Cotangent space

Formal definitions

Definition as linear functionals

Let ${\displaystyle \mathcal{M}}$ be a smooth manifold and let ${\displaystyle x}$ be a point in ${\displaystyle \mathcal{M}}$. Let ${\displaystyle T_x\mathcal{M}}$ be the tangent space at ${\displaystyle x}$. Then the cotangent space at ${\displaystyle x}$ is defined as the dual space of ${\displaystyle T_x\mathcal{M}}$:

${\displaystyle T_x^{\ast }\mathcal{M}=(T_x\mathcal{M}{)}^{\ast }}$

Concretely, elements of the cotangent space are linear functionals on ${\displaystyle T_x\mathcal{M}}$. That is, every element ${\displaystyle \alpha \in T_x^{\ast }\mathcal{M}}$ is a linear map

${\displaystyle \alpha :T_x\mathcal{M}\to F}$

where ${\displaystyle F}$ is the underlying field of the vector space being considered, for example, the field of real numbers. The elements of ${\displaystyle T_x^{\ast }\mathcal{M}}$ 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 ${\displaystyle \mathcal{M}}$. Informally, we will say that two smooth functions f and g are equivalent at a point ${\displaystyle x}$ if they have the same first-order behavior near ${\displaystyle x}$, analogous to their linear Taylor polynomials; two functions f and g have the same first order behavior near ${\displaystyle x}$ if and only if the derivative of the function f − g vanishes at ${\displaystyle x}$. The cotangent space will then consist of all the possible first-order behaviors of a function near ${\displaystyle x}$.

Let ${\displaystyle \mathcal{M}}$ be a smooth manifold and let ${\displaystyle x}$ be a point in ${\displaystyle \mathcal{M}}$. Let ${\displaystyle I_x}$be the ideal of all functions in ${\displaystyle C^{\mathrm{\infty }}(\mathcal{M})}$ vanishing at ${\displaystyle x}$, and let ${\displaystyle I_x^2}$ be the set of functions of the form $\sum _i{f}_i{g}_i$, where ${\displaystyle f_i,g_i\in I_x}$. Then ${\displaystyle I_x}$ and ${\displaystyle I_x^2}$ are both real vector spaces and the cotangent space can be defined as the quotient space ${\displaystyle T_x^{\ast }\mathcal{M}=I_x/I_x^2}$ by showing that the two spaces are 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 ${\displaystyle M}$ be a smooth manifold and let ${\displaystyle f\in C^{\mathrm{\infty }}(M)}$ be a smooth function. The differential of ${\displaystyle f}$ at a point ${\displaystyle x}$ is the map

${\displaystyle \mathrm{d}{f}_x(X_x)=X_x(f)}$

where ${\displaystyle X_x}$ is a tangent vector at ${\displaystyle x}$, thought of as a derivation. That is ${\displaystyle X(f)={\mathcal{L}}_{X}f}$ is the Lie derivative of ${\displaystyle f}$ in the direction ${\displaystyle X}$, and one has ${\displaystyle \mathrm{d}f(X)=X(f)}$. Equivalently, we can think of tangent vectors as tangents to curves, and write

${\displaystyle \mathrm{d}{f}_x({\gamma }^{\prime }(0))=(f\circ \gamma {)}^{\prime }(0)}$

In either case, ${\displaystyle \mathrm{d}{f}_x}$ is a linear map on ${\displaystyle T_{x}M}$ and hence it is a tangent covector at ${\displaystyle x}$.

We can then define the differential map ${\displaystyle \mathrm{d}:C^{\mathrm{\infty }}(M)\to T_x^{\ast }(M)}$ at a point ${\displaystyle x}$ as the map which sends ${\displaystyle f}$ to ${\displaystyle \mathrm{d}{f}_x}$. Properties of the differential map include:

1. ${\displaystyle \mathrm{d}}$ is a linear map: ${\displaystyle \mathrm{d}(af+bg)=a\mathrm{d}f+b\mathrm{d}g}$ for constants ${\displaystyle a}$ and ${\displaystyle b}$,
2. ${\displaystyle \mathrm{d}(fg{)}_x=f(x)\mathrm{d}{g}_x+g(x)\mathrm{d}{f}_x}$

The differential map provides the link between the two alternate definitions of the cotangent space given above. Since for all ${\displaystyle f\in I_x^2}$ there exist ${\displaystyle g_i,h_i\in I_x}$ such that $f=\sum _i{g}_i{h}_i$, we have,

$\begin{array}{rcl}\mathrm{d}{f}_x& =& \sum _i\mathrm{d}(g_i{h}_i{)}_x\\ & =& \sum _i(g_i(x)\mathrm{d}(h_i{)}_x+\mathrm{d}(g_i{)}_x{h}_i(x))\\ & =& \sum _i(0\mathrm{d}(h_i{)}_x+\mathrm{d}(g_i{)}_{x}0)\\ & =& 0\end{array}$

So that all function in ${\displaystyle I_x^2}$ have differential zero, it follows that for every two functions ${\displaystyle f\in I_x^2}$, ${\displaystyle g\in I_x}$, we have ${\displaystyle \mathrm{d}(f+g)=\mathrm{d}(g)}$. We can now construct an isomorphism between ${\displaystyle T_x^{\ast }\mathcal{M}}$ and ${\displaystyle I_x/I_x^2}$ by sending linear maps ${\displaystyle \alpha }$ to the corresponding cosets ${\displaystyle \alpha +I_x^2}$. 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 ${\displaystyle f:M\to N}$ between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces

${\displaystyle f_{\ast }^{}:T_{x}M\to T_{f(x)}N}$

every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction:

${\displaystyle f^{\ast }:T_{f(x)}^{\ast }N\to T_x^{\ast }M.}$

The pullback is naturally defined as the dual (or transpose) of the pushforward. Unraveling the definition, this means the following:

${\displaystyle (f^{\ast }\theta )(X_x)=\theta (f_{\ast }^{}{X}_x),}$

where ${\displaystyle \theta \in T_{f(x)}^{\ast }N}$ and ${\displaystyle X_x\in T_{x}M}$. 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 ${\displaystyle g}$ be a smooth function on ${\displaystyle N}$ vanishing at ${\displaystyle f(x)}$. Then the pullback of the covector determined by ${\displaystyle g}$ (denoted ${\displaystyle \mathrm{d}g}$) is given by

${\displaystyle f^{\ast }\mathrm{d}g=\mathrm{d}(g\circ f).}$

That is, it is the equivalence class of functions on ${\displaystyle M}$ vanishing at ${\displaystyle x}$ determined by ${\displaystyle g\circ f}$.

Exterior powers

The ${\displaystyle k}$-th exterior power of the cotangent space, denoted ${\displaystyle {\mathrm{\Lambda }}^k(T_x^{\ast }\mathcal{M})}$, is another important object in differential and algebraic geometry. Vectors in the ${\displaystyle k}$-th exterior power, or more precisely sections of the ${\displaystyle k}$-th exterior power of the cotangent bundle, are called differential ${\displaystyle k}$-forms. They can be thought of as alternating, multilinear maps on ${\displaystyle k}$ tangent vectors. For this reason, tangent covectors are frequently called one-forms.