Editing Tangent space

{{Use American English|date=March 2019}}{{Short description|Assignment of vector fields to manifolds}}
In [[mathematics]], the '''tangent space''' of a [[manifold]] is a generalization of {{em|[[tangent line]]s}} to curves in [[Plane (mathematics)|two-dimensional space]] and {{em|[[Tangent#Tangent plane to a surface|tangent planes]]}} to surfaces in [[three-dimensional space]] in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.

== Informal description ==

[[Image:Image Tangent-plane.svg|thumb|A pictorial representation of the tangent space of a single point <math> x </math> on a [[sphere]]. A vector in this tangent space represents a possible velocity (of something moving on the sphere) at <math> x </math>. After moving in that direction to a nearby point, the velocity would then be given by a vector in the tangent space of that point—a different tangent space that is not shown.]]

In [[differential geometry]], one can attach to every point <math> x </math> of a [[differentiable manifold]] a ''tangent space''—a real [[vector space]] that intuitively contains the possible directions in which one can tangentially pass through <math> x </math>. The elements of the tangent space at <math> x </math> are called the ''[[tangent vector]]s'' at <math> x </math>. This is a generalization of the notion of a [[Vector (mathematics and physics)|vector]], based at a given initial point, in a [[Euclidean space]]. The [[dimension of a vector space|dimension]] of the tangent space at every point of a [[connected space|connected]] manifold is the same as that of the [[manifold]] itself.

For example, if the given manifold is a <math> 2 </math>-[[sphere]], then one can picture the tangent space at a point as the plane that 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 [[embedding|embedded]] [[submanifold]] of [[Euclidean space]], then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining [[parallel transport]]. Many authors in [[differential geometry]] and [[general relativity]] use it.<ref>{{cite book |last=do Carmo |first=Manfredo P. |title=Differential Geometry of Curves and Surfaces|year=1976 |publisher=Prentice-Hall }}: </ref><ref>{{cite book |last=Dirac |first=Paul A. M. |title=General Theory of Relativity |orig-year=1975 |year=1996 |publisher=Princeton University Press |isbn=0-691-01146-X }}</ref> More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.

In [[algebraic geometry]], in contrast, there is an intrinsic definition of the ''tangent space at a point'' of an [[algebraic variety]] <math> V </math> that gives a vector space with dimension at least that of <math> V </math> itself. The points <math> p </math> at which the dimension of the tangent space is exactly that of <math> V </math> are called ''non-singular'' points; the others are called ''singular'' points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of <math> V </math> are those where the "test to be a manifold" fails. See [[Zariski tangent space]].

Once the tangent spaces of a manifold have been introduced, one can define [[vector field]]s, which are abstractions of the velocity field of particles moving in 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.

All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the ''[[tangent bundle]]'' of the manifold.

== 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.

== Properties ==

If <math> M </math> is an open subset of <math> \mathbb{R}^{n} </math>, then <math> M </math> is a <math> C^{\infty} </math> manifold in a natural manner (take coordinate charts to be [[Identity function|identity maps]] on open subsets of <math> \mathbb{R}^{n} </math>), and the tangent spaces are all naturally identified with <math> \mathbb{R}^{n} </math>.

=== Tangent vectors as directional derivatives ===

Another way to think about tangent vectors is as [[directional derivative]]s. Given a vector <math> v </math> in <math> \mathbb{R}^{n} </math>, one defines the corresponding directional derivative at a point <math> x \in \mathbb{R}^{n} </math> by
:<math>
\forall f \in {C^{\infty}}(\mathbb{R}^{n}): \qquad
  (D_{v} f)(x) :=
  \left. \frac{\mathrm{d}}{\mathrm{d}{t}} [f(x + t v)] \right|_{t = 0}
= \sum_{i = 1}^{n} v^{i} {\frac{\partial f}{\partial x^{i}}}(x).
</math>
This map is naturally a derivation at <math> x </math>. Furthermore, every derivation at a point in <math> \mathbb{R}^{n} </math> is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if <math> v </math> is a tangent vector to <math> M </math> at a point <math> x </math> (thought of as a derivation), then define the directional derivative <math> D_{v} </math> in the direction <math> v </math> by
:<math>
\forall f \in {C^{\infty}}(M): \qquad
{D_{v}}(f) := v(f).
</math>
If we think of <math> v </math> as the initial velocity of a differentiable curve <math> \gamma </math> initialized at <math> x </math>, i.e., <math> v = \gamma'(0) </math>, then instead, define <math> D_{v} </math> by
:<math>
\forall f \in {C^{\infty}}(M): \qquad
{D_{v}}(f) := (f \circ \gamma)'(0).
</math>

