In differential geometry, the cotangent space is a vector space associated with a point
on a smooth (or differentiable) manifold
; one can define a cotangent space for every point on a smooth manifold.
The elements of the cotangent space are called cotangent vectors or tangent covectors.
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 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.
: Concretely, elements of the cotangent space are linear functionals on
is the underlying field of the vector space being considered, for example, the field of real numbers.
are called cotangent vectors.
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
Informally, we will say that two smooth functions f and g are equivalent at a point
, analogous to their linear Taylor polynomials; two functions f and g have the same first order behavior near
The cotangent space will then consist of all the possible first-order behaviors of a function near
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.
Properties of the differential map include: The differential map provides the link between the two alternate definitions of the cotangent space given above.
Since there is a unique linear map for a given kernel and slope, this is an isomorphism, establishing the equivalence of the two definitions.
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
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.
Then the pullback of the covector determined by
) is given by That is, it is the equivalence class of functions on
-th exterior power of the cotangent space, denoted
, is another important object in differential and algebraic geometry.
-th exterior power, or more precisely sections of the
-th exterior power of the cotangent bundle, are called differential
They can be thought of as alternating, multilinear maps on
For this reason, tangent covectors are frequently called one-forms.