Cotangent space

In differential geometry , a branch of mathematics , the cotangential space is a vector space that is assigned to a point of a differentiable manifold . It is the dual space of the corresponding tangent space . ${\displaystyle M}$

definition

Let be a differentiable manifold and its tangent space at the point . Then the cotangent space is defined as the dual space of . That means, the cotangent space consists of all linear forms on the tangent space . ${\displaystyle M}$ ${\displaystyle T_{p}M}$ ${\displaystyle p\in M}$ ${\displaystyle T_{p}M}$ ${\displaystyle T_{p}M}$

Alternative definition

In the following another approach is shown in which the dual space is defined directly, without reference to the tangent space.

Let it be a -dimensional differentiable manifold. Further the set of all smooth curves is through ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle \Gamma _{p}}$ ${\displaystyle p\in M}$

${\displaystyle c\colon (-\epsilon ,\epsilon )\to M\,,\qquad c(0)=p}$

and the set of all smooth functions that are defined in a neighborhood of : ${\displaystyle C_{p}^{\infty }}$ ${\displaystyle U_{p}}$ ${\displaystyle p}$

${\displaystyle f\colon U_{p}\to \mathbb {R} }$.

One denotes the following equivalence relation ${\displaystyle \sim _{p}}$ ${\displaystyle C_{p}^{\infty }}$

${\displaystyle f\sim _{p}g\qquad :\Leftrightarrow \qquad \exists U_{p}}$Environment of with , ${\displaystyle p}$ ${\displaystyle f|U_{p}=g|U_{p}}$

then the factor space is the vector space of the seeds over . over ${\displaystyle {\mathcal {F}}_{p}:=C_{p}^{\infty }/\sim _{p}}$ ${\displaystyle p}$

${\displaystyle \langle [f]_{p},c\rangle :={\frac {\operatorname {d} }{\operatorname {d} t}}{\Big |}_{t=0}f\circ c(t)}$

a formal pairing is then defined which is linear in the first component. Now is ${\displaystyle \langle \cdot ,\cdot \rangle :{\mathcal {F}}_{p}\times \Gamma _{p}\to \mathbb {R} }$

${\displaystyle {\mathcal {N}}_{p}:=\{[n]_{p}\in {\mathcal {F}}_{p}|\forall c\in \Gamma _{p}:\langle [n]_{p},c\rangle =0\}}$

a linear subspace of , more precisely the null space with respect to and ${\displaystyle {\mathcal {F}}_{p}}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

${\displaystyle T_{p}^{*}M:={\mathcal {F}}_{p}/{\mathcal {N}}_{p}}$

is the -dimensional cotangent space in the point . One also writes for the cotangent vector . ${\displaystyle n}$ ${\displaystyle p\in M}$ ${\displaystyle [[f]_{p}]}$ ${\displaystyle df_{p}}$

Relation to the tangential space

With the above definition one can define an equivalence relation as follows: ${\displaystyle \Gamma _{p}}$ ${\displaystyle \sim }$

${\displaystyle \gamma _{1}\sim \gamma _{2}\qquad \Leftrightarrow \qquad \forall df_{p}\in T_{p}^{*}M:\langle df_{p},\gamma _{1}\rangle =\langle df_{p},\gamma _{2}\rangle }$

The factor space just describes the -dimensional tangent space . ${\displaystyle T_{p}M:=\Gamma _{p}/\sim }$ ${\displaystyle n}$

If now form a basis of , one can choose a representative for each basis vector . is a differentiable map and for each one can have a curve ${\displaystyle dx_{1},\ldots ,dx_{n}}$ ${\displaystyle T_{p}^{*}M}$ ${\displaystyle x_{i}\in C_{p}^{\infty }}$ ${\displaystyle x=(x_{1},\ldots ,x_{n})\colon M\to \mathbb {R} ^{n}}$ ${\displaystyle i=1,\ldots ,n}$

${\displaystyle {\begin{matrix}\gamma _{i}\colon &(-\epsilon ;\epsilon )&\to &M\\&t&\mapsto &x^{-1}(t\cdot e_{i})\end{matrix}}}$

define, where the -th unit vector is im . Because of ${\displaystyle e_{i}}$ ${\displaystyle i}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \langle dx_{i},[\gamma _{j}]\rangle =\delta _{ij}}$

are and dual to each other and you write for too . ${\displaystyle T_{p}M}$ ${\displaystyle T_{p}^{*}M}$ ${\displaystyle [\gamma _{i}]={dx_{i}}^{*}}$ ${\displaystyle {\tfrac {\partial }{\partial x_{i}}}}$

Justification of the spellings

Be , , any function and for the curves , where the canonical basis vectors. Then in the above notations: ${\displaystyle M=\mathbb {R} ^{n}}$ ${\displaystyle p\in \mathbb {R} ^{n}}$ ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle \gamma _{i}\colon t\mapsto p+t\cdot e_{i}}$ ${\displaystyle e_{i}}$

${\displaystyle \langle [f]_{p},[\gamma _{i}]\rangle ={\frac {\operatorname {d} }{\operatorname {d} t}}{\Big |}_{t=0}f\circ \gamma _{i}=\lim _{h\to 0}{\frac {f(p+h\cdot e_{i})-f(p)}{h}}={\frac {\partial }{\partial x_{i}}}f(p)}$

Thus the spelling is justified. ${\displaystyle [\gamma _{i}]={\tfrac {\partial }{\partial x_{i}}}}$

The linear mapping is also the total differential . So the spelling is also justified. ${\displaystyle T_{p}M=\mathbb {R} ^{n}}$ ${\displaystyle \langle [[f]_{p}],\cdot \rangle \colon T_{p}M\to \mathbb {R} }$ ${\displaystyle df(p)}$ ${\displaystyle [[f]_{p}]=df_{p}}$

literature

• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .
• R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 .