Questions tagged [co-tangent-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. 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 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 each real vector spaces of the same dimension and therefore isomorphic to each other. 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.

Here is a direct definition of a cotangent space as a linear functional: Let M be a smooth manifold and let x be a point in M. Let T$_x$M be the tangent space at x. Then the cotangent space at x is the dual space of T$_x$M : T$_x^{\*}$M = (T$_x$M)*. Concretely, elements of the cotangent space are linear functionals on T$_x$M. That is, every element α ∈ T$_x^{\*}$M is a linear map α : T$_x$M → F where F is the underlying field of the vector space being considered. The elements of T$_x^{\*}$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 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, analogous to their linear Taylor polynomials, i.e., two functions f and g have the same first-order behavior near x if and only if the derivative of the function f – g vanishes at x. The cotangent space will then consist of all the possible first-order behaviors of a function near x.

To formulate such a definition, let M be a smooth manifold, and let x be a point in M. Let I$_x$ be the ideal of all functions in C$^∞$(M) vanishing at x, and let I$_x^2$ be the set of functions of the form $\sum_i$ f$_i$g$_i$ where f$_i,$ g$_i$ ∈ I$_x.$ Then I$_x$ and I$_x^2$ are real vector spaces and the cotangent space is defined as the quotient space T$_x^{\*}$M = I$_x$ / I$_x^2.$ That 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.