Tangent vector

Let be a differentiable manifold, and let be the algebra of smooth real-valued functions on it. A tangent vector to at is an -linear mapping such that

(a1)

For this definition one can equally well (in fact, better) use the ring of germs of smooth functions on at .

The tangent vectors to at form a vector space over of dimension . It is denoted by .

Let , , where is a system of coordinates on near . The -th partial derivative at with respect to is the tangent vector

where the right hand-side is the usual partial derivative of the function in the variables , at the point . One has (the Kronecker delta) and the form a basis for .

This basis for determined by the coordinate system is often denoted by .

A cotangent vector at is an -linear mapping such that the cotangent space at is the dual vector space to . The dual basis to is denoted by . One has

The disjoint union of the tangent spaces , , together with the projection , if , can be given the structure of a differentiable vector bundle, the tangent bundle.

Similarly, the cotangent spaces form a vector bundle dual to , called the cotangent bundle. The sections of are the vector fields on , the sections of are differentiable -forms on .

Let be a mapping of differentiable manifolds and let be the induced mapping . For a tangent vector at , composition with gives an -linear mapping which is a tangent vector to at . This defines a homomorphism of vector spaces and a vector bundle morphism .

In case and with global coordinates and , respectively, is given by differentiable functions and at each ,

so that the matrix of with respect to the basis of and the basis of is given by the Jacobi matrix of at .

Now, let be an imbedded manifold. Let , be a smooth curve in , . Then

(a2)

All tangent vectors in arise in this way. Identifying the vector (a2) with the -vector , viewed as a directed line segment starting in , one recovers the intuitive picture of the tangent space as the -plane in tangent to in .

A vector field on a manifold can be defined as a derivation (cf. Derivation in a ring) in the -algebra , . Composition with the evaluation mappings , , yields a family of tangent vectors , so that "becomes" a section of the tangent bundle. Given local coordinates , can locally be written as

and if a function in local coordinates is given by , then is the function given in local coordinates by the expression

showing once more the convenience of the " / x" notation for tangent vectors. (Of course, in practice one uses a bit more abuse of notation and writes instead of .)

Let be the -algebra of germs of smooth functions at (cf. Germ). Let be the ideal of germs that vanish at zero, and the ideal generated by all products for . If are local coordinates at so that , is generated as an ideal in by , and by the , . In fact, the quotient ring is the power series ring in variables over . Here is the ideal of flat function germs. (A smooth function is flat at a point if it vanishes there with all its derivatives (an example is at ); the "Taylor expansion mapping" is surjective, a very special consequence of the Whitney extension theorem.)

Now, let be a tangent vector of at . Then by (a1) for all constant functions in . Also , again by (a1). Thus, each defines an element in , which is of dimension because has dimension (and that element uniquely determines ). Moreover, the tangent vectors clearly define linearly independent elements in (because ). Thus,

the dual space of . This point of view is more generally applicable and serves as the definition of tangent space in analytic and algebraic geometry, cf. Analytic space; Zariski tangent space.

References

[a1]	R. Hermann, "Geometry, physics, and systems" , M. Dekker (1973)
[a2]	R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)
[a3]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78
[a4]	S.P. Novikov, A.T. Fomenko, "Basic elements of differential geometry and topology" , Kluwer (1991) (Translated from Russian)
[a5]	Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian)