# Cotangent space

In differential geometry, one can attach to every point ${\displaystyle x}$ of a smooth (or differentiable) manifold, ${\displaystyle {\mathcal {M}}}$, a vector space called the cotangent space at ${\displaystyle x}$. Typically, the cotangent space, ${\displaystyle T_{x}^{*}\!{\mathcal {M}}}$ is defined as the dual space of the tangent space at ${\displaystyle x}$, ${\displaystyle T_{x}{\mathcal {M}}}$, 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, 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 ${\displaystyle {\mathcal {M}}}$  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 x is defined as the dual space of ${\displaystyle T_{x}{\mathcal {M}}}$ :

${\displaystyle T_{x}^{*}\!{\mathcal {M}}=(T_{x}{\mathcal {M}})^{*}}$

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

${\displaystyle \alpha :T_{x}{\mathcal {M}}\rightarrow F}$

where ${\displaystyle F}$  is the underlying field of the vector space being considered, for example, the field of real numbers. The elements of ${\displaystyle T_{x}^{*}\!{\mathcal {M}}}$  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 M be a smooth manifold and let x be a point in ${\displaystyle {\mathcal {M}}}$ . Let ${\displaystyle I_{x}}$ be the ideal of all functions in ${\displaystyle C^{\infty }\!({\mathcal {M}})}$  vanishing at ${\displaystyle x}$ , and let ${\displaystyle I_{x}^{2}}$  be the set of functions of the form ${\displaystyle \sum _{i}f_{i}g_{i}\,}$ , where ${\displaystyle f_{i},g_{i}\in I_{x}}$ . Then ${\displaystyle I_{x}}$  and ${\displaystyle I_{x}^{2}}$  are real vector spaces and the cotangent space is defined as the quotient space ${\displaystyle T_{x}^{*}\!{\mathcal {M}}=I_{x}/I_{x}^{2}}$ .

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 f ∈ C(M) be a smooth function. The differential of f at a point x is the map

dfx(Xx) = Xx(f)

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

dfx(γ′(0)) = (f o γ)′(0)

In either case, dfx is a linear map on TxM and hence it is a tangent covector at x.

We can then define the differential map d : C(M) → Tx*M at a point x as the map which sends f to dfx. Properties of the differential map include:

1. d is a linear map: d(af + bg) = a df + b dg for constants a and b,
2. d(fg)x = f(x)dgx + g(x)dfx,

The differential map provides the link between the two alternate definitions of the cotangent space given above. Given a function fIx (a smooth function vanishing at x) we can form the linear functional dfx as above. Since the map d restricts to 0 on Ix2 (the reader should verify this), d descends to a map from Ix / Ix2 to the dual of the tangent space, (TxM)*. 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 f : MN between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces

${\displaystyle f_{*}^{}\colon T_{x}M\to T_{f(x)}N}$

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

${\displaystyle f^{*}\colon T_{f(x)}^{*}N\to T_{x}^{*}M}$

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

${\displaystyle (f^{*}\theta )(X_{x})=\theta (f_{*}^{}X_{x})}$

where θ ∈ Tf(x)*N and XxTxM. 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 dg) is given by

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

That is, it is the equivalence class of functions on M vanishing at x determined by g o f.

## Exterior powers

The k-th exterior power of the cotangent space, denoted Λk(Tx*M), is another important object in differential geometry. Vectors in the kth exterior power, or more precisely sections of the k-th exterior power of the cotangent bundle, are called 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

• Abraham, Ralph H.; Marsden, Jerrold E. (1978), Foundations of mechanics, London: Benjamin-Cummings, ISBN 978-0-8053-0102-1
• Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-25907-7
• Lee, John M. (2003), Introduction to smooth manifolds, Springer Graduate Texts in Mathematics, 218, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95448-6
• Misner, Charles W.; Thorne, Kip; Wheeler, John Archibald (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0