Editing Tangent space (section)

== Formal definitions ==

The informal description above relies on a manifold's ability to be embedded into an ambient vector space <math> \mathbb{R}^{m} </math> so that the tangent vectors can "stick out" of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself.<ref name = "Isham2002">{{cite book|author = Chris J. Isham|title = Modern Differential Geometry for Physicists|date = 1 January 2002|publisher = Allied Publishers|isbn = 978-81-7764-316-9|pages = 70–72|url = https://books.google.com/books?id=DCn9bjBe27oC&pg=PA70}}</ref>

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

=== Definition via tangent curves ===

In the embedded-manifold picture, a tangent vector at a point <math> x </math> is thought of as the ''velocity'' of a [[Curve#Topology|curve]] passing through the point <math> x </math>. We can therefore define a tangent vector as an equivalence class of curves passing through <math> x </math> while being tangent to each other at <math> x </math>.

Suppose that <math> M </math> is a <math> C^{k} </math> [[differentiable manifold]] (with [[smoothness]] <math> k \geq 1 </math>) and that <math> x \in M </math>. Pick a [[Manifold#Charts, atlases, and transition maps|coordinate chart]] <math> \varphi: U \to \mathbb{R}^{n} </math>, where <math> U </math> is an [[open set|open subset]] of <math> M </math> containing <math> x </math>. Suppose further that two curves <math> \gamma_{1},\gamma_{2}: (- 1,1) \to M </math> with <math> {\gamma_{1}}(0) = x = {\gamma_{2}}(0) </math> are given such that both <math> \varphi \circ \gamma_{1},\varphi \circ \gamma_{2}: (- 1,1) \to \mathbb{R}^{n} </math> are differentiable in the ordinary sense (we call these ''differentiable curves initialized at <math> x </math>''). Then <math> \gamma_{1} </math> and <math> \gamma_{2} </math> are said to be ''equivalent'' at <math> 0 </math> if and only if the derivatives of <math> \varphi \circ \gamma_{1} </math> and <math> \varphi \circ \gamma_{2} </math> at <math> 0 </math> coincide. This defines an [[equivalence relation]] on the set of all differentiable curves initialized at <math> x </math>, and [[equivalence class]]es of such curves are known as ''tangent vectors'' of <math> M </math> at <math> x </math>. The equivalence class of any such curve <math> \gamma </math> is denoted by <math> \gamma'(0) </math>. The ''tangent space'' of <math> M </math> at <math> x </math>, denoted by <math> T_{x} M </math>, is then defined as the set of all tangent vectors at <math> x </math>; it does not depend on the choice of coordinate chart <math> \varphi: U \to \mathbb{R}^{n} </math>.

[[Image:Tangentialvektor.svg|thumb|left|200px|The tangent space <math> T_{x} M </math> and a tangent vector <math> v \in T_{x} M </math>, along a curve traveling through <math> x \in M </math>.]]

To define vector-space operations on <math> T_{x} M </math>, we use a chart <math> \varphi: U \to \mathbb{R}^{n} </math> and define a [[Map (mathematics)|map]] <math> \mathrm{d}{\varphi}_{x}: T_{x} M \to \mathbb{R}^{n} </math> by <math display="inline"> {\mathrm{d}{\varphi}_{x}}(\gamma'(0)) := \frac{\mathrm{d}(\varphi \circ \gamma)}{\mathrm{d}{t}}(0), </math> where <math>\gamma \in \gamma'(0) </math>. The map <math> \mathrm{d}{\varphi}_{x} </math> turns out to be [[bijective]] and may be used to transfer the vector-space operations on <math> \mathbb{R}^{n} </math> over to <math> T_{x} M </math>, thus turning the latter set into an <math> n </math>-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart <math> \varphi: U \to \mathbb{R}^{n} </math> and the curve <math> \gamma </math> being used, and in fact it does not.

=== Definition via derivations ===

Suppose now that <math> M </math> is a <math> C^{\infty} </math> manifold. A real-valued function <math> f: M \to \mathbb{R} </math> is said to belong to <math> {C^{\infty}}(M) </math> if and only if for every coordinate chart <math> \varphi: U \to \mathbb{R}^{n} </math>, the map <math> f \circ \varphi^{- 1}: \varphi[U] \subseteq \mathbb{R}^{n} \to \mathbb{R} </math> is infinitely differentiable. Note that <math> {C^{\infty}}(M) </math> is a real [[associative algebra]] with respect to the [[pointwise product]] and sum of functions and scalar multiplication.

