Cotangent space

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.

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.

Let M be a smooth manifold and let fC(M) be a smooth function. The differential of f at a point x is the map

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:

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.

Just as every differentiable map f : MN between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces

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

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

where θTf(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

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

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.