=== Basis of the tangent space at a point ===

For a <math> C^{\infty} </math> manifold <math> M </math>, if a chart <math> \varphi = (x^{1},\ldots,x^{n}): U \to \mathbb{R}^{n} </math> is given with <math> p \in U </math>, then one can define an ordered basis <math display="inline"> \left\{ \left. \frac{\partial}{\partial x^{1}} \right|_{p} , \dots , \left. \frac{\partial}{\partial x^{n}} \right|_{p} \right\} </math> of <math> T_{p} M </math> by
:<math>
\forall i \in \{ 1,\ldots,n \}, ~ \forall f \in {C^{\infty}}(M): \qquad
{ \left. \frac{\partial}{\partial x^{i}} \right|_{p}}(f) :=
\left( \frac{\partial}{\partial x^{i}} \Big( f \circ \varphi^{- 1} \Big) \right) \Big( \varphi(p) \Big) .
</math>
Then for every tangent vector <math> v \in T_{p} M </math>, one has
:<math>
v = \sum_{i = 1}^{n} v^{i} \left. \frac{\partial}{\partial x^{i}} \right|_{p}.
</math>
This formula therefore expresses <math> v </math> as a linear combination of the basis tangent vectors <math display="inline"> \left. \frac{\partial}{\partial x^{i}} \right|_{p} \in T_{p} M </math> defined by the coordinate chart <math> \varphi: U \to \mathbb{R}^{n} </math>.<ref>{{cite web|title = An Introduction to Differential Geometry|first = Eugene|last = Lerman|page = 12| url = https://faculty.math.illinois.edu/~lerman/518/f11/8-19-11.pdf}}</ref>

=== The derivative of a map ===

{{main|Pushforward (differential)}}

Every smooth (or differentiable) map <math> \varphi: M \to N </math> between smooth (or differentiable) manifolds induces natural [[linear map]]s between their corresponding tangent spaces:
:<math>
\mathrm{d}{\varphi}_{x}: T_{x} M \to T_{\varphi(x)} N.
</math>
If the tangent space is defined via differentiable curves, then this map is defined by
:<math>
{\mathrm{d}{\varphi}_{x}}(\gamma'(0)) := (\varphi \circ \gamma)'(0).
</math>
If, instead, the tangent space is defined via derivations, then this map is defined by
:<math>
[\mathrm{d}{\varphi}_{x}(D)](f) := D(f \circ \varphi).
</math>

The linear map <math> \mathrm{d}{\varphi}_{x} </math> is called variously the ''derivative'', ''total derivative'', ''differential'', or ''pushforward'' of <math> \varphi </math> at <math> x </math>. It is frequently expressed using a variety of other notations:
:<math>
D \varphi_{x}, \qquad (\varphi_{*})_{x}, \qquad \varphi'(x).
</math>
In a sense, the derivative is the best linear approximation to <math> \varphi </math> near <math> x </math>. Note that when <math> N = \mathbb{R} </math>, then the map <math> \mathrm{d}{\varphi}_{x}: T_{x} M \to \mathbb{R} </math> coincides with the usual notion of the [[Differential (calculus)|differential]] of the function <math> \varphi </math>. In [[local coordinates]] the derivative of <math> \varphi </math> is given by the [[Jacobian matrix and determinant|Jacobian]].

An important result regarding the derivative map is the following:
{{math theorem|math_statement=If <math> \varphi: M \to N </math> is a [[local diffeomorphism]] at <math> x </math> in <math> M </math>, then <math> \mathrm{d}{\varphi}_{x}: T_{x} M \to T_{\varphi(x)} N </math> is a linear [[isomorphism]]. Conversely, if <math>\varphi : M\to N</math> is continuously differentiable and <math>\mathrm{d}{\varphi}_{x}</math> is an isomorphism, then there is an [[open set|open neighborhood]] <math> U </math> of <math> x </math> such that <math> \varphi </math> maps <math> U </math> diffeomorphically onto its image.}}
This is a generalization of the [[inverse function theorem]] to maps between manifolds.

== See also ==
* [[Coordinate-induced basis]]
* [[Cotangent space]]
* [[Differential geometry of curves]]
* [[Exponential map (Riemannian geometry)|Exponential map]]
* [[Vector space]]

== Notes ==
<references />

== References ==
* {{citation|first = Jeffrey M.|last = Lee|title = Manifolds and Differential Geometry|series = [[Graduate Studies in Mathematics]]|volume = 107|publisher = American Mathematical Society|location = Providence|year = 2009}}.
* {{citation|first = Peter W.|last = Michor|title = Topics in Differential Geometry|series = Graduate Studies in Mathematics|volume = 93|publisher = American Mathematical Society|location = Providence|year = 2008}}.
* {{citation|first = Michael|last = Spivak|title = Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus|year = 1965|author1-link = Michael Spivak|publisher = W. A. Benjamin, Inc.|isbn = 978-0-8053-9021-6|url = https://archive.org/details/SpivakM.CalculusOnManifolds_201703}}.

==External links==
* [http://mathworld.wolfram.com/TangentPlane.html Tangent Planes] at MathWorld

{{Manifolds}}

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[[Category:Differential topology]]
[[Category:Differential geometry]]