In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.

Informal description

A pictorial representation of the tangent space of a single point, x, on a sphere. A vector in this tangent space can represent a possible velocity at x. Moving in that direction, one's velocity would be given by a vector in the tangent space of a nearby point—a different tangent space, not shown.

In differential geometry, one can attach to every point x of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible "directions" in which one can pass through 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 The elements of the tangent space are called tangent vectors at x. All the tangent spaces have the same dimension, equal to the dimension of the manifold. In Mathematics, the dimension of a Vector space V is the cardinality (i

For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which touches the sphere at that point and is perpendicular to the sphere's radius through the point. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space one can picture the tangent space in this literal fashion. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

In algebraic geometry, in contrast, there is an intrinsic definition of tangent space at a point P of a variety V, that gives a vector space of dimension at least that of V. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety The points P at which the dimension is exactly that of V are called the non-singular points; the others are singular points. For example, a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of V are those where the 'test to be a manifold' fails. See Zariski tangent space. In Algebraic geometry, the Zariski tangent space is a construction that defines a Tangent space, at a point P on an Algebraic variety V

Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the tangent bundle of the manifold. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the

Formal definitions

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

Definition as directions of curves

Suppose M is a C^k manifold (k ≥ 1) and x is a point in M. Pick a chart φ : U → Rⁿ where U is an open subset of M containing x. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in Suppose two curves γ₁ : (-1,1) → M and γ₂ : (-1,1) → M with γ₁(0) = γ₂(0) = x are given such that φ o γ₁ and φ o γ₂ are both differentiable at 0. Then γ₁ and γ₂ are called tangent at 0 if the ordinary derivatives of φ o γ₁ and φ o γ₂ at 0 coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at x. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X The equivalence class of the curve γ is written as γ'(0). The tangent space of M at x, denoted by T_xM, is defined as the set of all tangent vectors; it does not depend on the choice of chart φ.

To define the vector space operations on T_xM, we use a chart φ : U → Rⁿ and define the map (dφ)_x : T_xM → Rⁿ by (dφ)_x(γ'(0)) = (φ o γ)(0). In Mathematics and related technical fields the term map or mapping is often a Synonym for function. It turns out that this map is bijective and can thus be used to transfer the vector space operations from Rⁿ over to T_xM, turning the latter into an n-dimensional real vector space. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not.

Definition via derivations

Suppose M is a C^∞ manifold. A real-valued function f : M → R belongs to C^∞(M) if f o φ^-1 is infinitely often differentiable for every chart φ : U → Rⁿ. C^∞(M) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive The pointwise product of two functions is another function obtained by multiplying the image of the two functions at each value in the domain

Pick a point x in M. A derivation at x is a linear map D : C^∞(M) → R which has the property that for all f, g in C^∞(M):

D(fg) = D(f)·g(x) + f(x)·D(g)

modeled on the product rule of calculus. In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable These derivations form a real vector space in a natural manner; this is the tangent space T_xM.

The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is D(f) = (f o γ)'(0) (where the derivative is taken in the ordinary sense, since f o γ is a function from (-1,1) to R).

Generalizations of this definition are possible, for instance to complex manifolds and algebraic varieties. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety However, instead of examining derivations D from the full algebra of functions, one must instead work at the level of germs of functions. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable The reason is that the structure sheaf may not be fine for such structures. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on In Mathematics, injective sheaves of Abelian groups are used to construct the resolutions needed to define Sheaf cohomology (and other Derived functors For instance, let X be an algebraic variety with structure sheaf F. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on Then the Zariski tangent space at a point p∈X is the collection of K-derivations D:F_p→K, where K is the groundfield and F_p is the stalk of F at p. In Algebraic geometry, the Zariski tangent space is a construction that defines a Tangent space, at a point P on an Algebraic variety V

Definition via the cotangent space

Again we start with a C^∞ manifold M and a point x in M. Consider the ideal I in C^∞(M) consisting of all functions f such that f(x) = 0. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Then I and I ² are real vector spaces, and T_xM may be defined as the dual space of the quotient space I / I ². In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N This latter quotient space is also known as the cotangent space of M at x. In Differential geometry, one can attach to every point x of a smooth (or differentiable Manifold a Vector space called the cotangent space

While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the varieties considered in algebraic geometry. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

If D is a derivation, then D(f) = 0 for every f in I², and this means that D gives rise to a linear map I / I² → R. Conversely, if r : I / I² → R is a linear map, then D(f) = r((f - f(x)) + I ²) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space.

Properties

If M is an open subset of Rⁿ, then M is a C^∞ manifold in a natural manner (take the charts to be the identity maps), and the tangent spaces are all naturally identified with Rⁿ. This article is about the Identity Map software design pattern

Tangent vectors as directional derivatives

One way to think about tangent vectors is as directional derivatives. Given a vector v in Rⁿ one defines the directional derivative of a smooth map f : Rⁿ→R at a point x by

This map is naturally a derivation. Moreover, it turns out that every derivation of C^∞(Rⁿ) is of this form. So there is a one-to-one map between vectors (thought of as tangent vectors at a point) and derivations.

Since tangent vectors to a general manifold can be defined as derivations it is natural to think of them as directional derivatives. Specifically, if v is a tangent vector of M at a point x (thought of as a derivation) then define the directional derivative in the direction v by

where f : M → R is an element of C^∞(M). If we think of v as the direction of a curve, v = γ'(0), then we write

The derivative of a map

Main article: Pushforward (differential)

Every smooth (or differentiable) map φ : M → N between smooth (or differentiable) manifolds induces natural linear maps between the corresponding tangent spaces:

If the tangent space is defined via curves, the map is defined as

If instead the tangent space is defined via derivations, then

The linear map dφ_x is called variously the derivative, total derivative, differential, or pushforward of φ at x. Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that It is frequently expressed using a variety of other notations:

In a sense, the derivative is the best linear approximation to φ near x. Note that when N = R, the map dφ_x : T_xM→R coincides with the usual notion of the differential of the function φ. In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings In local coordinates the derivative of f is given by the Jacobian. Local coordinates are measurement indices into a local Coordinate system or a local Coordinate space. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

An important result regarding the derivative map is the following:

Theorem. If φ : M → N is a local diffeomorphism at x in M then dφ_x : T_xM → T_φ(x)N is a linear isomorphism. In Mathematics, a local diffeomorphism is a Smooth map f: M &rarr N between Smooth manifolds such that for every point In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Conversely, if dφ_x is an isomorphism then there is an open neighborhood U of x such that φ maps U diffeomorphically onto its image. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

This is a generalization of the inverse function theorem to maps between manifolds. In Mathematics, the inverse function theorem gives sufficient conditions for a Vector-valued function to be Invertible on an Open region containing