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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. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...
Informal description
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible "directions" in which one can pass through p. The elements of the tangent space are called tangent vectors at p. All the tangent spaces have the same dimension, equal to the dimension of the manifold. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...
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. 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. A sphere (< Greek σφαίÏα) is a perfectly symmetrical geometrical object. ...
Fig. ...
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
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. 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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally). ...
Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. 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. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...
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 derivatives with respect to that variable. ...
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 differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
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 Ck manifold (k ≥ 1) and p is a point in M. Pick a chart φ : U → Rn where U is an open subset of M containing p. Suppose two curves γ1 : (-1,1) → M and γ2 : (-1,1) → M with γ1(0) = γ2(0) = p 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 p. The equivalence class of the curve γ is written as γ'(0). The tangent space of M at p, denoted by TpM, is defined as the set of all tangent vectors; it does not depend on the choice of chart φ. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
To define the vector space operations on TpM, we use a chart φ : U → Rn and define the map (dφ)p : TpM → Rn by (dφ)p(γ'(0)) = (φ o γ)'(0). It turns out that this map is bijective and can thus be used to transfer the vector space operations from Rn over to TpM, turning the latter into an n-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not. In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...
Definition via derivations Suppose M is a C∞ manifold. A real-valued function g : M → R belongs to C∞(M) if g 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 and associative manner. ...
The pointwise product of two functions is another function, obtained by multiplying the results of the two functions. ...
Pick a point p in M. A derivation at p is a linear map D : C∞(M) → R which has the property that for all g, h in C∞(M): In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
- D(gh) = D(g)·h(p) + g(p)·D(h)
modeled on the product rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space TpM. In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...
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(g) = (g o γ)'(0) (where the derivative is taken in the ordinary sense, since g o γ is a function from (-1,1) to R).
Definition via the cotangent space Again we start with a C∞ manifold M and a point p in M. Consider the ideal I in C∞(M) consisting of all functions g such that g(p) = 0. Then I and I 2 are real vector spaces, and TpM may be defined as the dual space of the quotient space I / I 2. This latter quotient space is also known as the cotangent space of M at p. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...
In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ...
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. In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
If D is a derivation, then D(g) = 0 for every g 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(g) = r((g - g(p)) + 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. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...
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 p 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 p (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: Push forward In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...
Every differentiable map f : M → N between Ck manifolds induces natural linear maps between the corresponding tangent spaces: In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
defined by if the tangent space is defined via curves and by if the tangent space is defined via derivations.
The linear map (df)p is called variously the derivative, total derivative, differential, or pushforward of f at p. It is frequently expressed using a variety of other notations:
In a sense, the derivative is the best linear approximation to f near p. Note that when N = R, the map (df)p : TpM→R coincides with the usual notion of the differential of the function f. In local coordinates the derivative of f is given by the Jacobian. In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ...
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 f : M → N is a local diffeomorphism at p in M then (df)p : TpM → Tf(p)N is a linear isomorphism. Conversely, if (df)p is an isomorphism then there is an open neighborhood U of p such that f maps U diffeomorphically onto its image.
This is a generalization of the inverse function theorem to maps between manifolds. In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...
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