Cotangent space

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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.

Contents

Properties

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.

Formal definitions

Definition as linear functionals

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

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

Where F is the underlying field of the vector space being considered. In most cases, this is the field of Real Numbers. The elements of Tx*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 M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x. The cotangent space will then consist of all the possible first-order behaviors of a function near x.

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