Editing Cotangent space

{{Use American English|date=March 2019}}{{Short description|Dual space to the tangent space in differential geometry}}
In [[differential geometry]], the '''cotangent space''' is a [[vector space]] associated with a point <math>x</math> on a [[smooth manifold|smooth (or differentiable) manifold]] <math>\mathcal M</math>; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, <math>T^*_x\!\mathcal M</math> is defined as the [[dual space]] of the [[tangent space]] at ''<math>x</math>'', <math>T_x\mathcal M</math>, although there are more direct definitions (see [[#Alternative_definition|below]]). The elements of the cotangent space are called '''cotangent vectors''' or '''tangent covectors'''.

==Properties==
All cotangent spaces at points on a connected manifold have the same [[dimension of a vector space|dimension]], equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the [[cotangent bundle]] of the manifold.

The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore [[isomorphic]] to each other via many possible isomorphisms. The introduction of a [[Riemannian metric]] or a [[symplectic form]] gives rise to a [[natural isomorphism]] between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.

==Formal definitions==

===Definition as linear functionals===

Let <math>\mathcal M</math> be a smooth manifold and let <math>x</math> be a point in <math>\mathcal M</math>. Let <math>T_x\mathcal M</math> be the [[tangent space]] at <math>x</math>. Then the cotangent space at <math>x</math> is defined as the [[dual space]] of {{nowrap|<math>T_x\mathcal M</math>:}}

:<math>T^*_x\!\mathcal M = (T_x \mathcal M)^*</math>

Concretely, elements of the cotangent space are [[linear functional]]s on <math>T_x\mathcal M</math>. That is, every element <math>\alpha\in T^*_x\mathcal M</math> is a [[linear map]]
:<math>\alpha:T_x\mathcal M \to F</math>

where <math>F</math> is the underlying [[field (mathematics)|field]] of the vector space being considered, for example, the field of [[real number]]s. The elements of <math>T^*_x\!\mathcal M</math> are called cotangent vectors.

===Alternative definition===

In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of [[equivalence class]]es of smooth functions on <math>\mathcal M</math>. Informally, we will say that two smooth functions ''f'' and ''g'' are equivalent at a point <math>x</math> if they have the same first-order behavior near <math>x</math>, analogous to their linear Taylor polynomials; two functions ''f'' and ''g'' have the same first order behavior near <math>x</math> if and only if the derivative of the function ''f'' − ''g'' vanishes at <math>x</math>. The cotangent space will then consist of all the possible first-order behaviors of a function near <math>x</math>.

Let <math>\mathcal M</math> be a smooth manifold and let <math>x</math> be a point in <math>\mathcal M</math>. Let <math>I_x</math>be the [[ideal (ring theory)|ideal]] of all functions in <math>C^\infty\! (\mathcal M)</math> vanishing at <math>x</math>, and let <math>I_x^2</math> be the set of functions of the form <math display="inline">\sum_i f_i g_i</math>, where <math>f_i, g_i \in I_x</math>. Then <math>I_x</math> and <math>I_x^2</math> are both real vector spaces and the cotangent space can be defined as the [[Quotient space (linear algebra)|quotient space]] <math>T^*_x\!\mathcal M = I_x/I^2_x</math> by showing that the two spaces are [[isomorphism|isomorphic]] to each other.

This formulation is analogous to the construction of the cotangent space to define the [[Zariski tangent space]] in algebraic geometry. The construction also generalizes to [[locally ringed space]]s.

==The differential of a function==

Let <math>M</math> be a smooth manifold and let <math>f\in C^\infty(M)</math> be a [[smooth function]]. The differential of <math>f</math> at a point <math>x</math> is the map
:<math>\mathrm d f_x(X_x) = X_x(f)</math>
where <math>X_x</math> is a [[Differential geometry of curves|tangent vector]] at <math>x</math>, thought of as a derivation. That is <math>X(f)=\mathcal{L}_Xf</math> is the [[Lie derivative]] of <math>f</math> in the direction <math>X</math>, and one has <math>\mathrm df(X)=X(f)</math>.  Equivalently, we can think of tangent vectors as tangents to curves, and write
:<math>\mathrm d f_x(\gamma'(0))=(f\circ\gamma)'(0)</math>
In either case, <math>\mathrm df_x</math> is a linear map on <math>T_xM</math> and hence it is a tangent covector at <math>x</math>.

