have a [[symplectic structure]], the signed area form on each [[tangent space]] induced by the ambient Euclidean [[inner product]].<ref group="note">This form is clearly nondegenerate, and it must be closed because it is top-dimensional with respect to the [[surface]]; this reflects the [[exceptional isomorphism]] of Lie groups Sp(2, '''R''') ≅ SL(2, '''R''') between the [[symplectic group]] (corresponding to symplectic structure) and the [[special linear group]] (corresponding to orientable structure). Note that a symplectic structure requires an additional integrability condition, beyond this isomorphism of groups: it is not just a [[G-structure]].</ref> Every [[Riemann surface]] is an example of such a surface, and hence a [[symplectic manifold]], when considered as a [[real manifold]].
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| === Lie groups ===
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| {{details|Lie group}}
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| A [[Lie group]] is ''C''<sup>∞</sup> manifold that also carries a [[group (mathematics)|group]] structure whose product and inversion operations are smooth as maps of manifolds. These objects arise naturally in describing symmetries.
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| == Generalizations ==
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| The [[category theory|category]] of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. [[Diffeological space]]s use a different notion of chart known as a "plot". [[Frölicher space]]s and [[orbifold]]s are other attempts.
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| A [[rectifiable set]] generalizes the idea of a piece-wise smooth or [[rectifiable curve]] to higher dimensions; however, rectifiable sets are not in general manifolds.
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| ==See also==
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| *[[Affine connection]]
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| *[[Atlas (topology)]]
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| *[[Christoffel symbols]]
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| *[[Differential geometry]]
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| *[[Introduction to mathematics of general relativity]]
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| *[[List of formulas in Riemannian geometry]]
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| *[[Riemannian geometry]]
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| *[[Space (mathematics)]]
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| {{reflist|2}}
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| ==Bibliography==
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| {{refbegin}}
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| * {{cite journal| author = [[Simon Donaldson|Donaldson, Simon]]| title= An application of gauge theory to four-dimensional topology| journal = Journal of Differential Geometry |volume = 18 |issue = 2 |year = 1983 | pages = 279–315| ref = harv}}
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| * {{cite book | first = Robin | last = Hartshorne | year = 1977 | title = Algebraic Geometry | publisher = Springer-Verlag | isbn = 0-387-90244-9}}
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| * {{springer|title=Differentiable manifold|id=p/d031790}}
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| * {{Cite journal|doi=10.1007/BF02565940|last=Kervaire|first1=Michel A.|title=A manifold which does not admit any differentiable structure|journal=Coment. Math. Helv.|volume=34|issue=1|pages=257–270|year=1960|ref=harv|postscript=<!--None-->}}.
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| * {{cite book |author = Kobayashi, S. |year = 1972 |title = Transformation groups in differential geometry | publisher = Springer}}
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| * {{citation|first=Jeffrey M.|last=Lee|title=Manifolds and Differential Geometry|series=Graduate Studies in Mathematics|volume=Vol. 107 |publisher=American Mathematical Society|publication-place=Providence|year=2009}} .
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| * {{cite book| first = Tullio| last = Levi-Civita | title = The absolute differential calculus (calculus of tensors) | year = 1927}}
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| * {{cite book | authorlink1=S. MacLane | author1 = MacLane, S. | author2 = [[Ieke Moerdijk|Moerdijk]], I. | title = Sheaves in Geometry and Logic | publisher = Springer | year = 1992 | isbn = 0-387-97710-4}}
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| * {{cite journal| title = On Manifolds Homeomorphic to the 7-Sphere |first = John |last = Milnor| journal = [[Annals of Mathematics]]| volume = 64 |year = 1956 |pages = 399–405| ref = harv | doi=10.2307/1969983 | jstor=1969983}}
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| * {{cite book| first = Andrew | last = Ranicki| title = Algebraic and Geometric Surgery | publisher = Oxford Mathematical Monographs, Clarendon Press| year = 2002 | isbn = 0-19-850924-3}}
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| * {{cite book| author = Ricci-Curbastro, Gregorio; Levi-Civita, Tullio | title = Die Methoden des absoluten Differentialkalkuls |year = 1901}}
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| * {{cite journal|first = Gregorio|last = Ricci-Curbastro| title = Delle derivazioni covarianti e controvarianti e del loro uso nella analisi applicata (Italian)| year =1888|ref = harv}}
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| * {{cite journal|first = Bernhard|last=Riemann|title = Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses which lie at the Bases of Geometry) | journal = Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen| volume=13 |year = 1867|ref = harv}} [http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ Available online at Trinity College Dublin]
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| * {{cite journal|first = Zlil| last = Sela | title = The isomorphism problem for hyperbolic groups. I| journal = [[Annals of Mathematics]]| volume = 141 | year = 1995 | pages = 217–283 | doi = 10.2307/2118520|issue = 2|publisher = Annals of Mathematics|ref = harv|jstor = 2118520}}
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| * {{cite book|first=Shlomo|last=Sternberg|title=Lectures on Differential Geometry|year=1964|publisher=Prentice-Hall}}
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| | |
| *{{cite web
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| |url = http://mathworld.wolfram.com/SmoothManifold.html
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| |title = Smooth Manifold
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| |accessdate = 2008-03-04
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| |author =
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| |last = Weisstein
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| |first = Eric W.
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| }}
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| * {{cite book| first = Hermann| last = Weyl| title = Die Idee der Riemannschen Fläche | publisher = Teubner| year = 1955}}
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| * {{cite journal | first = Hassler| last = Whitney| title = Differentiable Manifolds | journal = [[Annals of Mathematics]] | volume =37 | year= 1936| pages = 645–680| doi = 10.2307/1968482 | issue = 3 | publisher = Annals of Mathematics | ref = harv | jstor = 1968482}}
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| {{refend}}
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| {{DEFAULTSORT:Differentiable Manifold}}
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| [[Category:Smooth manifolds| ]]
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| [[ko:미분다양체]]
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| [[he:יריעה חלקה]]
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| [[pl:Rozmaitość różniczkowalna]]
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| [[ru:Дифференцируемое многообразие]]
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| [[uk:Диференційовний многовид]]
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