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| In [[mathematics]], the '''tangent space''' of a [[manifold]] facilitates the generalization of vectors from [[affine space]]s to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.
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| == Informal description ==
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| [[Image:Image Tangent-plane.svg|thumb|A pictorial representation of the tangent space of a single point, ''x'', on a [[sphere]]. A vector in this tangent space can represent a possible velocity at ''x''. After moving in that direction to another nearby point, one's velocity would then be given by a vector in the tangent space of that nearby point—a different tangent space, not shown.]]
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| In [[differential geometry]], one can attach to every point ''x'' of a [[differentiable manifold]] a '''tangent space''', a real [[vector space]] that intuitively contains the possible "directions" at which one can tangentially pass through ''x''. The elements of the tangent space are called '''tangent vectors''' at ''x''. This is a generalization of the notion of a [[bound vector]] in a [[Euclidean space]]. All the tangent spaces of a [[connected space | connected]] manifold have the same [[dimension of a vector space|dimension]], equal to the dimension of the [[manifold]].
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| For example, if the given manifold is a 2-[[sphere]], one can picture the tangent space at a point as the plane which touches the sphere at that point and is [[perpendicular]] to the sphere's radius through the point. More generally, if a given manifold is thought of as an [[embedding|embedded]] [[submanifold]] of [[Euclidean space]] one can picture the tangent space in this literal fashion.{{dubious|date=February 2012}}
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| In [[algebraic geometry]], in contrast, there is an intrinsic definition of '''tangent space at a point P''' of a [[algebraic variety|variety]] ''V'', that gives a vector space of dimension at least that of ''V''. The points P at which the dimension is exactly that of ''V'' are called the '''non-singular''' points; the others are '''singular''' points. For example, a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of ''V'' are those where the 'test to be a manifold' fails. See [[Zariski tangent space]].
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| Once tangent spaces have been introduced, one can define [[vector field]]s, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized [[ordinary differential equation]] on a manifold: a solution to such a differential equation is a differentiable [[curve]] on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.
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| All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension of the original manifold, called the [[tangent bundle]] of the manifold.
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| == Formal definitions ==
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| There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via velocities of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.
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| === Definition as velocities of curves ===
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| Suppose ''M'' is a C<sup>''k''</sup> manifold (''k'' ≥ 1) and ''x'' is a point in ''M''. Pick a [[chart (topology)|chart]] φ : ''U'' → '''R'''<sup>''n''</sup> where ''U'' is an [[open set|open subset]] of ''M'' containing ''x''. Suppose two curves γ<sub>1</sub> : (-1,1) → ''M'' and γ<sub>2</sub> : (-1,1) → ''M'' with γ<sub>1</sub>(0) = γ<sub>2</sub>(0) = ''x'' are given such that φ ∘ γ<sub>1</sub> and φ ∘ γ<sub>2</sub> are both differentiable at 0. Then γ<sub>1</sub> and γ<sub>2</sub> are called ''equivalent at 0'' if the ordinary derivatives of φ ∘ γ<sub>1</sub> and φ ∘ γ<sub>2</sub> at 0 coincide. This defines an [[equivalence relation]] on such curves, and the [[equivalence class]]es are known as the tangent vectors of ''M'' at ''x''. The equivalence class of the curve γ is written as γ'(0). The tangent space of ''M'' at ''x'', denoted by T<sub>''x''</sub>''M'', is defined as the set of all tangent vectors; it does not depend on the choice of chart φ.
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| [[Image:Tangentialvektor.svg|thumb|left|200px|The tangent space <math>\scriptstyle T_xM</math> and a tangent vector <math>\scriptstyle v\in T_xM</math>, along a curve traveling through <math>\scriptstyle x\in M</math>]]
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| To define the vector space operations on T<sub>''x''</sub>''M'', we use a chart φ : ''U'' → '''R'''<sup>''n''</sup> and define the [[Map (mathematics)|map]] (dφ)<sub>''x''</sub> : T<sub>''x''</sub>''M'' → '''R'''<sup>''n''</sup> by (dφ)<sub>''x''</sub>(γ'(0)) = <math>\scriptstyle\frac{d}{dt}</math>(φ ∘ γ)(0). It turns out that this map is [[bijective]] and can thus be used to transfer the vector space operations from '''R'''<sup>''n''</sup> over to T<sub>''x''</sub>''M'', turning the latter into an ''n''-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not.
