Cotangent space - Citizendia

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. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below). In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since The elements of the cotangent space are called cotangent vectors or tangent covectors.

1 Properties
2 Formal definitions
- 2.1 Definition as linear functionals
- 2.2 Alternative definition
3 The differential of a function
4 The pullback of a smooth map
5 Exterior powers
6 References

Properties

All cotangent spaces on a manifold have the same dimension, equal to the dimension of the manifold. In Mathematics, the dimension of a Vector space V is the cardinality (i All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold. In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces

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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal

Formal definitions

Definition as linear functionals

Let M be a smooth manifold and let x be a point in M. Let T_xM be the tangent space at x. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since Then the cotangent space at x is defined as the dual space of T_xM:

T_x^*M = (T_xM)^*

Concretely, elements of the cotangent space are linear functionals on T_xM. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional That is, every element α ∈ T_x^*M is a linear map

α : T_xM→R

The elements of T_x^*M are called cotangent vectors. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

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. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X 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.

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_x² be the set of functions of the form $\sum_i f_i g_i\,$ , where f_i, g_i ∈ I_x. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Then I_x and I_x² are real vector spaces and the cotangent space is defined as the quotient space T_x^*M = I_x / I_x². In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N

The differential of a function

Let M be a smooth manifold and let f ∈ C^∞(M) be a smooth function. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability The differential of f at a point x is the map

df_x(X_x) = X_x(f)

where X_x is a tangent vector at x, thought of as a derivation. That is $X(f)=\mathcal{L}_Xf$ is the Lie derivative of f in the direction X, and one has df(X)=X(f). In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the Equivalently, we can think of tangent vectors as tangents to curves, and write

df_x(γ′(0)) = (f o γ)′(0)

In either case, df_x is a linear map on T_xM and hence it is a tangent covector at x.

We can then define the differential map d : C^∞(M) → T_x^*M at a point x as the map which sends f to df_x. Properties of the differential map include:

d is a linear map: d(af + bg) = a df + b dg for constants a and b,
d(fg)_x = f(x)dg_x + g(x)df_x,

The differential map provides the link between the two alternate definitions of the cotangent bundle given above. Given a function f ∈ I_x (a smooth function vanishing at x) we can form the linear functional df_x as above. Since the map d restricts to 0 on I_x² (the reader should verify this), d descends to a map from I_x / I_x² to the dual of the tangent space, (T_xM)^*. One can show that this map is an isomorphism, establishing the equivalence of the two definitions.

The pullback of a smooth map

Just as every differentiable map f : M → N between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces

$f_{*}^{}\colon T_x M \to T_{f(x)} N$

every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction:

$f^{*}\colon T_{f(x)}^{*} N \to T_{x}^{*} M$

The pullback is naturally defined as the dual (or transpose) of the pushforward. Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some Unraveling the definition, this means the following:

$(f^{*}\theta)(X_x) = \theta(f_{*}^{}X_x)$

where θ ∈ T_f(x)^*N and X_x ∈ T_xM. Note carefully where everything lives.

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. Let g be a smooth function on N vanishing at f(x). Then the pullback of the covector determined by g (denoted dg) is given by

$f^{*}\mathrm dg = \mathrm d(g \circ f)$

That is, it is the equivalence class of functions on M vanishing at x determined by g o f.

Exterior powers

The k^th exterior power of the cotangent space, denoted Λ^k(T_x^*M), is another important object in differential geometry. Vectors in the k^th exterior power are called differential k-forms. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is They can be thought of as alternating, multilinear maps on k tangent vectors. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable For this reason, tangent covectors are frequently called one-forms. In Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space

References

John M. Lee, Introduction to Smooth Manifolds, (2003) Springer Graduate Texts in Mathematics 218.
Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-63654-4.
Ralph Abraham and Jerrold E. Ralph H Abraham (b July 4 1936, Burlington Vermont) is an American Mathematician. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X.
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W. H. Freeman, New York; ISBN 0-7167-0344-0.

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