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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, one can attach to every point x of a smooth (or differentiable) manifold, \mathcal M, a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
called the cotangent space at ''x''. Typically, the cotangent space, T^*_x\!\mathcal M is defined as the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of the
tangent space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
at ''x'', T_x\mathcal M, although there are more direct definitions (see 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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, 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 In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
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 \mathcal M be a smooth manifold and let x be a point in \mathcal M. Let T_x\mathcal M be the
tangent space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
at x. Then the cotangent space at ''x'' is defined as the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of :T^*_x\!\mathcal M = (T_x \mathcal M)^* Concretely, elements of the cotangent space are linear functionals on T_x\mathcal M. That is, every element \alpha\in T^*_x\mathcal M is a linear map :\alpha:T_x\mathcal M \to F where F is the underlying field (mathematics), field of the vector space being considered, for example, the field of real numbers. The elements of T^*_x\!\mathcal M 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 classes of smooth functions on \mathcal M. Informally, we will say that two smooth functions ''f'' and ''g'' are equivalent at a point x if they have the same first-order behavior near x, analogous to their linear Taylor polynomials; two functions ''f'' and ''g'' have the same first order behavior near x if and only if the derivative of the function ''f'' − ''g'' vanishes at x. The cotangent space will then consist of all the possible first-order behaviors of a function near x. Let ''M'' be a smooth manifold and let ''x'' be a point in \mathcal M. Let I_xbe the ideal (ring theory), ideal of all functions in C^\infty\! (\mathcal M) vanishing at x, and let I_x^2 be the set of functions of the form \sum_i f_i g_i, where f_i, g_i \in I_x. Then I_x and I_x^2 are both real vector spaces and the cotangent space can be defined as the Quotient space (linear algebra), quotient space T^*_x\!\mathcal M = I_x/I^2_x 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 spaces.


The differential of a function

Let ''M'' be a smooth manifold and let be a smooth function. The differential of ''f'' at a point ''x'' is the map :d''f''''x''(''X''''x'') = ''X''''x''(''f'') where ''X''''x'' is a Differential geometry of curves, tangent vector at ''x'', thought of as a derivation. That is X(f)=\mathcal_Xf is the Lie derivative of ''f'' in the direction ''X'', and one has . Equivalently, we can think of tangent vectors as tangents to curves, and write :d''f''''x''(''γ''(0)) = (''f'' ∘ ''γ'')′(0) In either case, d''f''''x'' is a linear map on ''T''''x''''M'' and hence it is a tangent covector at ''x''. We can then define the differential map at a point ''x'' as the map which sends ''f'' to d''f''''x''. Properties of the differential map include: # d is a linear map: d(''af'' + ''bg'') = ''a'' d''f'' + ''b'' d''g'' for constants ''a'' and ''b'', # d(''fg'')''x'' = ''f''(''x'') d''g''''x'' + ''g''(''x'') d''f''''x'', The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function (a smooth function vanishing at ''x'') we can form the linear functional d''f''''x'' as above. Since the map d restricts to 0 on ''I''''x''2 (the reader should verify this), d descends to a map from to the dual of the tangent space, (''T''''x''''M'')*. One can show that this map is an isomorphism, establishing the equivalence of the two definitions.


The pullback of a smooth map

Just as every differentiable map between manifolds induces a linear map (called the ''pushforward'' or ''derivative'') between the tangent spaces :f_^\colon T_x M \to T_ N every such map induces a linear map (called the ''pullback (differential geometry), pullback'') between the cotangent spaces, only this time in the reverse direction: :f^\colon T_^ N \to T_^ M . The pullback is naturally defined as the dual (or transpose) of the pushforward (differential), pushforward. Unraveling the definition, this means the following: :(f^\theta)(X_x) = \theta(f_^X_x) , where and . 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 ''g'' be a smooth function on ''N'' vanishing at ''f''(''x''). Then the pullback of the covector determined by ''g'' (denoted d''g'') is given by :f^\mathrm dg = \mathrm d(g \circ f). That is, it is the equivalence class of functions on ''M'' vanishing at ''x'' determined by .


Exterior powers

The ''k''-th exterior power of the cotangent space, denoted Λ''k''(''T''''x''*''M''), is another important object in differential geometry. Vectors in the ''k''th exterior power, or more precisely sections of the ''k''-th exterior power of the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
, are called differential form, differential ''k''-forms. They can be thought of as alternating, multilinear maps on ''k'' tangent vectors. For this reason, tangent covectors are frequently called ''one-forms''.


References

* * * * {{DEFAULTSORT:Cotangent Space Differential topology Tensors