A ''[[Derivation (abstract algebra)|derivation]]'' at <math> x \in M </math> is defined as a [[linear map]] <math> D: {C^{\infty}}(M) \to \mathbb{R} </math> that satisfies the Leibniz identity
<math display="block">
\forall f,g \in {C^{\infty}}(M): \qquad
D(f g) = D(f) \cdot g(x) + f(x) \cdot D(g),
</math>
which is modeled on the [[product rule]] of calculus.

(For every identically constant function <math>f=\text{const},</math> it follows that <math> D(f)=0 </math>).

Denote <math> T_{x} M </math> the set of all derivations at <math> x. </math> Setting
* <math> (D_1+D_2)(f) := {D}_1(f) + {D}_2(f) </math> and
* <math> (\lambda \cdot D)(f) := \lambda \cdot D(f) </math>
turns <math> T_{x} M </math> into a vector space.

==== Generalizations ====
Generalizations of this definition are possible, for instance, to [[complex manifold]]s and [[algebraic variety|algebraic varieties]]. However, instead of examining derivations <math> D </math> from the full algebra of functions, one must instead work at the level of [[germ (mathematics)|germs]] of functions. The reason for this is that the [[structure sheaf]] may not be [[injective sheaf#Fine sheaves|fine]] for such structures. For example, let <math> X </math> be an algebraic variety with [[structure sheaf]] <math> \mathcal{O}_{X} </math>. Then the [[Zariski tangent space]] at a point <math> p \in X </math> is the collection of all <math> \mathbb{k} </math>-derivations <math> D: \mathcal{O}_{X,p} \to \mathbb{k} </math>, where <math> \mathbb{k} </math> is the [[ground field]] and <math> \mathcal{O}_{X,p} </math> is the [[stalk (sheaf)|stalk]] of <math> \mathcal{O}_{X} </math> at <math> p </math>.

=== Equivalence of the definitions ===
For <math>x \in M</math> and a differentiable curve <math> \gamma: (- 1,1) \to M </math> such that <math>\gamma (0) = x,</math> define <math> {D_{\gamma}}(f) := (f \circ \gamma)'(0) </math> (where the derivative is taken in the ordinary sense because <math> f \circ \gamma </math> is a function from <math> (- 1,1) </math> to <math> \mathbb{R} </math>). One can ascertain that <math>D_{\gamma}(f)</math> is a derivation at the point <math>x,</math> and that equivalent curves yield the same derivation. Thus, for an equivalence class <math> \gamma'(0), </math> we can define <math> {D_{\gamma'(0)}}(f) := (f \circ \gamma)'(0), </math> where the curve <math>\gamma \in \gamma'(0) </math> has been chosen arbitrarily. The map <math> \gamma'(0) \mapsto D_{\gamma'(0)} </math> is a vector space isomorphism between the space of the equivalence classes <math> \gamma'(0) </math> and the space of derivations at the point <math>x.</math>

=== Definition via cotangent spaces === 

Again, we start with a <math> C^\infty </math> manifold <math> M </math> and a point <math> x \in M </math>. Consider the [[ideal (ring theory)|ideal]] <math> I </math> of <math> C^\infty(M) </math> that consists of all smooth functions <math> f </math> vanishing at <math> x </math>, i.e., <math> f(x) = 0 </math>. Then <math> I </math> and <math> I^2 </math> are both real vector spaces, and the [[quotient space (linear algebra)|quotient space]] <math> I / I^2 </math> can be shown to be [[isomorphism| isomorphic]] to the [[cotangent space]] <math> T^{*}_x M </math> through the use of [[Taylor's theorem]]. The tangent space <math> T_x M </math> may then be defined as the [[dual space]] of <math> I / I^2 </math>.

While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the [[algebraic variety|varieties]] considered in [[algebraic geometry]].

If <math> D </math> is a derivation at <math> x </math>, then <math> D(f) = 0 </math> for every <math> f \in I^2 </math>, which means that <math> D </math> gives rise to a linear map <math> I / I^2 \to \mathbb{R} </math>. Conversely, if <math> r: I / I^2 \to \mathbb{R} </math> is a linear map, then <math> D(f) := r\left((f - f(x)) + I^2\right) </math> defines a derivation at <math> x </math>. This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.