We can then define the differential map <math>\mathrm d:C^\infty(M)\to T_x^*(M)</math> at a point <math>x</math> as the map which sends <math>f</math> to <math>\mathrm df_x</math>. Properties of the differential map include:

# <math>\mathrm d</math> is a linear map: <math>\mathrm d(af+bg)=a\mathrm df + b\mathrm dg</math> for constants <math>a</math> and <math>b</math>,
# <math>\mathrm d(fg)_x=f(x)\mathrm dg_x+g(x)\mathrm df_x</math>

The differential map provides the link between the two alternate definitions of the cotangent space given above. Since for all <math> f \in I^2_x </math> there exist <math>g_i, h_i \in I_x</math> such that <math display="inline">f=\sum_i g_i h_i</math>, we have,
 
<math display>
\begin{array}{rcl}
\mathrm d f_x & = & \sum_i \mathrm d (g_i h_i)_x \\
 & = & \sum_i (g_i(x)\mathrm d(h_i)_x+\mathrm d(g_i)_x h_{i}(x)) \\
 & = & \sum_i (0\mathrm d(h_i)_x+\mathrm d(g_i)_x 0) \\
 & = & 0
\end{array} </math> 

So that all function in <math>I^2_x </math> have differential zero, it follows that for every two functions <math>f \in I^2_x</math>, <math>g \in I_x</math>, we have <math>\mathrm d (f+g)=\mathrm d (g)</math>. We can now construct an [[isomorphism]] between <math>T^*_x\!\mathcal M</math> and <math>I_x/I^2_x</math> by sending linear maps <math>\alpha</math> to the corresponding cosets <math>\alpha + I^2_x</math>. Since there is a unique linear map for a given kernel and slope, this is an isomorphism, establishing the equivalence of the two definitions.

==The pullback of a smooth map==

Just as every differentiable map <math>f:M\to N</math> between manifolds induces a linear map (called the ''pushforward'' or ''derivative'') between the tangent spaces
:<math>f_{*}^{}\colon T_x M \to T_{f(x)} N</math>
every such map induces a linear map (called the ''[[pullback (differential geometry)|pullback]]'') between the cotangent spaces, only this time in the reverse direction:
:<math>f^{*}\colon T_{f(x)}^{*} N \to T_{x}^{*} M .</math>
The pullback is naturally defined as the dual (or transpose) of the [[pushforward (differential)|pushforward]]. Unraveling the definition, this means the following:
:<math>(f^{*}\theta)(X_x) = \theta(f_{*}^{}X_x) ,</math>
where <math>\theta\in T_{f(x)}^*N</math> and <math>X_x\in T_xM</math>. Note carefully where everything lives.

If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let <math>g</math> be a smooth function on <math>N</math> vanishing at <math>f(x)</math>. Then the pullback of the covector determined by <math>g</math> (denoted <math>\mathrm d g</math>) is given by
:<math>f^{*}\mathrm dg = \mathrm d(g \circ f).</math>
That is, it is the equivalence class of functions on <math>M</math> vanishing at <math>x</math> determined by <math>g\circ f</math>.

==Exterior powers==

The <math>k</math>-th [[exterior power]] of the cotangent space, denoted <math>\Lambda^k(T_x^*\mathcal{M})</math>, is another important object in differential and algebraic geometry. Vectors in the <math>k</math>-th exterior power, or more precisely sections of the <math>k</math>-th exterior power of the [[cotangent bundle]], are called [[differential form|differential <math>k</math>-forms]]. They can be thought of as alternating, [[multilinear map]]s on <math>k</math> tangent vectors. 
For this reason, tangent covectors are frequently called ''[[one-form]]s''.

== References ==

* {{Citation | last1=Abraham | first1=Ralph H. | author1-link=Ralph Abraham (mathematician) | last2=Marsden | first2=Jerrold E. | author2-link=Jerrold E. Marsden | title=Foundations of mechanics | publisher=Benjamin-Cummings | location=London | isbn=978-0-8053-0102-1 | year=1978}}
* {{Citation | last1=Jost | first1=Jürgen | title=Riemannian Geometry and Geometric Analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=4th | isbn=978-3-540-25907-7 | year=2005}}
* {{Citation | last1=Lee | first1=John M. | title=Introduction to smooth manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Graduate Texts in Mathematics | isbn=978-0-387-95448-6 | year=2003 | volume=218}}
* {{Citation | last1=Misner | first1=Charles W. | author1-link=Charles W. Misner | last2=Thorne | first2=Kip | author2-link=Kip Thorne | last3=Wheeler | first3=John Archibald | author3-link=John Archibald Wheeler | title=[[Gravitation (book)|Gravitation]] | publisher=W. H. Freeman | isbn=978-0-7167-0344-0 | year=1973}}

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[[Category:Differential topology]]
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