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| === Definition via derivations ===
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| Suppose ''M'' is a C<sup>∞</sup> manifold. A real-valued function ƒ: ''M'' → '''R''' belongs to C<sup>∞</sup>(''M'') if ƒ ∘ φ<sup>−1</sup> is infinitely differentiable for every chart φ : ''U'' → '''R'''<sup>''n''</sup>. C<sup>∞</sup>(''M'') is a real [[associative algebra]] for the [[pointwise product]] and sum of functions and scalar multiplication.
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| Pick a point ''x'' in ''M''. A ''[[Derivation (abstract algebra)|derivation]]'' at ''x'' is a [[linear map]] ''D'' : C<sup>∞</sup>(''M'') → '''R''' that has the property that for all ƒ, ''g'' in C<sup>∞</sup>(''M''):
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| :<math>D(fg) = D(f)\cdot g(x) + f(x)\cdot D(g)</math>
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| modeled on the [[product rule]] of calculus. These derivations form a real vector space if we define addition and scalar multiplication for derivations by
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| <math>(D_1 + D_2)(f) = D_1(f) + D_2(f)</math> and
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| <math>(\lambda D)(f) = \lambda D(f)</math>
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| This is the tangent space T<sub>''x''</sub>''M''.
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| The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is ''D''(ƒ) = (ƒ ∘ γ)'(0) (where the derivative is taken in the ordinary sense, since ƒ ∘ γ is a function from (-1,1) to '''R''').
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| : <math> \gamma'(0) \longmapsto D_\gamma </math> where <math> D_\gamma(f) = \frac{d}{dt}(f \circ \gamma)(t=0) = (f \circ \gamma)'(0)</math>.
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| Generalizations of this definition are possible, for instance to [[complex manifold]]s and [[algebraic variety|algebraic varieties]]. However, instead of examining derivations ''D'' from the full algebra of functions, one must instead work at the level of [[germ (mathematics)|germs]] of functions. The reason is that the [[structure sheaf]] may not be [[injective sheaf|fine]] for such structures. For instance, let ''X'' be an algebraic variety with [[structure sheaf]] ''O''<sub>X</sub>. Then the [[Zariski tangent space]] at a point ''p''∈''X'' is the collection of ''K''-derivations ''D'':''O''<sub>X,p</sub>→''K'', where ''K'' is the [[ground field]] and ''O''<sub>X,p</sub> is the stalk of ''O''<sub>X</sub> at ''p''.
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| === Definition via the cotangent space ===
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| Again we start with a C<sup>∞</sup> manifold, ''M'', and a point, ''x'', in ''M''. Consider the [[ideal (ring theory)|ideal]], ''I'', in C<sup>∞</sup>(''M'') consisting of all functions, ƒ, such that ƒ(''x'') = 0. That is, of functions defining curves, surfaces, etc. passing through ''x''. Then ''I'' and ''I''<sup> 2</sup> are real vector spaces, and T<sub>''x''</sub>''M'' may be defined as the [[dual space]] of the [[quotient space (linear algebra)|quotient space]] ''I'' / ''I''<sup> 2</sup>. This latter quotient space is also known as the [[cotangent space]] of ''M'' at ''x''.
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| While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the [[algebraic variety|varieties]] considered in [[algebraic geometry]].
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| If ''D'' is a derivation at ''x'', then ''D''(ƒ) = 0 for every ƒ in ''I'' <sup>2</sup>, and this means that ''D'' gives rise to a linear map ''I'' / ''I'' <sup>2</sup> → '''R'''. Conversely, if ''r'' : ''I'' / ''I'' <sup>2</sup> → '''R''' is a linear map, then ''D''(ƒ) = ''r''((ƒ - ƒ(''x'')) + ''I''<sup> 2</sup>) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space. | |
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| == Properties ==
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| If ''M'' is an open subset of '''R'''<sup>''n''</sup>, then ''M'' is a C<sup>∞</sup> manifold in a natural manner (take the charts to be the [[Identity function|identity maps]]), and the tangent spaces are all naturally identified with '''R'''<sup>''n''</sup>.
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| === Tangent vectors as directional derivatives ===
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| Another way to think about tangent vectors is as [[directional derivative]]s. Given a vector ''v'' in '''R'''<sup>''n''</sup> one defines the directional derivative of a smooth map ƒ: '''R'''<sup>''n''</sup>→'''R''' at a point ''x'' by
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| :<math> D_v f(x) = \frac{d}{dt}f(x+tv)\big|_{t=0}=\sum_{i=1}^{n}v^i\frac{\partial f}{\partial x^i}(x).</math>
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| This map is naturally a derivation. Moreover, it turns out that every derivation of C<sup>∞</sup>('''R'''<sup>''n''</sup>) is of this form. So there is a one-to-one map between vectors (thought of as tangent vectors at a point) and derivations.
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| Since tangent vectors to a general manifold can be defined as derivations it is natural to think of them as directional derivatives. Specifically, if ''v'' is a tangent vector of ''M'' at a point ''x'' (thought of as a derivation) then define the directional derivative in the direction ''v'' by
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| :<math> D_v(f) = v(f)\,</math> | |
| where ƒ: ''M'' → '''R''' is an element of C<sup>∞</sup>(''M'').
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| If we think of ''v'' as the direction of a curve, ''v'' = γ'(0), then we write
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| :<math> D_v(f) = (f\circ\gamma)'(0).</math>
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| === The derivative of a map ===
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| {{main|Pushforward (differential)}}
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| Every smooth (or differentiable) map ''φ'' : ''M'' → ''N'' between smooth (or differentiable) manifolds induces natural [[linear map]]s between the corresponding tangent spaces:
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| :<math> \mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N.</math>
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| If the tangent space is defined via curves, the map is defined as
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| :<math> \mathrm d\varphi_x(\gamma'(0)) = (\varphi\circ\gamma)'(0).</math>
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| If instead the tangent space is defined via derivations, then
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| :<math> \mathrm d\varphi_x(X)(f) = X(f\circ \varphi).</math>
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| The linear map d''φ''<sub>''x''</sub> is called variously the ''derivative'', ''total derivative'', ''differential'', or ''pushforward'' of ''φ'' at ''x''. It is frequently expressed using a variety of other notations:
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| :<math> D\varphi_x,\quad (\varphi_*)_x,\quad \varphi'(x).</math>
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| In a sense, the derivative is the best linear approximation to ''φ'' near ''x''. Note that when ''N'' = '''R''', the map d''φ''<sub>''x''</sub> : T<sub>''x''</sub>''M''→'''R''' coincides with the usual notion of the [[Differential (calculus)|differential]] of the function ''φ''. In [[local coordinates]] the derivative of ƒ is given by the [[Jacobian matrix and determinant|Jacobian]].
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| An important result regarding the derivative map is the following:
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| :'''Theorem'''. If ''φ'' : ''M'' → ''N'' is a [[local diffeomorphism]] at ''x'' in ''M'' then d''φ''<sub>''x''</sub> : T<sub>''x''</sub>''M'' → T<sub>''φ''(''x'')</sub>''N'' is a linear [[isomorphism]]. Conversely, if d''φ''<sub>''x''</sub> is an isomorphism then there is an [[open set|open neighborhood]] ''U'' of ''x'' such that ''φ'' maps ''U'' diffeomorphically onto its image.
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| This is a generalization of the [[inverse function theorem]] to maps between manifolds.
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| == See also ==
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| * [[Exponential map]]
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| * [[Differential geometry of curves]]
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| * [[Cotangent space]]
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| == References ==
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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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| * {{citation|first=Peter W.|last=Michor|title=Topics in Differential Geometry|series=Graduate Studies in Mathematics|volume=Vol. 93|publisher=American Mathematical Society|publication-place=Providence|year=2008}} .
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| * {{Citation | last1=Spivak | first1=Michael | author1-link=Michael Spivak | title=Calculus on Manifolds | publisher=[[HarperCollins]] | isbn=978-0-8053-9021-6 | year=1965}}
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| == External links ==
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| * [http://mathworld.wolfram.com/TangentPlane.html Tangent Planes] at MathWorld
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| {{DEFAULTSORT:Tangent Space}}
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| [[Category:Differential topology]]
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| [[Category:Differential geometry]]
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