Cotangent space

In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors.

All cotangent spaces 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.

Let M be a smooth manifold and let x be a point in M. Let T_xM be the tangent space at x. Then the cotangent space at x is defined as the dual space of T_xM:

Concretely, elements of the cotangent space are linear functionals on T_xM. That is, every element α ∈ T_x^*M is a linear map

where F is the underlying field of the vector space being considered. For example, the field of real numbers. The elements of T_x^*M are called cotangent vectors.

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