Tangent vector

Let $M$ be a differentiable manifold, and let $F(M)$ be the algebra of smooth real-valued functions on it. A tangent vector to $M$ at $m\in M$ is an $\mathbb{R}$-linear mapping $v:F(M)\rightarrow \mathbb{R}$ such that

\begin{equation}\tag{a1} v(fg)=f(m)v(g)+g(m)v(f) \end{equation}

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

The tangent vectors to $M$ at $m\in M$ form a vector space over $\mathbb{R}$ of dimension $n=\dim (M)$. It is denoted by $T_m M$.

Let $\phi :U\rightarrow \mathbb{R}$, $m\mapsto (x_1(m),\dots ,x_n(m))$, where $(x_1,\dots ,x_n)$ is a system of coordinates on $M$ near $m$. The $i$-th partial derivative at $m$ with respect to $\phi$ is the tangent vector

\begin{equation} (D_{x_i})(m)(f)=\left.\frac{\partial (f\phi^{-1})}{\partial x_i}\right|_{\phi(m)}, \end{equation}

where the right hand-side is the usual partial derivative of the function $f\phi^{-1}:\mathbb{R}^n\rightarrow \mathbb{R}$ in the variables $x_1,\dots ,x_n$, at the point $\phi(m)$. One has $D_{x_i}(m)(x_j)=\delta_{ij}$ (the Kronecker delta) and the $D_{x_i}(m)$ form a basis for $T_m M$.

This basis for $T_m M$ determined by the coordinate system $(x_1,\dots ,x_n)$ is often denoted by $\{\partial/\partial x_1,\dots ,\aprtial/\partial x_n\}$.

A cotangent vector at $m\in M$ is an $\mathbb{R}$-linear mapping $T_m M\rightarrow \mathbb{R}$ such that the cotangent space $T_m^* M$ at $38$ is the dual vector space to $T_m M$. The dual basis to $(\partial/\partial x_1,\dots ,\aprtial/\partial x_n)$ is denoted by $dx_1,\dots ,dx_n$. One has

\begin{equation} dx_i(v)=v(x_i),\qquad v\in T_m M. \end{equation}

The disjoint union $TM$ of the tangent spaces $T_m M$, $m\in M$, together with the projection $\pi :TM\rightarrow M$, $\pi(v)=m$ if $v\in T_m M$, can be given the structure of a differentiable vector bundle, the tangent bundle.

Similarly, the cotangent spaces $T_m^* M$ form a vector bundle $T^*M$ dual to $TM$, called the cotangent bundle. The sections of $TM$ are the vector fields on $M$, the sections of $T^*M$ are differentiable $1$-forms on $M$.

Let $\alpha: M\rightarrow N$ be a mapping of differentiable manifolds and let $\alpha^* F(N)\rightarrow F(M)$ be the induced mapping $g\mapsto g\alpha$. For a tangent vector $v:F(M)\rightarrow \mathbb{R}$ at $m$, composition with $\alpha^*$ gives an $\mathbb{R}$-linear mapping $v\alpha^*:F(N)\rightarrow\mathbb{R})$ which is a tangent vector to $N$ at $\alpha(m)$. This defines a homomorphism of vector spaces $T\alpha(m):T_m M\rightarrow T_{\alpha(m)}N$ and a vector bundle morphism $T\alpha:TM\rightarrow TN$.

In case $70$ and $71$ with global coordinates $72$ and $73$, respectively, $74$ is given by $75$ differentiable functions and at each $76$,

$77$

so that the matrix of $78$ with respect to the basis $79$ of $80$ and the basis $81$ of $82$ is given by the Jacobi matrix of $83$ at $84$.

Now, let $85$ be an imbedded manifold. Let $86$, $87$ be a smooth curve in $88$, $89$. Then

$90$

(a2)

All tangent vectors in $91$ arise in this way. Identifying the vector (a2) with the $92$-vector $93$, viewed as a directed line segment starting in $94$, one recovers the intuitive picture of the tangent space $95$ as the $96$-plane in $97$ tangent to $98$ in $99$.

A vector field on a manifold $100$ can be defined as a derivation (cf. Derivation in a ring) in the $101$-algebra $102$, $103$. Composition with the evaluation mappings $104$, $105$, yields a family of tangent vectors $106$, so that $107$ "becomes" a section of the tangent bundle. Given local coordinates $108$, $109$ can locally be written as

$110$

and if a function $111$ in local coordinates is given by $112$, then $113$ is the function given in local coordinates by the expression

$114$

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 $115$ instead of $116$.)

Let $117$ be the $118$-algebra of germs of smooth functions at $119$ (cf. Germ). Let $120$ be the ideal of germs that vanish at zero, and $121$ the ideal generated by all products $122$ for $123$. If $124$ are local coordinates at $125$ so that $126$, $127$ is generated as an ideal in $128$ by $129$, and $130$ by the $131$, $132$. In fact, the quotient ring $133$ is the power series ring in $134$ variables over $135$. Here $136$ 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 $137$ at $138$); the "Taylor expansion mapping" $139$ is surjective, a very special consequence of the Whitney extension theorem.)

Now, let $140$ be a tangent vector of $141$ at $142$. Then $143$ by (a1) for all constant functions in $144$. Also $145$, again by (a1). Thus, each $146$ defines an element in $147$, which is of dimension $148$ because $149$ has dimension $150$ (and that element uniquely determines $151$). Moreover, the tangent vectors $152$ clearly define $153$ linearly independent elements in $154$ (because $155$). Thus,

$156$

the dual space of $157$. 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) MR0494183 Zbl 0285.58001
[a2]	R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964) MR0169148 Zbl 0132.16003
[a3]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 MR0448362 Zbl 0356.57001
[a4]	S.P. Novikov, A.T. Fomenko, "Basic elements of differential geometry and topology" , Kluwer (1991) (Translated from Russian) MR1135798 Zbl 0711.53001
[a5]	Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian) MR1450091 MR0824983 MR0591670 Zbl 0836.57001 Zbl 0834.57001 Zbl 0478.57001