Cotangent space

In differential geometry, the cotangent space is a vector space associated with a point ${\displaystyle x}$ on a smooth (or differentiable) manifold ${\displaystyle {\mathcal {M}}}$; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, ${\displaystyle T_{x}^{*}\!{\mathcal {M}}}$ is defined as the dual space of the tangent space at ${\displaystyle x}$, ${\displaystyle T_{x}{\mathcal {M}}}$, although there are more direct definitions (see below). 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 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 ${\displaystyle {\mathcal {M}}}$ be a smooth manifold and let ${\displaystyle x}$ be a point in ${\displaystyle {\mathcal {M}}}$. Let ${\displaystyle T_{x}{\mathcal {M}}}$ be the tangent space at ${\displaystyle x}$. Then the cotangent space at ${\displaystyle x}$ is defined as the dual space of ${\displaystyle T_{x}{\mathcal {M}}}$:

Concretely, elements of the cotangent space are linear functionals on ${\displaystyle T_{x}{\mathcal {M}}}$. That is, every element ${\displaystyle \alpha \in T_{x}^{*}{\mathcal {M}}}$ is a linear map

where ${\displaystyle F}$ is the underlying field of the vector space being considered, for example, the field of real numbers. The elements of ${\displaystyle T_{x}^{*}\!{\mathcal {M}}}$ 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 ${\displaystyle {\mathcal {M}}}$. Informally, we will say that two smooth functions f and g are equivalent at a point ${\displaystyle x}$ if they have the same first-order behavior near ${\displaystyle x}$, analogous to their linear Taylor polynomials; two functions f and g have the same first order behavior near ${\displaystyle x}$ if and only if the derivative of the function f − g vanishes at ${\displaystyle x}$. The cotangent space will then consist of all the possible first-order behaviors of a function near ${\displaystyle x}$.

Let ${\displaystyle {\mathcal {M}}}$ be a smooth manifold and let ${\displaystyle x}$ be a point in ${\displaystyle {\mathcal {M}}}$. Let ${\displaystyle I_{x}}$be the ideal of all functions in ${\displaystyle C^{\infty }\!({\mathcal {M}})}$ vanishing at ${\displaystyle x}$, and let ${\displaystyle I_{x}^{2}}$ be the set of functions of the form ${\textstyle \sum _{i}f_{i}g_{i}}$, where ${\displaystyle f_{i},g_{i}\in I_{x}}$. Then ${\displaystyle I_{x}}$ and ${\displaystyle I_{x}^{2}}$ are both real vector spaces and the cotangent space can be defined as the quotient space ${\displaystyle T_{x}^{*}\!{\mathcal {M}}=I_{x}/I_{x}^{2}}$ by showing that the two spaces are isomorphic to each other.