In differential geometry, the cotangent space is a vector space on a smooth (or differentiable) manifold is defined as the dual space of the tangent space at , 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

be the tangent space is defined as the dual space of

Concretely, elements of the cotangent space are linear functionals is a linear map

is the underlying field of the vector space being considered, for example, the field of real numbers 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. Informally, we will say that two smooth functions f and g, analogous to their linear Taylor polynomials; two functions f and g if and only if the derivative of the function f − g.

be the ideal are both real vector spaces and the cotangent space can be defined as the quotient space 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

be a smooth function is the map

is a tangent vector is the Lie derivative. Equivalently, we can think of tangent vectors as tangents to curves, and write

.

. Properties of the differential map include:

1. ,

, we have,

. We can now construct an isomorphism. 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

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:

. Note carefully where everything lives.

) is given by

.

Exterior powers

-th exterior power-th exterior power of the cotangent bundle, are called k-forms. They can be thought of as alternating, multilinear maps tangent vectors. For this reason, tangent covectors are frequently called one-forms.

References